In 35 years of watching baseball I had never been to a postseason game, until this Saturday, when I was able to get two tickets to Game 2 of the National League Championship Series through a wonderful terrific beautiful friend with connections.

First of all, I salute whoever the free spirit was who slammed a Zima right before entering Miller Park.

The game started at 3pm; in late afternoon with the roof shut at Miller Park there’s a slant-line of sunlight across the field which is lovely to look at and probably terrible to hit in.

And indeed there wasn’t a lot of hitting to start with. Wade Miley, once a bad Oriole, now a good Brewer, never looked dominant, giving up lots of hard-hit balls including a shot by Jeremy Freese in the first that Lorenzo Cain hauled back in from over the wall, but somehow pitched 5 2/3 only allowing 2 hits (and collecting a single himself.) Hyun-jin Ryu matched him zero for zero. Every seat in Miller Park full, everyone attentive to the game, a level of attention I’ve never seen there. The guy behind us kept saying “NASTY, throw something NASTY.” CJ believes he sees Marlins Man in the front row — he’s right! Brewers get runners on second and third with one out, Dodgers intentionally walk Yelich to load the bases, (wave of boos), Braun delivers the RBI groundout but can’t score any more. Travis Shaw hits a solo shot to deepest center, the Brewers go up 3-0, and people start to smell win, but the Dodgers lineup has good hitters all the way down to #8 and the usually reliable Milwaukee bullpen starts to crack. Jeremy Jeffress comes in with runners on first and second and nobody out, immediately gives up a single to Joc Pederson, now they’re loaded, still nobody out, Brewers up 3-1. Manny Machado, on third base, keeps jumping off the bag, trying to distract Jeffress. But Jeffress strikes out Yasiel Puig, who’s so angry he smashes his bat over his knee. Crowd exults. Then he walks light-hitting catcher Austin Barnes to force in a run. Nobody’s up in the bullpen. Crowd panics. Yasmini Grandal comes in to hit in the pitcher’s spot and Jeffress somehow gets the double play ball and is out of it. But the next inning, Jeffress stays in a little too long; Chris Taylor leads off with a lucky little dink of an infield single and then Turner muscles a ball out to the short corner in left field; 4-3 Dodgers and it stays that way.

But the Brewers do threaten. 43,000 Brewers fans want to see Yelich get one more chance to be the hero. Hernan Perez draws a walk in the bottom of the ninth, steals while Cain strikes out. So Yelich gets to bat with 2 outs and a runner in scoring position. He grounds out. Crowd deflates. But that’s all you can ask of a baseball game, right? The hitter you want in the situation you want with the game on the line and whatever happens happens. Great baseball. Great team. I hope they win it all.  Maybe I’ll try to be there when they do.

It’s like asking if I like New York.  It’s big!  A lot of different things are in it.  Some things are monumental and wonderful, some things have an offhand arresting beauty, some things smell bad.

Minor thoughts after break — this book just came out 165 years ago and I want to spare you spoilers.

• • The court case, Jarndyce and Jarndyce, that makes up the center of the book, promises the person who masters it great power, but it insidiously and inevitably corrupts everyone who comes in contact with it.  In other words I sort of see it as the basis for the Ring in Tolkien! And in the end the person most fully corrupted by it ends up participating in its destruction.
  • A lot of space in this book is taken up by the question of “What is the right way to be a woman?  What is the right way to be a man?”  Dickens offers you a lot of pretty different examples of men he sees as properly manly; the gentle and endlessly patient John Jarndyce, the simple-yet-deeply-good-and-loves-his-mother-a-lot soldier George, the selfless do-gooder Woodcourt, the comically presented and finally viewed as noble Lord Dedlock, and so on.  But in this book there is one and only one way to be good at being a woman, which is reiterated again and again — not to make trouble.  Not to ask for anything, not to disagree, to make of oneself a kind of item of service.  Here’s Inspector Bucket, praising Esther Summerson in much the same kind of language she uses aspirationally about herself:
    “Lord! You’re no trouble at all. I never see a young woman in any station of society — and I’ve seen many elevated ones too — conduct herself like you have conducted yourself since you was called out of your bed. You’re a pattern, you know, that’s what you are,” said Mr. Bucket warmly; “you’re a pattern.”I told him I was very glad, as indeed I was, to have been no hindrance to him, and that I hoped I should be none now.“My dear,” he returned, “when a young lady is as mild as she’s game, and as game as she’s mild, that’s all I ask, and more than I expect. She then becomes a queen, and that’s about what you are yourself.”
  • The character of Skimpole is the best thing in this, maybe because he’s one of the few characters who’s not a fixed type.  He’s charming and innocent, until he’s not.  Bucket again:  “Whenever a person proclaims to you ‘In worldly matters I’m a child,’ you consider that that person is only a-crying off from being held accountable and that you have got that person’s number, and it’s Number One.”
  • There’s a spontaneous combustion in the middle of this book!  Better still, before the spontaneous combustion reveal, there’s twenty pages of characters wandering around saying things like “my goodness, it certainly does smell like greasy smoke tonight, they must be cooking mutton at the pub.”  In the very last scene before Krook’s charred body is discovered, another character, directly upstairs from Krook’s room, puts his hand on the windowsill and finds it covered with a foul oily yellow ichor.  The whole thing is extremely metal.

    In the section that follows there’s a weird and kind of boring catalog of contemporary accounts of spontaneous combustions, which is apparently there because scientists hassled Dickens about the ridiculousness of the scene and he felt the need to defend himself in the subsequently published chapter.

  • Lots of sentence fragments in this book, moody description in a telegraphic style I think of as more modern than the 1850s (did Dickens invent this?)  Here’s the opening:
    

    London. Michaelmas term lately over, and the Lord Chancellor sitting in Lincoln’s Inn Hall. Implacable November weather. As much mud in the streets as if the waters had but newly retired from the face of the earth, and it would not be wonderful to meet a Megalosaurus, forty feet long or so, waddling like an elephantine lizard up Holborn Hill. Smoke lowering down from chimney-pots, making a soft black drizzle, with flakes of soot in it as big as full-grown snowflakes — gone into mourning, one might imagine, for the death of the sun. Dogs, undistinguishable in mire. Horses, scarcely better; splashed to their very blinkers.

  • And here’s the closing, which lands again on this question of “what does it mean to be good at being a woman”:
    

    “My dear Dame Durden,” said Allan, drawing my arm through his, “do you ever look in the glass?”

    “You know I do; you see me do it.”

    “And don’t you know that you are prettier than you ever were?”

    “I did not know that; I am not certain that I know it now. But I know that my dearest little pets are very pretty, and that my darling is very beautiful, and that my husband is very handsome, and that my guardian has the brightest and most benevolent face that ever was seen, and that they can very well do without much beauty in me — even supposing —.

    I struggle with this paragraph on a purely literal level. I don’t understand the final sentence fragment. What is elided? Even supposing what?

We now turn to the local existence theory for the initial value problem for the incompressible Euler equations

For sake of discussion we will just work in the non-periodic domain , , although the arguments here can be adapted without much difficulty to the periodic setting. We will only work with solutions in which the pressure is normalised in the usual fashion:

Formally, the Euler equations (with normalised pressure) arise as the vanishing viscosity limit of the Navier-Stokes equations

that was studied in previous notes. However, because most of the bounds established in previous notes, either on the lifespan of the solution or on the size of the solution itself, depended on , it is not immediate how to justify passing to the limit and obtain either a strong well-posedness theory or a weak solution theory for the limiting equation (1). (For instance, weak solutions to the Navier-Stokes equations (or the approximate solutions used to create such weak solutions) have lying in for , but the bound on the norm is and so one could lose this regularity in the limit , at which point it is not clear how to ensure that the nonlinear term still converges in the sense of distributions to what one expects.)

Nevertheless, by carefully using the energy method (which we will do loosely following an approach of Bertozzi and Majda), it is still possible to obtain local-in-time estimates on (high-regularity) solutions to (3) that are uniform in the limit . Such a priori estimates can then be combined with a number of variants of these estimates obtain a satisfactory local well-posedness theory for the Euler equations. Among other things, we will be able to establish the Beale-Kato-Majda criterion – smooth solutions to the Euler (or Navier-Stokes) equations can be continued indefinitely unless the integral

becomes infinite at the final time , where is the vorticity field. The vorticity has the important property that it is transported by the Euler flow, and in two spatial dimensions it can be used to establish global regularity for both the Euler and Navier-Stokes equations in these settings. (Unfortunately, in three and higher dimensions the phenomenon of vortex stretching has frustrated all attempts to date to use the vorticity transport property to establish global regularity of either equation in this setting.)

There is a rather different approach to establishing local well-posedness for the Euler equations, which relies on the vorticity-stream formulation of these equations. This will be discused in a later set of notes.

— 1. A priori bounds —

We now develop some a priori bounds for very smooth solutions to Navier-Stokes that are uniform in the viscosity . Define an function to be a function that lies in every space; similarly define an function to be a function that lies in for every . Given divergence-free initial data , an mild solution to the Navier-Stokes initial value problem (3) is a solution that is an mild solution for all . From the (non-periodic version of) Corollary 40 of Notes 1, we know that for any divergence-free initial data , there is unique maximal Cauchy development , with infinite if is finite.

Here are our first bounds:

Theorem 1 (A priori bound) Let be an maximal Cauchy development to (3) with initial data .

• (i) For any integer , we have

  

  Furthermore, if for a sufficiently small constant depending only on , then

  

• (ii) For any and integer , one has

  

  

The hypothesis that is integer can be dropped by more heavily exploiting the theory of paraproducts, but we shall restrict attention to integer for simplicity.

We now prove this theorem using the energy method. Using the Navier-Stokes equations, we see that and all lie in for any ; an easy iteration argument then shows that the same is true for all higher derivatives of also. This will make it easy to justify the differentiation under the integral sign that we shall shortly perform.

Let be an integer. For each time , we introduce the energy-type quantity

Here we think of as taking values in the Euclidean space . This quantity is of course comparable to , up to constants depending on . It is easy to verify that is continuously differentiable in time, with derivative

where we suppress explicit dependence on in the integrand for brevity. We now try to bound this quantity in terms of . We expand the right-hand side in coordinates using (3) to obtain

where

For , we can integrate by parts to move the operator onto and use the divergence-free nature of to conclude that . Similarly, we may integrate by parts for to move one copy of over to the other factor in the integrand to conclude

so in particular (note that as we are seeking bounds that are uniform in , we can’t get much further use out of beyond this bound). Thus we have

Now we expand out using the Leibniz rule. There is one dangerous term, in which all the derivatives in fall on the factor, giving rise to the expression

But we can locate a total derivative to write this as

and then an integration by parts using as before shows that this term vanishes. Estimating the remaining contributions to using the triangle inequality, we arrive at the bound

At this point we now need a variant of Proposition 35 from Notes 1:

Exercise 2 Let be integers. For any , show that

(Hint: for or , use Hölder’s inequality. Otherwise, use a suitable Littlewood-Paley decomposition.)

and thus

By Gronwall’s inequality we conclude that

which when inserted into (4) yields the differential inequality

or equivalently

for some constant (strictly speaking one should work with for some small which one sends to zero later, if one wants to avoid the possibility that vanishes, but we will ignore this small technicality for sake of exposition.) Since , we conclude that stays bounded for a time interval of the form ; this, together with the blowup criterion that must go to infinity as , gives part (i).

As a consequence, we can now obtain local existence for the Euler equations from smooth data:

Corollary 3 (Local existence for smooth solutions) Let be divergence-free. Let be an integer, and set

Then there is a smooth solution , to (1) with all derivatives of in for appropriate . Furthermore, for any integer , one has

 

Proof: We use the compactness method, which will be more powerful here than in the last section because we have much higher regularity uniform bounds (but they are only local in time rather than global). Let be a sequence of viscosities going to zero. By the local existence theory for Navier-Stokes (Corollary 40 of Notes 1), for each we have a maximal Cauchy development , to the Navier-Stokes initial value problem (3) with viscosity and initial data . From Theorem 1(i), we have for all (if is small enough), and

for all . By Sobolev embedding, this implies that

and then by Theorem 1(ii) one has

for every integer . Thus, for each , is bounded in , uniformly in . By repeatedly using (3) and product estimates for Sobolev spaces, we see the same is true for , and for all higher derivatives of . In particular, all derivatives of are equicontinuous.

Using weak compactness (Proposition 2 of Notes 2), one can pass to a subsequence such that converge weakly to some limits , such that and all their derivatives lie in on ; in particular, are smooth. From the Arzelá-Ascoli theorem (and Proposition 3 of Notes 2), and converge locally uniformly to , and similarly for all derivatives of . One can then take limits in (3) and conclude that solve (1). The bound (5) follows from taking limits in (6).

Remark 4 We are able to easily pass to the zero viscosity limit here because our domain has no boundary. In the presence of a boundary, we cannot freely differentiate in space as casually as we have been doing above, and one no longer has bounds on higher derivatives on and near the boundary that are uniform in the viscosity. Instead, it is possible for the fluid to form a thin boundary layer that has a non-trivial effect on the limiting dynamics. We hope to return to this topic in a future set of notes.

We have constructed a local smooth solution to the Euler equations from smooth data, but have not yet established uniqueness or continuous dependence on the data; related to the latter point, we have not extended the construction to larger classes of initial data than the smooth class . To accomplish these tasks we need a further a priori estimate, now involving differences of two solutions, rather than just bounding a single solution:

Theorem 5 (A priori bound for differences) Let , let be an integer, and let be divergence-free with norm at most . Let

where is sufficiently small depending on . Let and be an solution to (1) with initial data (this exists thanks to Corollary 3), and let and be an solution to (1) with initial data . Then one has

 

and

 

Note the asymmetry between and in (8): this estimate requires control on the initial data in the high regularity space in order to be usable, but has no such requirement on the initial data . This asymmetry will be important in some later applications.

Proof: From Corollary 3 we have

and

Now we need bounds on the difference . Initially we have , where . To evolve later in time, we will need to use the energy method. Subtracting (1) for and , we have

By hypothesis, all derivatives of and lie in on , which will allow us to justify the manipulations below without difficulty. We introduce the low regularity energy for the difference:

Arguing as in the proof of Proposition 1, we see that

where

As before, the divergence-free nature of ensures that vanishes. For , we use the Leibniz rule and again extract out the dangerous term

which again vanishes by integration by parts. We then use the triangle inequality to bound

Using Exercise 2 and Hölder, we may bound this by

which by Sobolev embedding gives

Applying (9) and Gronwall’s inequality, we conclude that

for , and (7) follows.

Now we work with the high regularity energ

Arguing as before we have

Using Exercise 2 and Hölder, we may bound this by

Using Sobolev embedding we thus have

and hence by (9), (10), (7)

By the chain rule, we obtain

(one can work with in place of and then send later if one wishes to avoid a lack of differentiability at ). By Gronwall’s inequality, we conclude that

for all , and (8) follows.

By specialising (7) (or (8)) to the case where , we see the solution constructed in Corollary 3 is unique. Now we can extend to wider classes of initial data than initial data. The following result is essentially due to Kato and to Swann (with a similar result obtained by different methods by Ebin-Marsden):

Proposition 6 Let be an integer, and let be divergence-free. Set

where is sufficiently small depending on . Let be a sequence of divergence-free vector fields converging to in norm (for instance, one could apply Littlewood-Paley projections to ). Let , be the associated solutions to (1) provided by Corollary 3 (these are well-defined for large enough). Then and converge in norm on to limits , respectively, which solve (1) in a distributional sense.

Proof: We use a variant of Kato’s argument (see also the paper of Bona and Smith for a related technique). It will suffice to show that the form a Cauchy sequence in , since the algebra properties of then give the same for , and one can then easily take limits (in this relatively high regularity setting) to obtain the limiting solution that solves (1) in a distributional sense.

Let be a large dyadic integer. By Corollary 3, we may find an solution be the solution to the Euler equations (1) with initial data (which lies in ). From Theorem 5, one has

Applying the triangle inequality and then taking limit superior, we conclude that

But by Plancherel’s theorem and dominated convergence we see that

as , and hence

giving the claim.

Remark 7 Since the sequence can converge to at most one limit , we see that the solution to (1) is unique in the class of distributional solutions that are limits of smooth solutions (with initial data of those solutions converging to in ). However, this leaves open the possibility that there are other distributional solutions that do not arise as the limits of smooth solutions (or as limits of smooth solutions whose initial data only converge to in a weaker sense). It is possible to recover some uniqueness results for fairly weak solutions to the Euler equations if one also assumes some additional regularity on the fields (or on related fields such as the vorticity ). In two dimensions, for instance, there is a celebrated theorem of Yudovich that weak solutions to 2D Euler are unique if one has an bound on the vorticity. In higher dimensions one can also obtain uniqueness results if one assumes that the solution is in a high-regularity space such as , . See for instance this paper of Chae for an example of such a result.

Exercise 8 (Continuous dependence on initial data) Let be an integer, let , and set , where is sufficiently small depending on . Let be the closed ball of radius around the origin of divergence-free vector fields in . The above proposition provides a solution to the associated initial value problem. Show that the map from to is a continuous map from to .

Remark 9 The continuity result provided by the above exercise is not as strong as in Navier-Stokes, where the solution map is in fact Lipschitz continuous (see e.g., Exercise 43 of Notes 1). In fact for the Euler equations, which is classified as a “quasilinear” equation rather than a “semilinear” one due to the lack of the dissipative term in the equation, the solution map is not expected to be uniformly continuous on this ball, let alone Lipschitz continuous. See this previous blog post for some more discussion.

Exercise 10 (Maximal Cauchy development) Let be an integer, and let be divergence free. Show that there exists a unique and unique , with the following properties:

• (i) If and is divergence-free and converges to in norm, then for large enough, there is an solution to (1) with initial data on , and furthermore and converge in norm on to .
• (ii) If , then as .
• (iii) If , then we have the weak Beale-Kato-Majda criterion

  

Furthermore, show that do not depend on the particular choice of , in the sense that if belongs to both and for two integers then the time and the fields produced by the above claims are the same for both and .

We will refine part (iii) of the above exercise in the next section. It is a major open problem as to whether the case (i.e., finite time blowup) can actually occur. (It is important here that we have some spatial decay at infinity, as represented here by the presence of the norm; when the solution is allowed to diverge at spatial infinity, it is not difficult to construct smooth solutions to the Euler equations that blow up in finite time; see e.g., this article of Stuart for an example.)

Remark 11 The condition that recurs in the above results can be explained using the heuristics from Section 5 of Notes 1. Assume that a given time , the velocity field fluctuates at a spatial frequency , with the fluctuations being of amplitude . (We however permit the velocity field to contain a “bulk” low frequency component which can have much higher amplitude than ; for instance, the first component of might take the form where is a quantity much larger than .) Suppose one considers the trajectories of two particles whose separation at time zero is comparable to the wavelength of the frequency oscillation. Then the relative velocities of will differ by about , so one would expect the particles to stay roughly the same distance from each other up to time , and then exhibit more complicated and unpredictable behaviour after that point. Thus the natural time scale here is , so one only expects to have a reasonable local well-posedness theory in the regime

 

On the other hand, if lies in , and the frequency fluctuations are spread out over a set of volume , the heuristics from the previous notes predict that

The uncertainty principle predicts , and so

Thus we force the regime (11) to occur if , and barely have a chance of doing so in the endpoint case , but would not expect to have a local theory (at least using the sort of techniques deployed in this section) for .

Exercise 12 Use similar heuristics to explain the relevance of quantities of the form that occurs in various places in this section.

Because the solutions constructed in Exercise 10 are limits (in rather strong topologies) of smooth solutions, it is fairly easy to extend estimates and conservation laws that are known for smooth solutions to these slightly less regular solutions. For instance:

Exercise 13 Let be as in Exercise 10.

• (i) (Energy conservation) Show that for all .
• (ii) Show that

  

  

  for all .

Exercise 14 (Vanishing viscosity limit) Let the notation and hypotheses be as in Corollary 3. For each , let , be the solution to (3) with this choice of viscosity and with initial data . Show that as , and converge locally uniformly to , and similarly for all derivatives of and . (In other words, there is actually no need to pass to a subsequence as is done in the proof of Corollary 3.) Hint: apply the energy method to control the difference .

Exercise 15 (Local existence for forced Euler) Let be divergence-free, and let , thus is smooth and for any and any integer and , . Show that there exists and a smooth solution to the forced Euler equation

Note: one will first need a local existence theory for the forced Navier-Stokes equation. It is also possible to develop forced analogues of most of the other results in this section, but we will not detail this here.

— 2. The Beale-Kato-Majda blowup criterion —

In Exercise 10 we saw that we could continue solutions, to the Euler equations indefinitely in time, unless the integral became infinite at some finite time . There is an important refinement of this blowup criterion, due to Beale, Kato, and Majda, in which the tensor is replaced by the vorticity two-form (or vorticity, for short)

that is to say is essentially the anti-symmetric component of . Whereas is the tensor field

is the anti-symmetric tensor field

Remark 16 In two dimensions, is essentially a scalar, since and . As such, it is common in fluid mechanics to refer to the scalar field as the vorticity, rather than the two form . In three dimensions, there are three independent components of the vorticity, and it is common to view as a vector field rather than a two-form in this case (actually, to be precise would be a pseudovector field rather than a vector field, because it behaves slightly differently to vectors with respect to changes of coordinate). With this interpretation, the vorticity is now the curl of the velocity field . From a differential geometry viewpoint, one can view the two-form as an antisymmetric bilinear map from vector fields to scalar functions , and the relation between the vorticity two-form and the vorticity (pseudo-)vector field in is given by the relation

for arbitrary vector fields , where is the volume form on , which can be viewed in three dimensions as an antisymmetric trilinear form on vector fields. The fact that is a pseudovector rather than a vector then arises from the fact that the volume form changes sign upon applying a reflection.

The point is that vorticity behaves better under the Euler flow than the full derivative . Indeed, if one takes a smooth solution to the Euler equation in coordinates

and applies to both sides, one obtains

If one interchanges and then subtracts, the pressure terms disappear, and one is left with

which we can rearrange using the material derivative as

Writing and , this becomes the vorticity equation

Exercise 17 (Transport of vorticity) Let be a smooth solution to Euler equation in , and let be the vorticity two-form.

• (i) If , show that

  

• (ii) If , show that

  

  where is the vorticity pseudovector.

Remark 18 One can interpret the vorticity equation in the language of differential geometry, which is a more covenient formalism when working on more general Riemann manifolds than . To be consistent with the conventions of differential geometry, we now write the components of the velocity field as rather than (and the coordinates of as rather than ). Define the covelocity -form as

where is the Euclidean metric tensor (in the standard coordinates, is the Kronecker delta, though can take other values than if one uses a different coordinate system). Thus in coordinates, ; the covelocity field is thus the musical isomorphism applied to the velocity field. The vorticity -form can then be interpreted as the exterior derivative of the covelocity, thus

or in coordinates

The Euler equations can be rearranged as

 

where is the Lie derivative along , which for -forms is given in coordinates as

and is the modified pressure

If one takes exterior derivatives of both sides of (14) using the basic differential geometry identities and , one obtains the vorticity equation

where the Lie derivative for -forms is given in coordinates as

and so we recover (13) after some relabeling.

We now present the Beale-Kato-Majda condition.

Theorem 19 (Beale-Kato-Majda) Let be an integer, and let be divergence free. Let , be the maximal Cauchy development from Exercise 10, and let be the vorticity.

• (i) For any , one has

  

   

  

  

• (ii) If , then we have the Beale-Kato-Majda criterion

  

We would like to convert control on back to control of the full derivative . If one takes divergences of the vorticity using (12) and the divergence-free nature of , we see that

Thus, we can recover the derivative from the vorticity by the formula

where one can define via the Fourier transform as a multiplier bounded on every space.

and the claimed bound (15) would follow from Theorem 1(ii) (with one exponential to spare). Unfortunately, is not quite bounded on . Indeed, from Exercise 18 of Notes 1 we have the formula

for any test function and , where is the singular kernel

If one sets to be a (smooth approximation) to the signum restricted to an annulus , we conclude that the operator norm of is at least as large as

But one can calculate using polar coordinaates that this expression diverges like in the limit , , giving unboundedness.

As it turns out, though, the Gronwall argument used to establish Theorem 1(ii) can just barely tolerate an additional “logarithmic loss” of the above form, albeit at the cost of worsening the exponential term to a double exponential one. The key lemma is the following result that quantifies the logarithmic divergence indicated by the previous calculaation, and is similar in spirit to a well known inequality of Brezis and Wainger.

Lemma 20 (Near-boundedness of ) For any and , one has

 

and hence by the triangle inequality we may bound the left-hand side of (17) by

where we omit the domain and range from the function space norms for brevity.

By Bernstein’s inequality we have

Also, from Bernstein and Plancherel we have

and hence by geometric series we have

for any . This gives an acceptable contribution if we select . This leaves remaining values of to control, so if one can bound

for each , we will be done.

Observe from applying the scaling (that is, replacing with that to prove (18) for all it suffices to do so for . By Fourier analysis, the function is the convolution of with the inverse Fourier transform of the function

This function is a test function, so is a Schwartz function, and the claim now follows from Young’s inequality.

We return now to the proof of (15). We adapt the proof of Proposition 1(i). As in that proposition, we introduce the higher energy

We no longer have the viscosity term as , but that term was discarded anyway in the analysis. From (4) we have

Applying (16), (20) one thus has

From Exercise 13 one has

By the chain rule, one then has

and hence by Gronwall’s inequality one has

The claim (15) follows.

Remark 21 The Beale-Kato-Majda criterion can be sharpened a little bit, by replacing the sup norm with slightly smaller norms, such as the bounded mean oscillation (BMO) norm of , basically by improving the right-hand side of Lemma 20 slightly. See for instance this paper of Planchon and the references therein.

Remark 22 An inspection of the proof of Theorem 19 reveals that the same result holds if the Euler equations are replaced by the Navier-Stokes equations; the energy estimates acquire an additional “” term by doing so (as in the proof of Proposition 1), but the sign of that term is favorable.

We now apply the Beale-Kato-Majda criterion to obtain global well-posedness for the Euler equations in two dimensions:

Theorem 23 (Global well-posedness) Let be as in Exercise 10. If , then .

This theorem will be immediate from Theorem 19 and the following conservation law:

Proposition 24 (Conservation of vorticity distribution) Let be as in Exercise 10 with . Then one has

for all and .

Proof: By a limiting argument it suffices to show the claim for , thus we need to show

By another limiting argument we can take to be an solution. By the monotone convergence theorem (and Sobolev embedding), it suffices to show that

whenever is a test function that vanishes in a neighbourhood of the origin . Note that as and all its derivatives are in on for every , it is Lipschitz in space and time, which among other things implies that the level sets are compact for every , and so is smooth and compactly supported in . We may therefore may differentiate under the integral sign to obtain

where we omit explicit dependence on for brevity. By Exercise 17(i), the right-hand side is

which one can write as a total derivative

which vanishes thanks to integration by parts and the divergence-free nature of . The claim follows.

The above proposition shows that in two dimensions, is constant, and so the integral cannot diverge for finite . Applying Theorem 19, we obtain Theorem 23. We remark that global regularity for two-dimensional Euler was established well before the Beale-Kato-Majda theorem, starting with the work of Wolibner.

One can adapt this argument to the Navier-Stokes equations:

Exercise 25 Let be an integer, let , let be divergence-free, and let , be a maximal Cauchy development to the Navier-Stokes equations with initial data . Let be the vorticity.

• (i) Establish the vorticity equation .
• (ii) Show that for all and . (Note: to adapt the proof of Proposition 12, one should restrict attention to functions that are convex on the range of on, say, . The case of this inequality can also be established using the maximum principle for parabolic equations.)
• (iii) Show that .

Remark 26 There are other ways to establish global regularity for two-dimensional Navier-Stokes (originally due to Ladyzhenskaya); for instance, the bound on the vorticity in Exercise 25(ii), combined with energy conservation, gives a uniform bound on the velocity field, which can then be inserted into (the non-periodic version of) Theorem 38 of Notes 1.

Remark 27 If solve the Euler equations on some time interval with initial data , then the time-reversed fields solve the Euler equations on the reflected interval with initial data . Because of this time reversal symmetry, the local and global well-posedness theory for the Euler equations can also be extended backwards in time; for instance, in two dimensions any divergence free initial data leads to an solution to the Euler equations on the whole time interval . However, the Navier-Stokes equations are very much not time-reversible in this fashion.

I’m thinking about finite groups these days, from a Klein geometry perspective where we think of a group $G$ as a source of $G$-sets. Since the category of $G$-sets is a topos, this lets us translate concepts, facts and questions about groups into concepts, facts and questions about topoi. I’m not at all good at this, so here are a bunch of basic questions.

For any group $G$ the category of $G$-sets is a Boolean topos, which means basically that its internal logic obeys the principle of excluded middle.

• Which Boolean topoi are equivalent to the category of $G$-sets for some group $G$?

• Which are equivalent to the category of $G$-sets for a finite group $G$?

It might be easiest to start by characterizing the categories of $G$-sets where $G$ is a groupoid, and then add an extra condition to force $G$ to be a group.

The category $G Set$ comes with a forgetful functor $U: G Set \to Set$.

• Is the group of natural automorphisms of $U$ just $G$?

This should be easy to check, I’m just feeling lazy. If some result like this is true, how come people talk so much about the Tannaka–Krein reconstruction theorem and not so much about this simpler thing? (Maybe it’s just too obvious.)

Whenever we have a homomorphism $f \colon H \to G$ we get an obvious functor

$$f^\ast \colon G Set \to H Set$$

This is part of an essential geometric morphism, which means that it has both a right and left adjoint. By this means we can actually get a 2-functor from the 2-category of groups (yeah, it’s a 2-category since groups can be seen as one-object categories) to the 2-category $Topos_{ess}$ consisting of topoi, essential geometric morphisms and natural transformations. If I’m reading the $n$Lab correctly, this makes $G Set$ into a full sub-2-category of $Topos_{ess}$. This makes it all the more interesting to know which topoi are equivalent to categories of $G$-sets.

• What properties characterize essential geometric morphisms of the form $i^\ast \colon G \Set \to H \Set$ when $i \colon H \to G$ is the inclusion of a subgroup?

Whenever we have this, we get a transitive $G$-set $G/H$, which is thus a special object in $G Set$. These objects are just the atoms in $G Set$: that is, the objects whose only subobjects are themselves and the initial object. Indeed $G Set$ is an atomic topos, meaning that every object is a coproduct of atoms. That’s just a fancy way of saying that every $G$-set can be broken into orbits, which are transitive $G$-sets.

Next:

• What properties characterize essential geometric morphisms of the form $i^\ast \colon G \Set \to H \Set$ when $i \colon H \to G$ is the inclusion of a normal subgroup?

In this case $G/H$ is a group with a surjection $p \colon G \to G/H$, so we get another topos $(G/H)Set$ and essential geometric morphisms

$$Set \longrightarrow (G/H)Set \stackrel{p^\ast}{\longrightarrow} G Set \stackrel{i^\ast}{\longrightarrow} H Set \longrightarrow Set$$

• What properties characterize essential geometric morphisms of the form $p^*$ for $p$ a surjective homomorphism of groups?

• Is there a concept of ‘short exact sequence’ of essential geometric morphisms such that the above sequence is an example?

Well, my questions could go on all day, but this is enough for now!

Introduction and Motivation

One thing that I keep bumping up against as an engineer who loves functional a programming is category theory. It often seems like there are two kinds of functional programmers: people who came into functional programming via engineering, and people who came into functional programming via math. The problem is that a lot of the really interesting work in languages and libraries for functional programming are being built from the mathematical side, but for people on the engineering side, it’s impenetrable: it’s like it’s written in a whole different language, and even basic discussions about programming go off the rails, because the basic abstractions don’t make any sense if you don’t know category theory.

But how do you learn category theory? It seems impenetrable to mere humans. For example, one of the textbooks on category theory that several people told me was the most approachable starts chapter one with the line:

A group extension of an abelian group by an abelian group consists of a group together with an inclusion of as a normal subgroup and a surjective homomorphism that displays as the quotient group .

If you’re not a professional mathematician, then that is pure gobbledigook. But that seems to be typical of how initiates of category theory talk about it. But the basic concepts, while abstract, really aren’t all that tricky. In many ways, it feels a lot like set theory: there’s a simple conceptual framework, on which you can build extremely complicated formalisms. The difference is that while many people have spent years figuring out how to make the basics of set theory accessible to lay-people, but that effort hasn’t been applied to set theory.

What’s the point?

Ok, so why should you care about category theory?

Category theory is a different way of thinking, and it’s a language for talking about abstractions. The heart of engineering is abstraction. We take problems, and turn them into abstract structures. We look at the structures we create, and recognize commonalities between those structures, and then we create new abstractions based on the commonalities. The hardest part of designing a good library is identifying the right abstractions.

Category theory is a tool for talking about structures, which is particularly well suited to thinking about software. In category theory, we think in terms of arrows, where arrows are mappings between objects. We’ll see what that means in detail later, but the gist of it is that one example of arrows mapping between objects is functions mapping between data types in a computer program.

Category theory is built on thinking with orrows, and building structures using arrows. It’s about looking at mathematical constructions built with arrows, and in those structures, figuring out what the fundamental parts are. When we abstract enough, we can start to see that things that look very different are really just different realizations of the same underlying structure. Category theory gives us a language and a set of tools for doing that kind of abstraction – and then we can take the abstract structures that we identify, and turn them into code – into very generic libraries that express deep, fundamental structure.

Start with an Example: Monoids

Monoids in Code

We’ll get started by looking at a simple mathematical structure called a monoid, and how we can implement it in code; and then, we’ll move on to take an informal look at how it works in terms of categories.

Most of the categorical abstractions in Scala are implemented using something called a typeclass, so we’ll start by looking at typeclasses. Typeclasses aren’t a category theoretical notion, but they make it much, much easier to build categorical structures. And they do give us a bit of categorical flavor: a typeclass defines a kind of metatype – that is, a type of type – and we’ll see, that kind of self-reflective abstraction is a key part of category theory.

The easiest way to think about typeclasses is that they’re a kind of metatype – literally, as the name suggests, they define classes where the elements of those classes are types. So a typeclass provides an interface that a type must provide in order to be an instance of the metatype. Just like you can implement an interface in a type by providing implementations of its methods, you can implement a typeclass by providing implementations of its operations.

In Scala, you implement the operations of a typeclasses using a language construct called an implicit parameter. The implicit parameter attaches the typeclass operations to a meta-object that can be passed around the program invisibly, providing the typeclass’s operations.

Let’s take a look at an example. An operation that comes up very frequently in any kind of data processing code is reduction: taking a collection of values of some type, and combining them into a single value. Taking the sum of a list of integers, the product of an array of floats, and the concatenation of a list of strings are all examples of reduction. Under the covers, these are all similar: they’re taking an ordered group of values, and performing an operation on them. Let’s look at a couple of examples of this:

def reduceFloats(floats: List[Float]): Float =
    floats.foldRight(0)((x, y) => x + y)

def reduceStrings(strings: Seq[String]): String =
    strings.foldRight("")((x, y) => x.concat(y))

When you look at the code, they look very similar. They’re both just instantiations of the same structural pattern:

def reduceX(xes: List[X]): X =
    xes.foldRight(xIdentity)((a, b) => Xcombiner(a, b))

The types are different; the actual operation used to combine the values is different; the base value in the code is different. But they’re both built on the same pattern:

• There’s a type of values we want to combine: Float or String. Everything we care about in reduction is a connected with this type.
• There’s a collection of values that we want to combine, from left to right. In one case, that’s a List[Float], and in the other, it’s a Seq[String]. The type doesn’t matter, as long as we can iterate over it.
• There’s an identity value that we can use as a starting point for building the result; 0 for the floats, and "" (the empty string) for the strings.
• There’s an operation to combine two values: + for the floats, and concat for the strings.

We can capture that concept by writing an interface (a trait, in Scala terms) that captures it; that interface is called a typeclass. It happens that this concept of reducible values is called a monoid in abstract algebra, so that’s the name we’ll use.

trait Monoid[A]  {
    def empty: A
    def combine(x: A, y: A): A
}

We can read that as saying “A is a monoid if there are implementations of empty and combine that meet these constraints”. Given the declaration of the typeclass, we can implement it as an object which provides those operations for a particular type:

object FloatAdditionMonoid extends Monoid[Float] {
    def empty: Float = 0.0
    def combine(x: Float, y: Float): Float = x + y
}

object StringConcatMonoid extends Monoid[String] {
    def empty: String = ""
    def combine(x: String, y: String): String = x.concat(y)
}

FloatAdditionMonoid implements the typeclass Monoid for the type Float. And since we can write an implementation of Monoid for Float or String, we can say that the types Float and String are instances of the typeclass Monoid.

Using our implementation of Monoid, we can write a single, generic reduction operator now:

def reduce[A](values: Seq[A], monoid: Monoid[A]): A =
   values.foldRight(monoid.empty)(monoid.combine)

We can use that to reduce a list of floats:

reduce([1.0, 3.14, 2.718, 1.414, 1.732], FloatAdditionMonoid)

And we can do a bit better than that! We can set up an implicit, so that we don’t need to pass the monoid implementation around. In Scala, an implicit is a kind of dynamically scoped value. For a given type, there can be one implicit value of that type in effect at any point in the code. If a function takes an implicit parameter of that type, then the nearest definition in the execution stack will automatically be inserted if the parameter isn’t passed explicitly.

def reduce[A](values: Seq[A])(implicit A: Monoid[A]): A =
   list.foldRight(A.empty)(A.combine)

And as long as there’s a definition of the Monoid for a type A in scope, we can can use that now by just writing:

implicit object FloatAdditionMonoid extends Monoid[Float] {
    def empty: Float = 0.0
    def combine(x: Float, y: Float): Float = x + y
}

val floats: List[Float] = ...
val result = reduce(floats)

Now, anywhere that the FloatAdditionMonoid declaration is imported, you can call reduce on any sequence of floats, and the implicit value will automatically be inserted.

Using this idea of a monoid, we’ve captured the concept of reduction in a common abstraction. Our notion of reduction doesn’t care about whether we’re reducing strings by concatenation, integers by addition, floats by multiplication, sets by union. Those are all valid uses of the concept of a monoid, and they’re all easy to implement using the monoid typeclass. The concept of monoid isn’t a difficult one, but at the same time, it’s not necessarily something that most of us would have thought about as an abstraction.

We’ve got a typeclass for a monoid; now, we’ll try to connect it into category theory. It’s a bit tricky, so we won’t cover it all at once. We’ll look at it a little bit now, and we’ll come back to it in a later lesson, after we’ve absorbed a bit more.

From Sets to Arrows

For most of us, if we’ve heard of monoids, we’ve heard of them in terms of set theory and abstract algebra. So in that domain, what’s a monoid?

A monoid is a triple , where:

• is a set of values;
• is a value in ;
• is a total binary operator where:
  • is an identity of : For any value .
  • is associative: for any values

That’s all just a formal way of saying that a monoid is a set with a binary associative operator and an identity value. The set of integers can form a monoid with addition as the operator, and 0 as identity. Real numbers can be a monoid with multiplication and 1. Strings can be a monoid with concatenation as the operator, and empty string as identity.

But we can look at it in a different way, too, by thinking entirely in terms of function.
Let’s forget about the numbers as individual values, and instead, let’s think about them in functional terms. Every number is a function which adds itself to its parameter. So “2” isn’t a number, it’s a function which adds two to anything.

How can we tell that 2 is a function which adds two to things?

If we compose it with 3 (the function that adds three to things), we get 5 (the function that adds five to things). And how do we know that? Because it’s the same thing that we get if we compose 3 with 1, and then compose the result of that with 1 again. 3+1+1=5, and 3+2=5. We can also tell that it’s 2, because if we just take 1, and compose it with itself, what we’ll get back is the object that we call 2.

In this scheme, all of the numbers are related not by arithmetic, not by an underlying concept of quantity or cardinality or ordinality, but only by how they compose with each other. We can’t see anything else – all we have are these functions. But we can recognize that they are the natural numbers that we’re familiar with.

Looking at it this way, we can think of the world of natural numbers as a single point, which represents the set of all natural numbers. And around that point, we’ve got lots and lots of arrows, each of which goes from that point back to itself. Each of those arrows represents one number. The way we tell them apart is by understanding which arrow we get back when we compose them. Take any arrow from that point back to that point, and compose it with the arrow 0, and what do you get? The arrow you started with. Take any arrow that you want, and compose it with 2. What do you get? You get the same thing that you’d get if you composed it with 1, and then composed it with one again.

That dot, with those arrows, is a category.

What kind of advantage do we get in going from the algebraic notion of a set with a binary operation, to the categorical notion of an object with a bunch of composable arrows? It allows to understand a monoid purely as a structure, without having the think about what the objects are, or what the operator means.

Now, let’s jump back to our monoid typeclass for a moment.

trait Monoid[A]  {
    def empty: A
    def combine(x: A, y: A): A
}

We can understand this as being a programmable interface for the categorical object that we just described. All we need to do is read “:” as “is an arrow in”: It says that A is a monoid if:

• It has an element called empty which is an arrow in A.
• It has an operation called combine which, given any two arrows in A, composes them into a new arrow in A.

There are, of course, other conditions – combine needs to be associative, and empty needs to behave as the identity value. But just like when we write an interface for, say, a binary search tree, the interface only defines the structure not the ordering condition, the typeclass defines the functional structure of the categorical object, not the logical conditions.

This is what categories are really all about: tearing things down to a simple core, where everything is expressed in terms of arrows. It’s almost reasoning in functions, except that it’s even more abstract than that: the arrows don’t need to be functions – they just need to be composable mappings from things to things.

Deeper Into Arrows

We can abstract a bit more, and look at the entire construction, including the identity and associativity constraints entirely in terms of arrows. To really understand this, we’ll need to spend some time diving deeper into the actual theory of categories, but as a preview, we can describe a monoid with the following pair of diagrams (copied from wikipedia):

In these diagrams, any two paths between the same start and end-nodes are equivalent (up to isomorphism). When you understand how to read this diagrams, these really do define everything that we care about for monoids.

For now, we’ll just run through and name the parts – and then later, in another lesson, we’ll come back, and we’ll look at this in more detail.

• is an arrow from , which we’ll call a multiplication operator.
• is an arrow from , called unit.
• is an arrow from which represents the associativity property of the monoid.
• is a morphism which represents the left identity property of the monoid (that is, ), and is a morphism representing the right identity property .

This diagram, using these arrows, is a way of representing all of the key properties of a monoid via nothing but arrows and composition. It says, among other things, that:

• composes with multiplication to be .
  That is, applying multiplication to evaluates to (M \times M).
• composed with associativity can become .

So it’s a monoid – but it’s a higher level monoid. In this, isn’t just an object in a category: it’s an entire category. These arrows are arrows between categories in a category of categories.

What we’ll see when we get deeper into category theory is how powerful this kind of abstraction can get. We’ll often see a sequence of abstractions, where we start with a simple concept (like monoid), and find a way to express it in terms of arrows between objects in a category. But then, we’ll lift it up, and look at how we can see in not just as a relation between objects in a category, but as a different kind of relation between categories, by constructing the same thing using a category of categories. And then we’ll abstract even further, and construct the same thing using mappings between categories of categories.

Back in 2015, I reported some progress on this difficult problem in plane geometry. I’m happy to report some more.

First, remember the story. A subset of the plane has diameter 1 if the distance between any two points in this set is ≤ 1. A universal covering is a convex subset of the plane that can cover a translated, reflected and/or rotated version of every subset of the plane with diameter 1. In 1914, the famous mathematician Henri Lebesgue sent a letter to a fellow named Pál, challenging him to find the universal covering with the least area.

Pál worked on this problem, and 6 years later he published a paper on it. He found a very nice universal covering: a regular hexagon in which one can inscribe a circle of diameter 1. This has area

But he also found a universal covering with less area, by removing two triangles from this hexagon—for example, the triangles C₁C₂C₃ and E₁E₂E₃ here:

The resulting universal covering has area

In 1936, Sprague went on to prove that more area could be removed from another corner of Pál’s original hexagon, giving a universal covering of area

In 1992, Hansen took these reductions even further by removing two more pieces from Pál’s hexagon. Each piece is a thin sliver bounded by two straight lines and an arc. The first piece is tiny. The second is downright microscopic!

Hansen claimed the areas of these regions were 4 · 10^-11 and 6 · 10^-18. This turned out to be wrong. The actual areas are 3.7507 · 10^-11 and 8.4460 · 10^-21. The resulting universal covering had an area of

This tiny improvement over Sprague’s work led Klee and Wagon to write:

it does seem safe to guess that progress on [this problem], which has been painfully slow in the past, may be even more painfully slow in the future.

However, in 2015 Philip Gibbs found a way to remove about a million times more area than Hansen’s larger region: a whopping 2.233 · 10^-5. This gave a universal covering with area

Karine Bagdasaryan and I helped Gibbs write up a rigorous proof of this result, and we published it here:

• John Baez, Karine Bagdasaryan and Philip Gibbs,The Lebesgue universal
covering problem, Journal of Computational Geometry 6 (2015), 288–299.

Greg Egan played an instrumental role as well, catching various computational errors.

At the time Philip was sure he could remove even more area, at the expense of a more complicated proof. Since the proof was already quite complicated, we decided to stick with what we had.

But this week I met Philip at The philosophy and physics of Noether’s theorems, a wonderful workshop in London which deserves a full blog article of its own. It turns out that he has gone further: he claims to have found a vastly better universal covering, with area

This is an improvement of 2.178245 × 10^-5 over our earlier work—roughly equal to our improvement over Hansen.

You can read his argument here:

• Philip Gibbs, An upper bound for Lebesgue’s universal covering problem, 22 January 2018.

I say ‘claims’ not because I doubt his result—he’s clearly a master at this kind of mathematics!—but because I haven’t checked it and it’s easy to make mistakes, for example mistakes in computing the areas of the shapes removed.

It seems we are closing in on the final result; however, Philip Gibbs believes there is still room for improvement, so I expect it will take at least a decade or two to solve this problem… unless, of course, some mathematicians start working on it full-time, which could speed things up considerably.

This coming fall quarter, I am teaching a class on topics in the mathematical theory of incompressible fluid equations, focusing particularly on the incompressible Euler and Navier-Stokes equations. These two equations are by no means the only equations used to model fluids, but I will focus on these two equations in this course to narrow the focus down to something manageable. I have not fully decided on the choice of topics to cover in this course, but I would probably begin with some core topics such as local well-posedness theory and blowup criteria, conservation laws, and construction of weak solutions, then move on to some topics such as boundary layers and the Prandtl equations, the Euler-Poincare-Arnold interpretation of the Euler equations as an infinite dimensional geodesic flow, and some discussion of the Onsager conjecture. I will probably also continue to more advanced and recent topics in the winter quarter.

In this initial set of notes, we begin by reviewing the physical derivation of the Euler and Navier-Stokes equations from the first principles of Newtonian mechanics, and specifically from Newton’s famous three laws of motion. Strictly speaking, this derivation is not needed for the mathematical analysis of these equations, which can be viewed if one wishes as an arbitrarily chosen system of partial differential equations without any physical motivation; however, I feel that the derivation sheds some insight and intuition on these equations, and is also worth knowing on purely intellectual grounds regardless of its mathematical consequences. I also find it instructive to actually see the journey from Newton’s law

to the seemingly rather different-looking law

for incompressible Navier-Stokes (or, if one drops the viscosity term , the Euler equations).

Our discussion in this set of notes is physical rather than mathematical, and so we will not be working at mathematical levels of rigour and precision. In particular we will be fairly casual about interchanging summations, limits, and integrals, we will manipulate approximate identities as if they were exact identities (e.g., by differentiating both sides of the approximate identity), and we will not attempt to verify any regularity or convergence hypotheses in the expressions being manipulated. (The same holds for the exercises in this text, which also do not need to be justified at mathematical levels of rigour.) Of course, once we resume the mathematical portion of this course in subsequent notes, such issues will be an important focus of careful attention. This is a basic division of labour in mathematical modeling: non-rigorous heuristic reasoning is used to derive a mathematical model from physical (or other “real-life”) principles, but once a precise model is obtained, the analysis of that model should be completely rigorous if at all possible (even if this requires applying the model to regimes which do not correspond to the original physical motivation of that model). See the discussion by John Ball quoted at the end of these slides of Gero Friesecke for an expansion of these points.

Note: our treatment here will differ slightly from that presented in many fluid mechanics texts, in that it will emphasise first-principles derivations from many-particle systems, rather than relying on bulk laws of physics, such as the laws of thermodynamics, which we will not cover here. (However, the derivations from bulk laws tend to be more robust, in that they are not as reliant on assumptions about the particular interactions between particles. In particular, the physical hypotheses we assume in this post are probably quite a bit stronger than the minimal assumptions needed to justify the Euler or Navier-Stokes equations, which can hold even in situations in which one or more of the hypotheses assumed here break down.)

— 1. From Newton’s laws to the Euler and Navier-Stokes equations —

For obvious reasons, the derivation of the equations of fluid mechanics is customarily presented in the three dimensional setting (and sometimes also in the two-dimensional setting ), but actually the general dimensional case is not that much more difficult (and in some ways clearer, as it reveals that the derivation does not depend on any structures specific to three dimensions, such as the cross product), so for this derivation we will work in the spatial domain for arbitrary . One could also work with bounded domains , or periodic domains such as ; the derivation is basically the same, thanks to the local nature of the forces of fluid mechanics, except at the boundary where the situation is more subtle (and may be discussed in more detail in later posts). For sake of notational simplicity, we will assume that the time variable ranges over the entire real line ; again, since the laws of classical mechanics are local in time, one could just as well restrict to some sub-interval of this line, such as for some time .

Our starting point is Newton’s second law , which (partially) describes the motion of a particle of some fixed mass moving in the spatial domain . (Here we assume that the mass of a particle does not vary with time; in particular, our discussion will be purely non-relativistic in nature, though it is possible to derive a relativistic version of the Euler equations by variants of the arguments given here.) We write Newton’s second law as the ordinary differential equation

where is the trajectory of the particle (thus denotes the position of the particle at time ), and is the force applied to that particle. If we write for the coordinates of the vector-valued function , and similarly write for the components of , we therefore have

where we adopt in this section the convention that the indices are always understood to range from to .

If one has some collection particles instead of a single particle, indexed by some set of labels (e.g. the numbers from to , if there are a finite number of particles; for unbounded domains such as one can also imagine situations in which is infinite), then for each , the particle has some mass , some trajectory (with components ), and some force applied (with components , we thus have the equation of motion

or in components

In this section we adopt the convention that the indices are always understood to range over the set of labels for the particles; in particular, their role should not be confused with those of the coordinate indices .

Newton’s second law does not, by itself, completely specify the evolution of a system of particles, because it does not specify exactly how the forces depend on the current state of the system. For particles, the current state at a given time is given by the positions of all the particles , as well as their velocities ; we assume for simplicity that the particles have no further physical characteristics or internal structure of relevance that would require more state variables than these. (No higher derivatives need to be specified beyond the first, thanks to Newton’s second law. On the other hand, specifying position alone is insufficient to describe the state of the system; this was noticed as far back as Zeno in his paradox of the arrow, which in retrospect can be viewed as a precursor to Newton’s second law insofar as it demonstrated that the laws of motion needed to be second-order in time (in contrast, for instance, to Aristotlean physics, which was morally first-order in nature).) At a fundamental level, the dependency of forces on the current state are governed by the laws of physics for such forces; for instance, if the particles interact primarily through electrostatic forces, then one needs the laws of electrostatics to describe these forces. (In some cases, such as electromagnetic interactions, one cannot accurately model the situation purely in terms of interacting particles, and the equations of motion will then involve some additional mediating fields such as the electromagnetic field; but we will ignore this possibility in the current discussion for sake of simplicity.)

Fortunately, thanks to other laws of physics, and in particular Newton’s other two laws of motion, one can still obtain partial information about the forces without having to analyse the fundamental laws producing these forces. For instance, Newton’s first law of motion (when combined with the second) tells us that a single particle does not exert any force on itself; the net force on only arises from interaction with other particles , (for this discussion we neglect external forces, such as gravity, although one could easily incorporate such forces into this discussion; see Exercise 3 below). We will assume that the only forces present are pair interactions coming from individual pairs of particles ; it is theoretically possible that one could have more complicated interactions between, say, a triplet of particles that do not simply arise from the interactions between the three pairs , , , but we will not consider this possibility here. We also assume that the net force on a particle is just the sum of all the interacting forces (i.e., the force addition law contains no nonlinear terms). This gives us a decomposition

of the net force on a particle into the interaction force exerted on by another particle . Thus the equation of motion is now

Of course, this description is still incomplete, because we have not specified exactly what the interaction forces are. But one important constraint on these forces is provided by Newton’s third law

This already gives some restrictions on the possible dynamics. For instance, it implies (formally, at least) that the total momentum

(which takes values in ) is conserved in time:

We will also assume that the interaction force between a pair of particles is parallel to the displacement between the pair; in other words, we assume the torque created by this force vanishes, thus

Here is the exterior product on (which in three dimensions can be transformed if one wishes to the cross product, but is well defined in all dimensions). Algebraically, is the universal alternating bilinear form on ; in terms of the standard basis of , the wedge product of two vectors , is given by

and the vector space is the formal span of the basic wedge products for .

One consequence of the absence (4) of torque is the conservation of total angular momentum

(around the spatial origin ). Indeed, we may calculate

around any other point is also conserved in time, as is clear from repeating the above calculation, or by combining the existing conservation laws for (3) and (5).

Now we pass from particle mechanics to continuum mechanics, by considering the limiting (bulk) behaviour of many particle systems as the number of particles per unit volume goes to infinity. (In physically realistic scenarios, will be comparable to Avagadro’s constant, which seems large enough that such limits should be a good approximation to the truth.) To make such limits, we assume that the distribution of particles (and various properties of these particles, such as their velocities and net interaction forces) are approximated in a certain bulk sense by continuous fields. For instance, the mass distribution of a system of particles at a given time is given by the discrete measure

where denotes the Dirac measure (or distribution). We will assume that at each time , we have a “bulk” approximation

by some continuous measure , where the density function is some smooth function of time and space, and denotes Lebesgue measure on . What does bulk approximation mean? One could work with various notions of approximation, but we will adopt the viewpoint of the theory of distributions (as reviewed for instance in these old lecture notes of mine) and consider approximation against test functions in spacetime, thus we assume that

for all spacetime test functions . (One could also work with purely spatial test functions at each fixed time , or work with “infinitesimal” paralleopipeds or similar domains instead of using test functions; the arguments look slightly different when doing so, but the final equations of motion obtained are the same in all cases. See Exercise 1 for an example of this.) We will be deliberately vague as to what means, other than to say that the approximation should only be considered accurate (in the sense that it becomes exact in the limit ) when the test function exists at “macroscopic” (or “bulk”) spatial scales, in particular it should not oscillate with a wavelength that goes to zero as goes to infinity. (For instance, one certainly expects the approximation (7) to break down if one tries to test it on scales comparable to the mean spacing between particles.)

Applying (6) and evaluating the delta integrations, the approximation (8) becomes

In a physical liquid, particles in a given small region of space tend to move at nearly identical velocities (as opposed to gases, where Brownian motion effects lead one to expect velocities to be distributed stochastically, for instance in a Maxwellian distribution). To model this, we assume that there exists a smooth velocity field for which we have the approximation

for all particles and all times . (When stochastic effects are significant, the continuum limit of the fluid will be the Boltzmann equations rather than the Euler or Navier-Stokes equations; however the latter equations can still emerge as an approximation of the former in various regimes. See also Remark 7 below.)

Implicit in our model of many-particle interactions is the conservation of mass: each particle has a fixed mass , and no particle is created or destroyed by the evolution. This conservation of mass, when combined with the approximations (9) and (10), gives rise to a certain differential equation relating the density function and the velocity field . To see this, first observe from the fundamental theorem of calculus in time that

or equivalently (after applying (6) and evaluating the delta integrations)

for any test function (note that we make compactly supported in both space and time). By the chain rule and (10), we have

for any particle and any time , where denotes the partial derivative with respect to the time variable, denotes the spatial gradient (with denoting partial differentiation in the coordinate), and denotes the Euclidean inner product. (One could also use notation here that avoids explicit use of Euclidean structure, for instance writing in place of , but it is traditional to use Euclidean notation in fluid mechanics.) This particular combination of derivatives appears so often in the subject that we will give it a special name, the material derivative :

We have thus obtained the approximation

which on insertion back into (12) yields

The material derivative of a test function will still be a test function . Therefore we can use our field approximation (9) to conclude that

for all test functions . The left-hand side consists entirely of the limiting fields, so on taking limits we should therefore have the exact equation

We can integrate by parts to obtain

where is the adjoint material derivative

Since the test function was arbitrary, we conclude the continuity equation

or equivalently (and more customarily)

We will eventually specialise to the case of incompressible fluids in which the density is a non-zero constant in both space and time. (In some texts, one uses incompressibility to refer only to constancy of along trajectories: . But in this course we always use incompressibility to refer to homogeneous incompressibility, in which is constant in both space and time.) In this incompressible case, the continuity equation (15) simplifies to a divergence-free condition on the velocity:

For now, though, we allow for the possibility of compressibility by allowing to vary in space and time. We also note that by integrating (15) in space, we formally obtain conservation of the total mass

since on differentiating under the integral sign and then integrating by parts we formally have

Of course, this conservation law degenerates in the incompressible case, since the total mass is manifestly an infinite constant. (In periodic settings, for instance if one is working in instead of , the total mass is manifestly a finite constant in the incompressible case.)

Exercise 1

• (i) Assume that the spacetime bulk approximation (8) is replaced by the spatial bulk approximation

  

  for any time and any spatial test function . Give an alternate heuristic derivation of the continuity equation (15) in this case, without using any integration in time. (Feel free to differentiate under the integral sign.)

• (ii) Assume that the spacetime bulk approximation (8) is replaced by the spatial bulk approximation

  

  for any time and any “reasonable” set (e.g., a rectangular box). Give an alternate heuristic derivation of the continuity equation (15) in this case. (Feel free to introduce infinitesimals and argue non-rigorously with them.)

We can repeat the above analysis with the mass distribution (6) replaced by the momentum distribution

thus we now wish to exploit the conservation of momentum rather than conservation of mass. The measure is a vector-valued measure, or equivalently a vector of scalar measures

Instead of starting with the identity (11), we begin with the momentum counterpart

which on applying (17) and evaluating the delta integrations becomes

Using the product rule and (13), the left-hand side is approximately

Applying (1) and (10), we conclude

We can evaluate the second term using (9) to obtain

What about the first term? We can use symmetry and Newton’s third law (2) to write

Now we make the further physical assumption that the only significant interactions between particles are short-range interactions, in which and are very close to each other. With this hypothesis, it is then plausible to make the Taylor approximation

We thus have

We write this in coordinates as

for , where we use the Einstein convention that indices are implicitly summed over if they are repeated in an expression, and the stress on the particle at time is the rank -tensor defined by the formula

where denote the components of . Recall from the torque-free hypothesis (4) that is parallel to , thus we could write for some scalar . Thus we have

In particular, we see that the torque-free hypothesis makes the stress tensor symmetric:

To proceed further, we make the assumption (similar to (9)) that the stress tensor (or more precisely, the measure ) is approximated in the bulk by a smooth tensor field (with components for ), in the sense that

The tensor is known as the Cauchy stress tensor. Since is symmetric in , the right-hand side of (21) is also symmetric in , which by the arbitrariness of implies that the tensor is symmetric also:

This is also known as Cauchy’s second law of motion.

Inserting (19) into (21), we arrive at

for any test function and . Taking limits as we obtain the exact equation

and then integrating by parts we have

as the test function is arbitrary, we conclude the Cauchy momentum equation

From the Leibniz rule (in an adjoint form) we see that

using (14) we can thus also write the Cauchy momentum equation in the more conventional form

(This is a dynamical version of Cauchy’s first law of motion.)

To summarise so far, the unknown density field and velocity field obey two equations of motion: the continuity equation (15) (or (14)) and the momentum equation (22). As the former is a scalar equation and the latter is a vector equation, this is equations for unknowns, which looks good – so long as the stress tensor is known. However, the stress tensor is not given to us in advance, and so further physical assumptions on the underlying fluid are needed to derive additional equations to yield a more complete set of equations of motion.

One of the simplest such assumptions is isotropy – that, in the vicinity of a given particle at a given point in time, the distribution of the nearby particles (and of the forces ) is effectively rotationally symmetric, in the sense that rotation of the fluid around that particle does not significantly affect the net stresses acting on the particle. To give more mathematical meaning to this assumption, let us fix and , and let us set to be the spatial origin for simplicity. In particular the stress tensor now simplifies a little to

viewing this tensor as a symmetric matrix, we can also write

where we now think of the vector as a -dimensional column vector, and denotes the transpose of .

Imagine that we rotate the fluid around this spatial origin using some rotation matrix , thus replacing with (and hence is replaced with ). If we assume that interaction forces are rotationally symmetric, the interaction scalars should not be affected by this rotation. As such, would be replaced with . If we assume isotropy, though, this rotated fluid should generate the same stress as the original fluid, thus we have

for all rotation matrices , that is to say that commutes with all rotations. This implies that all eigenspaces of are rotation-invariant, but the only rotation-invariant subspaces of are and . Thus the spectral decomposition of the symmetric matrix only involves a single eigenspace , or equivalently is a multiple of the identity. In coordinates, we have

for some scalar (known as the pressure exerted on the particle ), where denotes the Kronecker delta (the negative sign here is for compatibility with other physical definitions of pressure). Passing from the individual stresses to the stress field , we see that is also rotationally invariant and thus is also a multiple of the identity, thus

for some field , which we call the pressure field, and which we assume to be smooth. The equations (15), (22) now become the Euler equations

This is still an underdetermined system, being equations for unknowns (two scalar fields and one vector field ). But if we assume incompressibility (normalising for simplicity), we obtain the incompressible Euler equations

Without incompressibility, one can still reach a determined system of equations if one postulates a relationship (known as an equation of state) between the pressure and the density (in some physical situations one also needs to introduce further thermodynamic variables, such as temperature or entropy, which also influence this relationship and obey their own equation of motion). Alternatively, one can proceed using an analysis of the energy conservation law, similar to how (15) arose from conservation of mass and (22) arose from conservation of momentum, though at the end of the day one would still need an equation of state connecting energy density to other thermodynamic variables. For details, see Exercise 4 below.

Now we consider relaxing the assumption of isotropy in the stress. Many fluids, in addition to experiencing isotropic stress coming from a scalar pressure field, also experience additional shear stress associated to strain in the fluid – distortion in the shape of the fluid arising from fluctuations in the velocity field. One can thus postulate a generalisation to (23) of the form

where is some function of the velocity at the point in spacetime, as well as the first derivatives , that arises from changes in shape. (Here, we assume here that the response to stress is autonomous – it does not depend directly on time, location, or other statistics of the fluid, such as pressure, except insofar as those variables are related to the quantities and . One can of course consider more complex models in which there is a dependence on such quantities or on higher derivatives , etc. of the velocity, but we will not do so here.)

It is a general principle in physics that functional relationships between physical quantities, such as the one in (27), can be very heavily constrained by requiring the relationship to be invariant with respect to various physically natural symmetries. This is certainly the case for (27). First of all, we can impose Galilean invariance: if one changes to a different inertial frame of reference, thus adding a constant vector to the velocity field (and not affecting the gradient at all), this should not actually introduce any new stresses on the fluid. This leads to the postulate

for any , and hence should not actually depend on the velocity and should only depend on the first derivative. Thus we now write for (thus is now a function from to ).

If then there should be no additional shear stress, so we should have . We now make a key assumption that the fluid is a Newtonian fluid, in that the linear term in the Taylor expansion of dominates, or in other words we assume that is a linear function of . (One can certainly study the mechanics of non-Newtonian fluids as well, in which depends nonlinearly on , or even on past values of , but these are no longer governed by the Navier-Stokes equations and will not be considered further here.) One can also think of as a linear map from the space of matrices (which is the space where takes values) to the space of matrices. In coefficients, this means we are postulating a relationship of the form

for some constants (recall we are using the Einstein summation conventions). This looks like a lot of unspecified constants, but again we can use physical principles to impose significant constraints. Firstly, because stress is symmetric in and , the coefficients must also be symmetric in and : . Next, let us for simplicity set to be the spatial origin , and consider a rotating velocity field of the form

for some constant-coefficient anti-symmetric matrix , or in coordinates

The derivative field is then just the anti-symmetric . This corresponds to fluids moving according to the rotating trajectories , where denotes the matrix exponential. (For instance, in two dimensions, the velocity field gives rise to trajectories , corresponding to counter-clockwise rotation around the origin.)

Exercise 2 When is anti-symmetric, show that the matrix is orthogonal for all , thus

for all , where denotes the identity matrix. (Hint: differentiate the expressions appearing in the above equation with respect to time.) Also show that is a rotation matrix, that is to say an orthogonal matrix of determinant .

As is orthogonal, it describes a rigid motion. Thus this rotating motion does not change the shape of the fluid, and so should not give rise to any shear stress. That is to say, the linear map should vanish when applied to an anti-symmetric matrix, or in coordinates . Another way of saying this is that only depends on through its symmetric component , known as the rate-of-strain tensor (or the deformation tensor). (The anti-symmetric part does not cause any strain, but instead measures the infinitesimal rotation the fluid; up to trivial factors such as , it is essentially the vorticity of the fluid, which will be an important field to study in subsequent notes.)

To constrain the behaviour of further, we introduce a hypothesis of rotational (and reflectional) symmetry. If one rotates the fluid by a rotation matrix around the origin, then if the original fluid had a velocity of at , the new fluid should have a velocity of at , thus the new velocity field is given by the formula

and the derivative of this velocity field at the origin is then related to the original derivative by the formula

Meanwhile, as discussed in the analysis of the isotropic case, the new stress at the origin is related to the original stress by the same relation:

This means that the linear map is rotationally equivariant in the sense that

for any matrix . Actually the same argument also applies for reflections, so one could also take in the orthogonal group rather than the special orthogonal group.

This severely constrains the possible behaviour of . First consider applying to the rank matrix , where is the first basis (column) vector of . The equivariant property (28) then implies that is invariant with respect to any rotation or reflection of the remaining coordinates . As in the isotropic analysis, this implies that the lower right minor of is a multiple of the identity; when it also implies that the upper right entries or lower left entries for also vanish (one can also obtain this by applying (28) with the reflection in the variable). Thus we have

for some constant scalars (known as the dynamic viscosity and second viscosity respectively); in matrix form we have

where is the identity matrix. Applying equivariance, we conclude that

for any unit vector ; applying the spectral theorem this implies that

for any symmetric matrix . Since is already known to vanish for non-symmetric matrices, upon decomposing a general matrix into the symmetric part and anti-symmetric part (with the latter having trace zero) we conclude that

for an arbitrary matrix . In particular

In the incompressible case , the second term vanishes, and this equation simply says that the shear stress is proportional to the (rate of) strain.

Inserting the above law back into (27) and then (15), (22), we obtain the Navier-Stokes equations

where is the spatial Laplacian. In the incompressible case, setting (this ratio is known as the kinematic viscosity), and also normalising for simplicity, this simplifies to the incompressible Navier-Stokes equations

Of course, the incompressible Euler equations arise as the special case when the viscosity is set to zero. For physical fluids, is positive, though it can be so small that the Euler equations serve as a good approximation. Negative values of are mathematically possible, but physically unrealistic for several reasons (for instance, the total energy of the fluid would increase over time, rather than dissipate over time) and also the equations become mathematically quite ill-behaved in this case (as they carry essentially all of the pathologies of the backwards heat equation).

Exercise 3 In a constant gravity field oriented in the direction , each particle will experience an external gravitational force , where is a fixed constant. Argue that the incompressible Euler equations in the presence of such a gravitational field should be modified to

and the incompressible Navier-Stokes equations should similarly be modified to

Exercise 4

• (i) Suppose that in an -particle system, the force between two particles is given by the conservative law

  

   

  for some smooth potential function , which we assume to obey the symmetry to be consistent with Newton’s third law. Show that the total energy

  

  is conserved in time (that is to say, the time derivative of this quantity vanishes). Here of course we use to denote the Euclidean magnitude of a vector .

• (ii) Assume furthermore that in the large limit, we have the velocity approximation (10), the stress approximation (21), and the isotropy condition (23). Assume all interactions are short-range. Assume we also have a potential energy approximation

  

   

  

  for all test functions and some smooth function (usually called the specific internal energy). By analysing the energy distribution

  

  

  in analogy with the previous analysis of the mass distribution and momentum distribution , make a heuristic derivation of the energy conservation law

  

  or equivalently

  

   

  where the energy density is defined by the formula

  

   

  (that is to say, is the sum of the kinetic energy density and the internal energy density ). Conclude in particular that the total energy

  

  is formally conserved in time.

• (iii) In addition to the previous assumptions, assume that there is a functional relationship between the specific energy density and the density , in the sense that there is a smooth function such that

  

   

  at all points in spacetime and all physically relevant configurations of particles. (In particular, we are in the compressible situation in which we allow the density to vary.) By dilating the position variables by a scaling factor , scaling the test function appropriately, and working out (via (9)) what the scaling does to the density function , generalise (30) to

  

  

  Formally differentiate this approximation with respect to at and use (29), (20), (21), (23) to heuristically derive the equation of state

  

   

• (iv) Give an alternate derivation of the energy conservation law (31) from the equations (24), (25), (33), (34). (Note that as the fluid here is compressible, one cannot use any equations in this post that rely on an incompressibility hypothesis.)
• (v) Suppose the functional relationship (33) uses a function of the form

  

  for some constant (the specific energy density at equilibrium) and some large and positive ; physically, this represents a fluid which is nearly incompressible in the sense that it is very resistant to having its density deviate from . Assuming that the pressure stays bounded and is large, heuristically derive the approximations

  

  and

  

  Formally, in the limit , we thus heuristically recover an incompressible fluid with a constant specific energy density . In particular the contributions of the specific energy to the energy conservation law (31) may be ignored in the incompressible case thanks to (24).

Remark 5 In the literature, the relationship between the functional relationship (33) and the equation of state (34) is usually derived instead using the laws of thermodynamics. However, as the above exercise demonstrates, it is also possible to recover this relationship from first principles. In the case of an (isoentropic) ideal gas, the laws of thermodynamics can be used to establish an equation of state of the form for some constants with , as well as the corresponding functional relationship , so that the internal energy density is . This is of course consistent with (33) and (34), after choosing appropriately.

Exercise 6 Suppose one has a (compressible or incompressible) fluid obeying the velocity approximation (10), the stress approximation (21), the isotropy condition (23), and the torque free condition (4). Assume also that all interactions are short range. Derive the angular momentum equation

for in two different ways:

• (a) From a heuristic analysis of the angular momentum distribution

  

  analogous to how the mass, momentum, and energy distributions were analysed previously; and

• (b) Directly from the system (24), (25).

(c) Conclude in particular that the total angular momentum

is formally conserved in time.

Remark 7 In this set of notes we made the rather strong assumption (10) that the velocities of particles could be approximated by a smooth function of their time and position. In practice, most fluids will violate this hypothesis due to thermal fluctuations in the velocity. However, one can still proceed with a similar derivation assuming that the velocities behave on the average like a smooth function , in the sense that

 

for any test function . The approximation (13) now has to be replaced by the more general

where

is the deviation of the particle velocity from its mean. The second term in (18) now needs to be replaced by the more complicated expression

 

From (35), (9) one has

for any test function . This allows us to heuristically drop the cross terms from (36) involving a single factor of , and simplify this expression (up to negligible errors) as

Repeating the analysis after (18), one eventually arrives again at (19), except that one has to add an additional term

to the stress tensor of a single particle . However, this term is still symmetric, and one can still continue most of the heuristic analysis in this post after suitable adjustments to the various physical hypotheses (for instance, assuming some form of the molecular chaos hypothesis to be able to neglect some correlation terms between and other quantities). We leave the details to the interested reader.

In the amplitudes field, we calculate probabilities for particles to interact.

We’re trying to improve on the old-school way of doing this, a kind of standard assembly line. First, you define your theory, writing down something called a Lagrangian. Then you start drawing Feynman diagrams, starting with the simplest “tree” diagrams and moving on to more complicated “loops”. Using rules derived from your Lagrangian, you translate these Feynman diagrams into a set of integrals. Do the integrals, and finally you have your answer.

Our field is a big tent, with many different approaches. Despite that, a kind of standard picture has emerged. It’s not the best we can do, and it’s certainly not what everyone is doing. But it’s in the back of our minds, a default to compare against and improve on. It’s the amplitudes assembly line: an “industrial” process that takes raw assumptions and builds particle physics probabilities.

1. Start with some simple assumptions about your particles (what mass do they have? what is their spin?) and your theory (minimally, it should obey special relativity). Using that, find the simplest “trees”, involving only three particles: one particle splitting into two, or two particles merging into one.
2. With the three-particle trees, you can now build up trees with any number of particles, using a technique called BCFW (named after its inventors, Ruth Britto, Freddy Cachazo, Bo Feng, and Edward Witten).
3. Now that you’ve got trees with any number of particles, it’s time to get loops! As it turns out, you can stitch together your trees into loops, using a technique called generalized unitarity. To do this, you have to know what kinds of integrals are allowed to show up in your result, and a fair amount of effort in the field goes into figuring out a better “basis” of integrals.
4. (Optional) Generalized unitarity will tell you which integrals you need to do, but those integrals may be related to each other. By understanding where these relations come from, you can reduce to a basis of fewer “master” integrals. You can also try to aim for integrals with particular special properties, quite a lot of effort goes in to improving this basis as well. The end goal is to make the final step as easy as possible:
5. Do the integrals! If you just want to get a number out, you can use numerical methods. Otherwise, there’s a wide variety of choices available. Methods that use differential equations are probably the most popular right now, but I’m a fan of other options.

Some people work to improve one step in this process, making it as efficient as possible. Others skip one step, or all of them, replacing them with deeper ideas. Either way, the amplitudes assembly line is the background: our current industrial machine, churning out predictions.

• The connection between particle spin and quantum statistics (fermions = half-integer spin, bosons = integer spin) is subtle, as I've mentioned before.  This week I happened upon a neat set of slides (pdf) by Jonathan Bain on this topic.  He looks at how we should think about why a pretty restrictive result from non-interacting relativistic quantum field theories has such profound, general implications.  He has a book on this, too.  
• There is a new book about the impact of condensed matter physics on the world and why it's the comparatively unsung branch of the discipline.   I have a copy on the way; once I read it I'll post a review.
• It's also worth reading about why mathematics as a discipline is viewed the way it is culturally.
• This is a really well-written article about turbulence, and why it's hard even though it's "just \(\mathbf{F} = m\mathbf{a}\)" for little blobs of fluid.
• Humanoid robots are getting more and more impressive.  I would really like to know the power consumption of one of those, though, given that the old ones used to have either big external power cables or on-board diesel engines.  The robot apocalypse is less scary if they have to recharge every ten minutes of operating time.
• I always wondered if fidget spinners were good for something.

It’s world mental health day. I’ve been meaning to do some more writing about social anxiety, and this seems like an appropriate day for that.

This isn’t easy to write about. A big part of social anxiety, to me, is that I’m afraid of how people will react to me. So talking about the things that are wrong with me is hard, and not exactly a lot of fun. But I try to do it, because I think it’s important. It’s useful for me to confront this; it’s important for other people with social anxiety to see and hear that they’re not alone; and it’s important to fight the general stigma against mental illness. I still struggle with my social anxiety – but I’m also happily married, with a great job and a successful career: I’m a walking demonstration of the fact that you can have mental illnesses like depression and social anxiety disorder, and still have a good, happy, full life.

In the past, I’ve tried to explain what it’s like to live with social anxiety. I’m going to try to expand on that a bit, and walk you through a particularly hard example of it that I’m trying to deal with right now.

What I’ve said before is that SA, for me, is a deeply seated belief that there’s something wrong with me, and whenever I’m socially interacting with people, I’m afraid that they’re going to realize what a freak I am.

That’s kind-of true, and it’s also kind-of not. This is difficult to put into words, because the actually feeling is almost a physical reaction, not a thought, so it’s not really linguistic. Yes, I am constantly on edge when I’m interacting socially. I am constantly afraid in social situations. The hard part to explain is that I don’t even know what I’m afraid of. There’s no specific bad outcome that I’m imagining. I can often relate the fear back to things that I’ve experienced in the past – but I don’t experience the fear and anxiety now as being fear/anxiety that those specific things, or things like them, will re-occur. I’m just afraid.

Here’s where I�