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In contrast to previous notes, in this set of notes we shall focus exclusively on Fourier analysis in the one-dimensional setting for simplicity of notation, although all of the results here have natural extensions to higher dimensions. Depending on the physical context, one can view the physical domain as representing either space or time; we will mostly think in terms of the former interpretation, even though the standard terminology of “time-frequency analysis”, which we will make more prominent use of in later notes, clearly originates from the latter.

In previous notes we have often performed various localisations in either physical space or Fourier space , for instance in order to take advantage of the uncertainty principle. One can formalise these operations in terms of the functional calculus of two basic operations on Schwartz functions , the *position operator* defined by

and the *momentum operator* , defined by

(The terminology comes from quantum mechanics, where it is customary to also insert a small constant on the right-hand side of (1) in accordance with de Broglie’s law. Such a normalisation is also used in several branches of mathematics, most notably semiclassical analysis and microlocal analysis, where it becomes profitable to consider the semiclassical limit , but we will not emphasise this perspective here.) The momentum operator can be viewed as the counterpart to the position operator, but in frequency space instead of physical space, since we have the standard identity

for any and . We observe that both operators are formally self-adjoint in the sense that

for all , where we use the Hermitian inner product

Clearly, for any polynomial of one real variable (with complex coefficients), the operator is given by the spatial multiplier operator

and similarly the operator is given by the Fourier multiplier operator

Inspired by this, if is any smooth function that obeys the derivative bounds

for all and (that is to say, all derivatives of grow at most polynomially), then we can define the spatial multiplier operator by the formula

one can easily verify from several applications of the Leibniz rule that maps Schwartz functions to Schwartz functions. We refer to as the *symbol* of this spatial multiplier operator. In a similar fashion, we define the Fourier multiplier operator associated to the symbol by the formula

For instance, any constant coefficient linear differential operators can be written in this notation as

however there are many Fourier multiplier operators that are not of this form, such as fractional derivative operators for non-integer values of , which is a Fourier multiplier operator with symbol . It is also very common to use spatial cutoffs and Fourier cutoffs for various bump functions to localise functions in either space or frequency; we have seen several examples of such cutoffs in action in previous notes (often in the higher dimensional setting ).

We observe that the maps and are ring homomorphisms, thus for instance

and

for any obeying the derivative bounds (2); also is formally adjoint to in the sense that

for , and similarly for and . One can interpret these facts as part of the functional calculus of the operators , which can be interpreted as densely defined self-adjoint operators on . However, in this set of notes we will not develop the spectral theory necessary in order to fully set out this functional calculus rigorously.

In the field of PDE and ODE, it is also very common to study *variable coefficient* linear differential operators

where the are now functions of the spatial variable obeying the derivative bounds (2). A simple example is the quantum harmonic oscillator Hamiltonian . One can rewrite this operator in our notation as

and so it is natural to interpret this operator as a combination of both the position operator and the momentum operator , where the *symbol* this operator is the function

Indeed, from the Fourier inversion formula

for any we have

and hence on multiplying by and summing we have

Inspired by this, we can introduce the *Kohn-Nirenberg quantisation* by defining the operator by the formula

whenever and is any smooth function obeying the derivative bounds

for all and (note carefully that the exponent in on the right-hand side is required to be uniform in ). This quantisation clearly generalises both the spatial multiplier operators and the Fourier multiplier operators defined earlier, which correspond to the cases when the symbol is a function of only or only respectively. Thus we have combined the physical space and the frequency space into a single domain, known as phase space . The term “time-frequency analysis” encompasses analysis based on decompositions and other manipulations of phase space, in much the same way that “Fourier analysis” encompasses analysis based on decompositions and other manipulations of frequency space. We remark that the Kohn-Nirenberg quantization is not the only choice of quantization one could use; see Remark 19 below.

In principle, the quantisations are potentially very useful for such tasks as inverting variable coefficient linear operators, or to localize a function simultaneously in physical and Fourier space. However, a fundamental difficulty arises: map from symbols to operators is now no longer a ring homomorphism, in particular

in general. Fundamentally, this is due to the fact that pointwise multiplication of symbols is a commutative operation, whereas the composition of operators such as and does not necessarily commute. This lack of commutativity can be measured by introducing the *commutator*

of two operators , and noting from the product rule that

(In the language of Lie groups and Lie algebras, this tells us that are (up to complex constants) the standard Lie algebra generators of the Heisenberg group.) From a quantum mechanical perspective, this lack of commutativity is the root cause of the uncertainty principle that prevents one from simultaneously localizing in both position and momentum past a certain point. Here is one basic way of formalising this principle:

Exercise 2 (Heisenberg uncertainty principle)For any and , show that(

Hint:evaluate the expression in two different ways and apply the Cauchy-Schwarz inequality.) Informally, this exercise asserts that the spatial uncertainty and the frequency uncertainty of a function obey the Heisenberg uncertainty relation .

Nevertheless, one still has the correspondence principle, which asserts that in certain regimes (which, with our choice of normalisations, corresponds to the high-frequency regime), quantum mechanics continues to behave like a commutative theory, and one can sometimes proceed as if the operators (and the various operators constructed from them) commute up to “lower order” errors. This can be formalised using the *pseudodifferential calculus*, which we give below the fold, in which we restrict the symbol to certain “symbol classes” of various orders (which then restricts to be pseudodifferential operators of various orders), and obtains approximate identities such as

where the error between the left and right-hand sides is of “lower order” and can in fact enjoys a useful asymptotic expansion. As a first approximation to this calculus, one can think of functions as having some sort of “phase space portrait” which somehow combines the physical space representation with its Fourier representation , and pseudodifferential operators behave approximately like “phase space multiplier operators” in this representation in the sense that

Unfortunately the uncertainty principle (or the non-commutativity of and ) prevents us from making these approximations perfectly precise, and it is not always clear how to even define a phase space portrait of a function precisely (although there are certain popular candidates for such a portrait, such as the FBI transform (also known as the Gabor transform in signal processing literature), or the Wigner quasiprobability distribution, each of which have some advantages and disadvantages). Nevertheless even if the concept of a phase space portrait is somewhat fuzzy, it is of great conceptual benefit both within mathematics and outside of it. For instance, the musical score one assigns a piece of music can be viewed as a phase space portrait of the sound waves generated by that music.

To complement the pseudodifferential calculus we have the basic *Calderón-Vaillancourt theorem*, which asserts that pseudodifferential operators of order zero are Calderón-Zygmund operators and thus bounded on for . The standard proof of this theorem is a classic application of one of the basic techniques in harmonic analysis, namely the exploitation of *almost orthogonality*; the proof we will give here will achieve this through the elegant device of the Cotlar-Stein lemma.

Pseudodifferential operators (especially when generalised to higher dimensions ) are a fundamental tool in the theory of linear PDE, as well as related fields such as semiclassical analysis, microlocal analysis, and geometric quantisation. There is an even wider class of operators that is also of interest, namely the Fourier integral operators, which roughly speaking not only approximately multiply the phase space portrait of a function by some multiplier , but also move the portrait around by a canonical transformation. However, the development of theory of these operators is beyond the scope of these notes; see for instance the texts of Hormander or Eskin.

This set of notes is only the briefest introduction to the theory of pseudodifferential operators. Many texts are available that cover the theory in more detail, for instance this text of Taylor.

Just a brief post to record some notable papers in my fields of interest that appeared on the arXiv recently.

- “A sharp square function estimate for the cone in “, by Larry Guth, Hong Wang, and Ruixiang Zhang. This paper establishes an optimal (up to epsilon losses) square function estimate for the three-dimensional light cone that was essentially conjectured by Mockenhaupt, Seeger, and Sogge, which has a number of other consequences including Sogge’s local smoothing conjecture for the wave equation in two spatial dimensions, which in turn implies the (already known) Bochner-Riesz, restriction, and Kakeya conjectures in two dimensions. Interestingly, modern techniques such as polynomial partitioning and decoupling estimates are not used in this argument; instead, the authors mostly rely on an induction on scales argument and Kakeya type estimates. Many previous authors (including myself) were able to get weaker estimates of this type by an induction on scales method, but there were always significant inefficiencies in doing so; in particular knowing the sharp square function estimate at smaller scales did not imply the sharp square function estimate at the given larger scale. The authors here get around this issue by finding an even stronger estimate that implies the square function estimate, but behaves significantly better with respect to induction on scales.
- “On the Chowla and twin primes conjectures over “, by Will Sawin and Mark Shusterman. This paper resolves a number of well known open conjectures in analytic number theory, such as the Chowla conjecture and the twin prime conjecture (in the strong form conjectured by Hardy and Littlewood), in the case of function fields where the field is a prime power which is fixed (in contrast to a number of existing results in the “large ” limit) but has a large exponent . The techniques here are orthogonal to those used in recent progress towards the Chowla conjecture over the integers (e.g., in this previous paper of mine); the starting point is an algebraic observation that in certain function fields, the Mobius function behaves like a quadratic Dirichlet character along certain arithmetic progressions. In principle, this reduces problems such as Chowla’s conjecture to problems about estimating sums of Dirichlet characters, for which more is known; but the task is still far from trivial.
- “Bounds for sets with no polynomial progressions“, by Sarah Peluse. This paper can be viewed as part of a larger project to obtain quantitative density Ramsey theorems of Szemeredi type. For instance, Gowers famously established a relatively good quantitative bound for Szemeredi’s theorem that all dense subsets of integers contain arbitrarily long arithmetic progressions . The corresponding question for polynomial progressions is considered more difficult for a number of reasons. One of them is that dilation invariance is lost; a dilation of an arithmetic progression is again an arithmetic progression, but a dilation of a polynomial progression will in general not be a polynomial progression with the same polynomials . Another issue is that the ranges of the two parameters are now at different scales. Peluse gets around these difficulties in the case when all the polynomials have distinct degrees, which is in some sense the opposite case to that considered by Gowers (in particular, she avoids the need to obtain quantitative inverse theorems for high order Gowers norms; which was recently obtained in this integer setting by Manners but with bounds that are probably not strong enough to for the bounds in Peluse’s results, due to a degree lowering argument that is available in this case). To resolve the first difficulty one has to make all the estimates rather uniform in the coefficients of the polynomials , so that one can still run a density increment argument efficiently. To resolve the second difficulty one needs to find a quantitative concatenation theorem for Gowers uniformity norms. Many of these ideas were developed in previous papers of Peluse and Peluse-Prendiville in simpler settings.
- “On blow up for the energy super critical defocusing non linear Schrödinger equations“, by Frank Merle, Pierre Raphael, Igor Rodnianski, and Jeremie Szeftel. This paper (when combined with two companion papers) resolves a long-standing problem as to whether finite time blowup occurs for the defocusing supercritical nonlinear Schrödinger equation (at least in certain dimensions and nonlinearities). I had a previous paper establishing a result like this if one “cheated” by replacing the nonlinear Schrodinger equation by a system of such equations, but remarkably they are able to tackle the original equation itself without any such cheating. Given the very analogous situation with Navier-Stokes, where again one can create finite time blowup by “cheating” and modifying the equation, it does raise hope that finite time blowup for the incompressible Navier-Stokes and Euler equations can be established… In fact the connection may not just be at the level of analogy; a surprising key ingredient in the proofs here is the observation that a certain blowup ansatz for the nonlinear Schrodinger equation is governed by solutions to the (compressible) Euler equation, and finite time blowup examples for the latter can be used to construct finite time blowup examples for the former.

Let be a divergence-free vector field, thus , which we interpret as a velocity field. In this post we will proceed formally, largely ignoring the analytic issues of whether the fields in question have sufficient regularity and decay to justify the calculations. The vorticity field is then defined as the curl of the velocity:

(From a differential geometry viewpoint, it would be more accurate (especially in other dimensions than three) to define the vorticity as the exterior derivative of the musical isomorphism of the Euclidean metric applied to the velocity field ; see these previous lecture notes. However, we will not need this geometric formalism in this post.)

Assuming suitable regularity and decay hypotheses of the velocity field , it is possible to recover the velocity from the vorticity as follows. From the general vector identity applied to the velocity field , we see that

and thus (by the commutativity of all the differential operators involved)

Using the Newton potential formula

and formally differentiating under the integral sign, we obtain the Biot-Savart law

This law is of fundamental importance in the study of incompressible fluid equations, such as the Euler equations

since on applying the curl operator one obtains the vorticity equation

and then by substituting (1) one gets an autonomous equation for the vorticity field . Unfortunately, this equation is non-local, due to the integration present in (1).

In a recent work, it was observed by Elgindi that in a certain regime, the Biot-Savart law can be approximated by a more “low rank” law, which makes the non-local effects significantly simpler in nature. This simplification was carried out in spherical coordinates, and hinged on a study of the invertibility properties of a certain second order linear differential operator in the latitude variable ; however in this post I would like to observe that the approximation can also be seen directly in Cartesian coordinates from the classical Biot-Savart law (1). As a consequence one can also initiate the beginning of Elgindi’s analysis in constructing somewhat regular solutions to the Euler equations that exhibit self-similar blowup in finite time, though I have not attempted to execute the entirety of the analysis in this setting.

Elgindi’s approximation applies under the following hypotheses:

- (i) (Axial symmetry without swirl) The velocity field is assumed to take the form
for some functions of the cylindrical radial variable and the vertical coordinate . As a consequence, the vorticity field takes the form

- (ii) (Odd symmetry) We assume that and , so that .

A model example of a divergence-free vector field obeying these properties (but without good decay at infinity) is the linear vector field

which is of the form (3) with and . The associated vorticity vanishes.

We can now give an illustration of Elgindi’s approximation:

Proposition 1 (Elgindi’s approximation)Under the above hypotheses (and assuing suitable regularity and decay), we have the pointwise boundsfor any , where is the vector field (5), and is the scalar function

Thus under the hypotheses (i), (ii), and assuming that is slowly varying, we expect to behave like the linear vector field modulated by a radial scalar function. In applications one needs to control the error in various function spaces instead of pointwise, and with similarly controlled in other function space norms than the norm, but this proposition already gives a flavour of the approximation. If one uses spherical coordinates

then we have (using the spherical change of variables formula and the odd nature of )

where

is the operator introduced in Elgindi’s paper.

*Proof:* By a limiting argument we may assume that is non-zero, and we may normalise . From the triangle inequality we have

and hence by (1)

In the regime we may perform the Taylor expansion

Since

we see from the triangle inequality that the error term contributes to . We thus have

where is the constant term

and are the linear term

By the hypotheses (i), (ii), we have the symmetries

The even symmetry (8) ensures that the integrand in is odd, so vanishes. The symmetry (6) or (7) similarly ensures that , so vanishes. Since , we conclude that

Using (4), the right-hand side is

where . Because of the odd nature of , only those terms with one factor of give a non-vanishing contribution to the integral. Using the rotation symmetry we also see that any term with a factor of also vanishes. We can thus simplify the above expression as

Using the rotation symmetry again, we see that the term in the first component can be replaced by or by , and similarly for the term in the second component. Thus the above expression is

giving the claim.

Example 2Consider the divergence-free vector field , where the vector potential takes the formfor some bump function supported in . We can then calculate

and

In particular the hypotheses (i), (ii) are satisfied with

One can then calculate

If we take the specific choice

where is a fixed bump function supported some interval and is a small parameter (so that is spread out over the range ), then we see that

(with implied constants allowed to depend on ),

and

which is completely consistent with Proposition 1.

One can use this approximation to extract a plausible ansatz for a self-similar blowup to the Euler equations. We let be a small parameter and let be a time-dependent vorticity field obeying (i), (ii) of the form

where and is a smooth field to be chosen later. Admittedly the signum function is not smooth at , but let us ignore this issue for now (to rigorously make an ansatz one will have to smooth out this function a little bit; Elgindi uses the choice , where ). With this ansatz one may compute

By Proposition 1, we thus expect to have the approximation

We insert this into the vorticity equation (2). The transport term will be expected to be negligible because , and hence , is slowly varying (the discontinuity of will not be encountered because the vector field is parallel to this singularity). The modulating function is similarly slowly varying, so derivatives falling on this function should be lower order. Neglecting such terms, we arrive at the approximation

and so in the limit we expect obtain a simple model equation for the evolution of the vorticity envelope :

If we write for the logarithmic primitive of , then we have and hence

which integrates to the Ricatti equation

which can be explicitly solved as

where is any function of that one pleases. (In Elgindi’s work a time dilation is used to remove the unsightly factor of appearing here in the denominator.) If for instance we set , we obtain the self-similar solution

and then on applying

Thus, we expect to be able to construct a self-similar blowup to the Euler equations with a vorticity field approximately behaving like

and velocity field behaving like

In particular, would be expected to be of regularity (and smooth away from the origin), and blows up in (say) norm at time , and one has the self-similarity

and

A self-similar solution of this approximate shape is in fact constructed rigorously in Elgindi’s paper (using spherical coordinates instead of the Cartesian approach adopted here), using a nonlinear stability analysis of the above ansatz. It seems plausible that one could also carry out this stability analysis using this Cartesian coordinate approach, although I have not tried to do this in detail.

I’ve just uploaded to the arXiv my paper “Quantitative bounds for critically bounded solutions to the Navier-Stokes equations“, submitted to the proceedings of the Linde Hall Inaugural Math Symposium. (I unfortunately had to cancel my physical attendance at this symposium for personal reasons, but was still able to contribute to the proceedings.) In recent years I have been interested in working towards establishing the existence of classical solutions for the Navier-Stokes equations

that blow up in finite time, but this time for a change I took a look at the other side of the theory, namely the conditional regularity results for this equation. There are several such results that assert that if a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular; taken in the contrapositive, they assert that if a solution blows up at a certain finite time , then certain norms of the solution must also go to infinity. Here are some examples (not an exhaustive list) of such blowup criteria:

- (Leray blowup criterion, 1934) If blows up at a finite time , and , then for an absolute constant .
- (Prodi–Serrin–Ladyzhenskaya blowup criterion, 1959-1967) If blows up at a finite time , and , then , where .
- (Beale-Kato-Majda blowup criterion, 1984) If blows up at a finite time , then , where is the vorticity.
- (Kato blowup criterion, 1984) If blows up at a finite time , then for some absolute constant .
- (Escauriaza-Seregin-Sverak blowup criterion, 2003) If blows up at a finite time , then .
- (Seregin blowup criterion, 2012) If blows up at a finite time , then .
- (Phuc blowup criterion, 2015) If blows up at a finite time , then for any .
- (Gallagher-Koch-Planchon blowup criterion, 2016) If blows up at a finite time , then for any .
- (Albritton blowup criterion, 2016) If blows up at a finite time , then for any .

My current paper is most closely related to the Escauriaza-Seregin-Sverak blowup criterion, which was the first to show a critical (i.e., scale-invariant, or dimensionless) spatial norm, namely , had to become large. This result now has many proofs; for instance, many of the subsequent blowup criterion results imply the Escauriaza-Seregin-Sverak one as a special case, and there are also additional proofs by Gallagher-Koch-Planchon (building on ideas of Kenig-Koch), and by Dong-Du. However, all of these proofs rely on some form of a compactness argument: given a finite time blowup, one extracts some suitable family of rescaled solutions that converges in some weak sense to a limiting solution that has some additional good properties (such as almost periodicity modulo symmetries), which one can then rule out using additional qualitative tools, such as unique continuation and backwards uniqueness theorems for parabolic heat equations. In particular, all known proofs use some version of the backwards uniqueness theorem of Escauriaza, Seregin, and Sverak. Because of this reliance on compactness, the existing proofs of the Escauriaza-Seregin-Sverak blowup criterion are qualitative, in that they do not provide any quantitative information on how fast the norm will go to infinity (along a subsequence of times).

On the other hand, it is a general principle that qualitative arguments established using compactness methods ought to have quantitative analogues that replace the use of compactness by more complicated substitutes that give effective bounds; see for instance these previous blog posts for more discussion. I therefore was interested in trying to obtain a quantitative version of this blowup criterion that gave reasonably good effective bounds (in particular, my objective was to avoid truly enormous bounds such as tower-exponential or Ackermann function bounds, which often arise if one “naively” tries to make a compactness argument effective). In particular, I obtained the following triple-exponential quantitative regularity bounds:

Theorem 1If is a classical solution to Navier-Stokes on with

and

for and .

As a corollary, one can now improve the Escauriaza-Seregin-Sverak blowup criterion to

for some absolute constant , which to my knowledge is the first (*very* slightly) supercritical blowup criterion for Navier-Stokes in the literature.

The proof uses many of the same quantitative inputs as previous arguments, most notably the Carleman inequalities used to establish unique continuation and backwards uniqueness theorems for backwards heat equations, but also some additional techniques that make the quantitative bounds more efficient. The proof focuses initially on points of concentration of the solution, which we define as points where there is a frequency for which one has the bound

for a large absolute constant , where is a Littlewood-Paley projection to frequencies . (This can be compared with the upper bound of for the quantity on the left-hand side that follows from (1).) The factor of normalises the left-hand side of (2) to be dimensionless (i.e., critical). The main task is to show that the dimensionless quantity cannot get too large; in particular, we end up establishing a bound of the form

from which the above theorem ends up following from a routine adaptation of the local well-posedness and regularity theory for Navier-Stokes.

The strategy is to show that any concentration such as (2) when is large must force a significant component of the norm of to also show up at many other locations than , which eventually contradicts (1) if one can produce enough such regions of non-trivial norm. (This can be viewed as a quantitative variant of the “rigidity” theorems in some of the previous proofs of the Escauriaza-Seregin-Sverak theorem that rule out solutions that exhibit too much “compactness” or “almost periodicity” in the topology.) The chain of causality that leads from a concentration (2) at to significant norm at other regions of the time slice is somewhat involved (though simpler than the much more convoluted schemes I initially envisaged for this argument):

- Firstly, by using Duhamel’s formula, one can show that a concentration (2) can only occur (with large) if there was also a preceding concentration
at some slightly previous point in spacetime, with also close to (more precisely, we have , , and ). This can be viewed as a sort of contrapositive of a “local regularity theorem”, such as the ones established by Caffarelli, Kohn, and Nirenberg. A key point here is that the lower bound in the conclusion (3) is precisely the same as the lower bound in (2), so that this backwards propagation of concentration can be iterated.

- Iterating the previous step, one can find a sequence of concentration points
with the propagating backwards in time; by using estimates ultimately resulting from the dissipative term in the energy identity, one can extract such a sequence in which the increase geometrically with time, the are comparable (up to polynomial factors in ) to the natural frequency scale , and one has . Using the “epochs of regularity” theory that ultimately dates back to Leray, and tweaking the slightly, one can also place the times in intervals (of length comparable to a small multiple of ) in which the solution is quite regular (in particular, enjoy good bounds on ).

- The concentration (4) can be used to establish a lower bound for the norm of the vorticity near . As is well known, the vorticity obeys the vorticity equation
In the epoch of regularity , the coefficients of this equation obey good bounds, allowing the machinery of Carleman estimates to come into play. Using a Carleman estimate that is used to establish unique continuation results for backwards heat equations, one can propagate this lower bound to also give lower bounds on the vorticity (and its first derivative) in annuli of the form for various radii , although the lower bounds decay at a gaussian rate with .

- Meanwhile, using an energy pigeonholing argument of Bourgain (which, in this Navier-Stokes context, is actually an enstrophy pigeonholing argument), one can locate some annuli where (a slightly normalised form of) the entrosphy is small at time ; using a version of the localised enstrophy estimates from a previous paper of mine, one can then propagate this sort of control forward in time, obtaining an “annulus of regularity” of the form in which one has good estimates; in particular, one has type bounds on in this cylindrical annulus.
- By intersecting the previous epoch of regularity with the above annulus of regularity, we have some lower bounds on the norm of the vorticity (and its first derivative) in the annulus of regularity. Using a Carleman estimate first introduced by Escauriaza, Seregin, and Sverak, as well as a second application of the Carleman estimate used previously, one can then propagate this lower bound back up to time , establishing a lower bound for the vorticity on the spatial annulus . By some basic Littlewood-Paley theory one can parlay this lower bound to a lower bound on the norm of the velocity ; crucially, this lower bound is uniform in .
- If is very large (triple exponential in !), one can then find enough scales with disjoint annuli that the total lower bound on the norm of provided by the above arguments is inconsistent with (1), thus establishing the claim.

The chain of causality is summarised in the following image:

It seems natural to conjecture that similar triply logarithmic improvements can be made to several of the other blowup criteria listed above, but I have not attempted to pursue this question. It seems difficult to improve the triple logarithmic factor using only the techniques here; the Bourgain pigeonholing argument inevitably costs one exponential, the Carleman inequalities cost a second, and the stacking of scales at the end to contradict the upper bound costs the third.

Let be some domain (such as the real numbers). For any natural number , let denote the space of symmetric real-valued functions on variables , thus

for any permutation . For instance, for any natural numbers , the elementary symmetric polynomials

will be an element of . With the pointwise product operation, becomes a commutative real algebra. We include the case , in which case consists solely of the real constants.

Given two natural numbers , one can “lift” a symmetric function of variables to a symmetric function of variables by the formula

where ranges over all injections from to (the latter formula making it clearer that is symmetric). Thus for instance

and

Also we have

With these conventions, we see that vanishes for , and is equal to if . We also have the transitivity

if .

The lifting map is a linear map from to , but it is not a ring homomorphism. For instance, when , one has

In general, one has the identity

for all natural numbers and , , where range over all injections , with . Combinatorially, the identity (2) follows from the fact that given any injections and with total image of cardinality , one has , and furthermore there exist precisely triples of injections , , such that and .

Example 1When , one haswhich is just a restatement of the identity

Note that the coefficients appearing in (2) do not depend on the final number of variables . We may therefore abstract the role of from the law (2) by introducing the real algebra of formal sums

where for each , is an element of (with only finitely many of the being non-zero), and with the formal symbol being formally linear, thus

and

for and scalars , and with multiplication given by the analogue

of (2). Thus for instance, in this algebra we have

and

Informally, is an abstraction (or “inverse limit”) of the concept of a symmetric function of an unspecified number of variables, which are formed by summing terms that each involve only a bounded number of these variables at a time. One can check (somewhat tediously) that is indeed a commutative real algebra, with a unit . (I do not know if this algebra has previously been studied in the literature; it is somewhat analogous to the abstract algebra of finite linear combinations of Schur polynomials, with multiplication given by a Littlewood-Richardson rule. )

For natural numbers , there is an obvious specialisation map from to , defined by the formula

Thus, for instance, maps to and to . From (2) and (3) we see that this map is an algebra homomorphism, even though the maps and are not homomorphisms. By inspecting the component of we see that the homomorphism is in fact surjective.

Now suppose that we have a measure on the space , which then induces a product measure on every product space . To avoid degeneracies we will assume that the integral is strictly positive. Assuming suitable measurability and integrability hypotheses, a function can then be integrated against this product measure to produce a number

In the event that arises as a lift of another function , then from Fubini’s theorem we obtain the formula

is an element of the formal algebra , then

Note that by hypothesis, only finitely many terms on the right-hand side are non-zero.

Now for a key observation: whereas the left-hand side of (6) only makes sense when is a natural number, the right-hand side is meaningful when takes a fractional value (or even when it takes negative or complex values!), interpreting the binomial coefficient as a polynomial in . As such, this suggests a way to introduce a “virtual” concept of a symmetric function on a fractional power space for such values of , and even to integrate such functions against product measures , even if the fractional power does not exist in the usual set-theoretic sense (and similarly does not exist in the usual measure-theoretic sense). More precisely, for arbitrary real or complex , we now *define* to be the space of abstract objects

with and (and now interpreted as formal symbols, with the structure of a commutative real algebra inherited from , thus

In particular, the multiplication law (2) continues to hold for such values of , thanks to (3). Given any measure on , we formally define a measure on with regards to which we can integrate elements of by the formula (6) (providing one has sufficient measurability and integrability to make sense of this formula), thus providing a sort of “fractional dimensional integral” for symmetric functions. Thus, for instance, with this formalism the identities (4), (5) now hold for fractional values of , even though the formal space no longer makes sense as a set, and the formal measure no longer makes sense as a measure. (The formalism here is somewhat reminiscent of the technique of dimensional regularisation employed in the physical literature in order to assign values to otherwise divergent integrals. See also this post for an unrelated abstraction of the integration concept involving integration over supercommutative variables (and in particular over fermionic variables).)

Example 2Suppose is a probability measure on , and is a random variable; on any power , we let be the usual independent copies of on , thus for . Then for any real or complex , the formal integralcan be evaluated by first using the identity

(cf. (1)) and then using (6) and the probability measure hypothesis to conclude that

For a natural number, this identity has the probabilistic interpretation

whenever are jointly independent copies of , which reflects the well known fact that the sum has expectation and variance . One can thus view (7) as an abstract generalisation of (8) to the case when is fractional, negative, or even complex, despite the fact that there is no sensible way in this case to talk about independent copies of in the standard framework of probability theory.

In this particular case, the quantity (7) is non-negative for every nonnegative , which looks plausible given the form of the left-hand side. Unfortunately, this sort of non-negativity does not always hold; for instance, if has mean zero, one can check that

and the right-hand side can become negative for . This is a shame, because otherwise one could hope to start endowing with some sort of commutative von Neumann algebra type structure (or the abstract probability structure discussed in this previous post) and then interpret it as a genuine measure space rather than as a virtual one. (This failure of positivity is related to the fact that the characteristic function of a random variable, when raised to the power, need not be a characteristic function of any random variable once is no longer a natural number: “fractional convolution” does not preserve positivity!) However, one vestige of positivity remains: if is non-negative, then so is

One can wonder what the point is to all of this abstract formalism and how it relates to the rest of mathematics. For me, this formalism originated implicitly in an old paper I wrote with Jon Bennett and Tony Carbery on the multilinear restriction and Kakeya conjectures, though we did not have a good language for working with it at the time, instead working first with the case of natural number exponents and appealing to a general extrapolation theorem to then obtain various identities in the fractional case. The connection between these fractional dimensional integrals and more traditional integrals ultimately arises from the simple identity

(where the right-hand side should be viewed as the fractional dimensional integral of the unit against ). As such, one can manipulate powers of ordinary integrals using the machinery of fractional dimensional integrals. A key lemma in this regard is

Lemma 3 (Differentiation formula)Suppose that a positive measure on depends on some parameter and varies by the formula

for some function . Let be any real or complex number. Then, assuming sufficient smoothness and integrability of all quantities involved, we have

for all that are independent of . If we allow to now depend on also, then we have the more general total derivative formula

again assuming sufficient amounts of smoothness and regularity.

*Proof:* We just prove (10), as (11) then follows by same argument used to prove the usual product rule. By linearity it suffices to verify this identity in the case for some symmetric function for a natural number . By (6), the left-hand side of (10) is then

Differentiating under the integral sign using (9) we have

and similarly

where are the standard copies of on :

By the product rule, we can thus expand (12) as

where we have suppressed the dependence on for brevity. Since , we can write this expression using (6) as

where is the symmetric function

But from (2) one has

and the claim follows.

Remark 4It is also instructive to prove this lemma in the special case when is a natural number, in which case the fractional dimensional integral can be interpreted as a classical integral. In this case, the identity (10) is immediate from applying the product rule to (9) to conclude thatOne could in fact derive (10) for arbitrary real or complex from the case when is a natural number by an extrapolation argument; see the appendix of my paper with Bennett and Carbery for details.

Let us give a simple PDE application of this lemma as illustration:

Proposition 5 (Heat flow monotonicity)Let be a solution to the heat equation with initial data a rapidly decreasing finite non-negative Radon measure, or more explicitlyfor al . Then for any , the quantity

is monotone non-decreasing in for , constant for , and monotone non-increasing for .

*Proof:* By a limiting argument we may assume that is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below.

For any , let denote the Radon measure

Then the quantity can be written as a fractional dimensional integral

Observe that

and thus by Lemma 3 and the product rule

where we use for the variable of integration in the factor space of .

To simplify this expression we will take advantage of integration by parts in the variable. Specifically, in any direction , we have

and hence by Lemma 3

Multiplying by and integrating by parts, we see that

where we use the Einstein summation convention in . Similarly, if is any reasonable function depending only on , we have

and hence on integration by parts

We conclude that

and thus by (13)

The choice of that then achieves the most cancellation turns out to be (this cancels the terms that are linear or quadratic in the ), so that . Repeating the calculations establishing (7), one has

and

where is the random variable drawn from with the normalised probability measure . Since , one thus has

This expression is clearly non-negative for , equal to zero for , and positive for , giving the claim. (One could simplify here as if desired, though it is not strictly necessary to do so for the proof.)

Remark 6As with Remark 4, one can also establish the identity (14) first for natural numbers by direct computation avoiding the theory of fractional dimensional integrals, and then extrapolate to the case of more general values of . This particular identity is also simple enough that it can be directly established by integration by parts without much difficulty, even for fractional values of .

A more complicated version of this argument establishes the non-endpoint multilinear Kakeya inequality (without any logarithmic loss in a scale parameter ); this was established in my previous paper with Jon Bennett and Tony Carbery, but using the “natural number first” approach rather than using the current formalism of fractional dimensional integration. However, the arguments can be translated into this formalism without much difficulty; we do so below the fold. (To simplify the exposition slightly we will not address issues of establishing enough regularity and integrability to justify all the manipulations, though in practice this can be done by standard limiting arguments.)

I was recently asked to contribute a short comment to Nature Reviews Physics, as part of a series of articles on fluid dynamics on the occasion of the 200th anniversary (this August) of the birthday of George Stokes. My contribution is now online as “Searching for singularities in the Navier–Stokes equations“, where I discuss the global regularity problem for Navier-Stokes and my thoughts on how one could try to construct a solution that blows up in finite time via an approximately discretely self-similar “fluid computer”. (The rest of the series does not currently seem to be available online, but I expect they will become so shortly.)

I was pleased to learn this week that the 2019 Abel Prize was awarded to Karen Uhlenbeck. Uhlenbeck laid much of the foundations of modern geometric PDE. One of the few papers I have in this area is in fact a joint paper with Gang Tian extending a famous singularity removal theorem of Uhlenbeck for four-dimensional Yang-Mills connections to higher dimensions. In both these papers, it is crucial to be able to construct “Coulomb gauges” for various connections, and there is a clever trick of Uhlenbeck for doing so, introduced in another important paper of hers, which is absolutely critical in my own paper with Tian. Nowadays it would be considered a standard technique, but it was definitely not so at the time that Uhlenbeck introduced it.

Suppose one has a smooth connection on a (closed) unit ball in for some , taking values in some Lie algebra associated to a compact Lie group . This connection then has a curvature , defined in coordinates by the usual formula

It is natural to place the curvature in a scale-invariant space such as , and then the natural space for the connection would be the Sobolev space . It is easy to see from (1) and Sobolev embedding that if is bounded in , then will be bounded in . One can then ask the converse question: if is bounded in , is bounded in ? This can be viewed as asking whether the curvature equation (1) enjoys “elliptic regularity”.

There is a basic obstruction provided by gauge invariance. For any smooth gauge taking values in the Lie group, one can gauge transform to

and then a brief calculation shows that the curvature is conjugated to

This gauge symmetry does not affect the norm of the curvature tensor , but can make the connection extremely large in , since there is no control on how wildly can oscillate in space.

However, one can hope to overcome this problem by *gauge fixing*: perhaps if is bounded in , then one can make bounded in *after* applying a gauge transformation. The basic and useful result of Uhlenbeck is that this can be done if the norm of is sufficiently small (and then the conclusion is that is small in ). (For large connections there is a serious issue related to the Gribov ambiguity.) In my (much) later paper with Tian, we adapted this argument, replacing Lebesgue spaces by Morrey space counterparts. (This result was also independently obtained at about the same time by Meyer and Riviére.)

To make the problem elliptic, one can try to impose the *Coulomb gauge condition*

(also known as the *Lorenz gauge* or *Hodge gauge* in various papers), together with a natural boundary condition on that will not be discussed further here. This turns (1), (2) into a divergence-curl system that is elliptic at the linear level at least. Indeed if one takes the divergence of (1) using (2) one sees that

and if one could somehow ignore the nonlinear term then we would get the required regularity on by standard elliptic regularity estimates.

The problem is then how to handle the nonlinear term. If we already knew that was small in the right norm then one can use Sobolev embedding, Hölder’s inequality, and elliptic regularity to show that the second term in (3) is small compared to the first term, and so one could then hope to eliminate it by perturbative analysis. However, proving that is small in this norm is exactly what we are trying to prove! So this approach seems circular.

Uhlenbeck’s clever way out of this circularity is a textbook example of what is now known as a “continuity” argument. Instead of trying to work just with the original connection , one works with the rescaled connections for , with associated rescaled curvatures . If the original curvature is small in norm (e.g. bounded by some small ), then so are all the rescaled curvatures . We want to obtain a Coulomb gauge at time ; this is difficult to do directly, but it is trivial to obtain a Coulomb gauge at time , because the connection vanishes at this time. On the other hand, once one has successfully obtained a Coulomb gauge at some time with small in the natural norm (say bounded by for some constant which is large in absolute terms, but not so large compared with say ), the perturbative argument mentioned earlier (combined with the qualitative hypothesis that is smooth) actually works to show that a Coulomb gauge can also be constructed and be small for all sufficiently close *nearby* times to ; furthermore, the perturbative analysis actually shows that the nearby gauges enjoy a slightly better bound on the norm, say rather than . As a consequence of this, the set of times for which one has a good Coulomb gauge obeying the claimed estimates is both open and closed in , and also contains . Since the unit interval is connected, it must then also contain . This concludes the proof.

One of the lessons I drew from this example is to not be deterred (especially in PDE) by an argument seeming to be circular; if the argument is still sufficiently “nontrivial” in nature, it can often be modified into a usefully non-circular argument that achieves what one wants (possibly under an additional qualitative hypothesis, such as a continuity or smoothness hypothesis).

I have just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows“, submitted to Pure and Applied Functional Analysis. This paper continues my attempts to establish “universality” properties of the Euler equations on Riemannian manifolds , as I conjecture that the freedom to set the metric ought to allow one to “program” such Euler flows to exhibit a wide range of behaviour, and in particular to achieve finite time blowup (if the dimension is sufficiently large, at least).

In coordinates, the Euler equations read

where is the pressure field and is the velocity field, and denotes the Levi-Civita connection with the usual Penrose abstract index notation conventions; we restrict attention here to the case where are smooth and is compact, smooth, orientable, connected, and without boundary. Let’s call an *Euler flow* on (for the time interval ) if it solves the above system of equations for some pressure , and an *incompressible flow* if it just obeys the divergence-free relation . Thus every Euler flow is an incompressible flow, but the converse is certainly not true; for instance the various conservation laws of the Euler equation, such as conservation of energy, will already block most incompressible flows from being an Euler flow, or even being approximated in a reasonably strong topology by such Euler flows.

However, one can ask if an incompressible flow can be *extended* to an Euler flow by adding some additional dimensions to . In my paper, I formalise this by considering warped products of which (as a smooth manifold) are products of with a torus, with a metric given by

for , where are the coordinates of the torus , and are smooth positive coefficients for ; in order to preserve the incompressibility condition, we also require the volume preservation property

though in practice we can quickly dispose of this condition by adding one further “dummy” dimension to the torus . We say that an incompressible flow is *extendible to an Euler flow* if there exists a warped product extending , and an Euler flow on of the form

for some “swirl” fields . The situation here is motivated by the familiar situation of studying axisymmetric Euler flows on , which in cylindrical coordinates take the form

The base component

of this flow is then a flow on the two-dimensional plane which is not quite incompressible (due to the failure of the volume preservation condition (2) in this case) but still satisfies a system of equations (coupled with a passive scalar field that is basically the square of the swirl ) that is reminiscent of the Boussinesq equations.

On a fixed -dimensional manifold , let denote the space of incompressible flows , equipped with the smooth topology (in spacetime), and let denote the space of such flows that are extendible to Euler flows. Our main theorem is

Theorem 1

- (i) (Generic inextendibility) Assume . Then is of the first category in (the countable union of nowhere dense sets in ).
- (ii) (Non-rigidity) Assume (with an arbitrary metric ). Then is somewhere dense in (that is, the closure of has non-empty interior).

More informally, starting with an incompressible flow , one usually cannot extend it to an Euler flow just by extending the manifold, warping the metric, and adding swirl coefficients, even if one is allowed to select the dimension of the extension, as well as the metric and coefficients, arbitrarily. However, many such flows can be *perturbed* to be extendible in such a manner (though different perturbations will require different extensions, in particular the dimension of the extension will not be fixed). Among other things, this means that conservation laws such as energy (or momentum, helicity, or circulation) no longer present an obstruction when one is allowed to perform an extension (basically this is because the swirl components of the extension can exchange energy (or momentum, etc.) with the base components in a basically arbitrary fashion.

These results fall short of my hopes to use the ability to extend the manifold to create universal behaviour in Euler flows, because of the fact that each flow requires a different extension in order to achieve the desired dynamics. Still it does seem to provide a little bit of support to the idea that high-dimensional Euler flows are quite “flexible” in their behaviour, though not completely so due to the generic inextendibility phenomenon. This flexibility reminds me a little bit of the flexibility of weak solutions to equations such as the Euler equations provided by the “-principle” of Gromov and its variants (as discussed in these recent notes), although in this case the flexibility comes from adding additional dimensions, rather than by repeatedly adding high-frequency corrections to the solution.

The proof of part (i) of the theorem basically proceeds by a dimension counting argument (similar to that in the proof of Proposition 9 of these recent lecture notes of mine). Heuristically, the point is that an arbitrary incompressible flow is essentially determined by independent functions of space and time, whereas the warping factors are functions of space only, the pressure field is one function of space and time, and the swirl fields are technically functions of both space and time, but have the same number of degrees of freedom as a function just of space, because they solve an evolution equation. When , this means that there are fewer unknown functions of space and time than prescribed functions of space and time, which is the source of the generic inextendibility. This simple argument breaks down when , but we do not know whether the claim is actually false in this case.

The proof of part (ii) proceeds by direct calculation of the effect of the warping factors and swirl velocities, which effectively create a forcing term (of Boussinesq type) in the first equation of (1) that is a combination of functions of the Eulerian spatial coordinates (coming from the warping factors) and the Lagrangian spatial coordinates (which arise from the swirl velocities, which are passively transported by the flow). In a non-empty open subset of , the combination of these coordinates becomes a non-degenerate set of coordinates for spacetime, and one can then use the Stone-Weierstrass theorem to conclude. The requirement that be topologically a torus is a technical hypothesis in order to avoid topological obstructions such as the hairy ball theorem, but it may be that the hypothesis can be dropped (and it may in fact be true, in the case at least, that is dense in all of , not just in a non-empty open subset).

These lecture notes are a continuation of the 254A lecture notes from the previous quarter.

We consider the Euler equations for incompressible fluid flow on a Euclidean space ; we will label as the “Eulerian space” (or “Euclidean space”, or “physical space”) to distinguish it from the “Lagrangian space” (or “labels space”) that we will introduce shortly (but the reader is free to also ignore the or subscripts if he or she wishes). Elements of Eulerian space will be referred to by symbols such as , we use to denote Lebesgue measure on and we will use for the coordinates of , and use indices such as to index these coordinates (with the usual summation conventions), for instance denotes partial differentiation along the coordinate. (We use superscripts for coordinates instead of subscripts to be compatible with some differential geometry notation that we will use shortly; in particular, when using the summation notation, we will now be matching subscripts with superscripts for the pair of indices being summed.)

In Eulerian coordinates, the Euler equations read

where is the velocity field and is the pressure field. These are functions of time and on the spatial location variable . We will refer to the coordinates as Eulerian coordinates. However, if one reviews the physical derivation of the Euler equations from 254A Notes 0, before one takes the continuum limit, the fundamental unknowns were not the velocity field or the pressure field , but rather the trajectories , which can be thought of as a single function from the coordinates (where is a time and is an element of the label set ) to . The relationship between the trajectories and the velocity field was given by the informal relationship

We will refer to the coordinates as (discrete) *Lagrangian coordinates* for describing the fluid.

In view of this, it is natural to ask whether there is an alternate way to formulate the continuum limit of incompressible inviscid fluids, by using a continuous version of the Lagrangian coordinates, rather than Eulerian coordinates. This is indeed the case. Suppose for instance one has a smooth solution to the Euler equations on a spacetime slab in Eulerian coordinates; assume furthermore that the velocity field is uniformly bounded. We introduce another copy of , which we call *Lagrangian space* or *labels space*; we use symbols such as to refer to elements of this space, to denote Lebesgue measure on , and to refer to the coordinates of . We use indices such as to index these coordinates, thus for instance denotes partial differentiation along the coordinate. We will use summation conventions for both the Eulerian coordinates and the Lagrangian coordinates , with an index being summed if it appears as both a subscript and a superscript in the same term. While and are of course isomorphic, we will try to refrain from identifying them, except perhaps at the initial time in order to fix the initialisation of Lagrangian coordinates.

Given a smooth and bounded velocity field , define a *trajectory map* for this velocity to be any smooth map that obeys the ODE

in view of (2), this describes the trajectory (in ) of a particle labeled by an element of . From the Picard existence theorem and the hypothesis that is smooth and bounded, such a map exists and is unique as long as one specifies the initial location assigned to each label . Traditionally, one chooses the initial condition

for , so that we label each particle by its initial location at time ; we are also free to specify other initial conditions for the trajectory map if we please. Indeed, we have the freedom to “permute” the labels by an arbitrary diffeomorphism: if is a trajectory map, and is any diffeomorphism (a smooth map whose inverse exists and is also smooth), then the map is also a trajectory map, albeit one with different initial conditions .

Despite the popularity of the initial condition (4), we will try to keep conceptually separate the Eulerian space from the Lagrangian space , as they play different physical roles in the interpretation of the fluid; for instance, while the Euclidean metric is an important feature of Eulerian space , it is not a geometrically natural structure to use in Lagrangian space . We have the following more general version of Exercise 8 from 254A Notes 2:

Exercise 1Let be smooth and bounded.

- If is a smooth map, show that there exists a unique smooth trajectory map with initial condition for all .
- Show that if is a diffeomorphism and , then the map is also a diffeomorphism.

Remark 2The first of the Euler equations (1) can now be written in the formwhich can be viewed as a continuous limit of Newton’s first law .

Call a diffeomorphism *(oriented) volume preserving* if one has the equation

for all , where the total differential is the matrix with entries for and , where are the components of . (If one wishes, one can also view as a linear transformation from the tangent space of Lagrangian space at to the tangent space of Eulerian space at .) Equivalently, is orientation preserving and one has a Jacobian-free change of variables formula

for all , which is in turn equivalent to having the same Lebesgue measure as for any measurable set .

The divergence-free condition then can be nicely expressed in terms of volume-preserving properties of the trajectory maps , in a manner which confirms the interpretation of this condition as an incompressibility condition on the fluid:

Lemma 3Let be smooth and bounded, let be a volume-preserving diffeomorphism, and let be the trajectory map. Then the following are equivalent:

- on .
- is volume-preserving for all .

*Proof:* Since is orientation-preserving, we see from continuity that is also orientation-preserving. Suppose that is also volume-preserving, then for any we have the conservation law

for all . Differentiating in time using the chain rule and (3) we conclude that

for all , and hence by change of variables

which by integration by parts gives

for all and , so is divergence-free.

To prove the converse implication, it is convenient to introduce the *labels map* , defined by setting to be the inverse of the diffeomorphism , thus

for all . By the implicit function theorem, is smooth, and by differentiating the above equation in time using (3) we see that

where is the usual material derivative

acting on functions on . If is divergence-free, we have from integration by parts that

for any test function . In particular, for any , we can calculate

and hence

for any . Since is volume-preserving, so is , thus

Thus is volume-preserving, and hence is also.

Exercise 4Let be a continuously differentiable map from the time interval to the general linear group of invertible matrices. Establish Jacobi’s formulaand use this and (6) to give an alternate proof of Lemma 3 that does not involve any integration in space.

Remark 5One can view the use of Lagrangian coordinates as an extension of the method of characteristics. Indeed, from the chain rule we see that for any smooth function of Eulerian spacetime, one hasand hence any transport equation that in Eulerian coordinates takes the form

for smooth functions of Eulerian spacetime is equivalent to the ODE

where are the smooth functions of Lagrangian spacetime defined by

In this set of notes we recall some basic differential geometry notation, particularly with regards to pullbacks and Lie derivatives of differential forms and other tensor fields on manifolds such as and , and explore how the Euler equations look in this notation. Our discussion will be entirely formal in nature; we will assume that all functions have enough smoothness and decay at infinity to justify the relevant calculations. (It is possible to work rigorously in Lagrangian coordinates – see for instance the work of Ebin and Marsden – but we will not do so here.) As a general rule, Lagrangian coordinates tend to be somewhat less convenient to use than Eulerian coordinates for establishing the basic analytic properties of the Euler equations, such as local existence, uniqueness, and continuous dependence on the data; however, they are quite good at clarifying the more algebraic properties of these equations, such as conservation laws and the variational nature of the equations. It may well be that in the future we will be able to use the Lagrangian formalism more effectively on the analytic side of the subject also.

Remark 6One can also write the Navier-Stokes equations in Lagrangian coordinates, but the equations are not expressed in a favourable form in these coordinates, as the Laplacian appearing in the viscosity term becomes replaced with a time-varying Laplace-Beltrami operator. As such, we will not discuss the Lagrangian coordinate formulation of Navier-Stokes here.

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