In the previous set of notes we developed a theory of “strong” solutions to the Navier-Stokes equations. This theory, based around viewing the Navier-Stokes equations as a perturbation of the linear heat equation, has many attractive features: solutions exist locally, are unique, depend continuously on the initial data, have a high degree of regularity, can be continued in time as long as a sufficiently high regularity norm is under control, and tend to enjoy the same sort of conservation laws that classical solutions do. However, it is a major open problem as to whether these solutions can be extended to be (forward) global in time, because the norms that we know how to control globally in time do not have high enough regularity to be useful for continuing the solution. Also, the theory becomes degenerate in the inviscid limit .

However, it is possible to construct “weak” solutions which lack many of the desirable features of strong solutions (notably, uniqueness, propagation of regularity, and conservation laws) but can often be constructed globally in time even when one us unable to do so for strong solutions. Broadly speaking, one usually constructs weak solutions by some sort of “compactness method”, which can generally be described as follows.

- Construct a sequence of “approximate solutions” to the desired equation, for instance by developing a well-posedness theory for some “regularised” approximation to the original equation. (This theory often follows similar lines to those in the previous set of notes, for instance using such tools as the contraction mapping theorem to construct the approximate solutions.)
- Establish some
*uniform*bounds (over appropriate time intervals) on these approximate solutions, even in the limit as an approximation parameter is sent to zero. (Uniformity is key;*non-uniform*bounds are often easy to obtain if one puts enough “mollification”, “hyper-dissipation”, or “discretisation” in the approximating equation.) - Use some sort of “weak compactness” (e.g., the Banach-Alaoglu theorem, the Arzela-Ascoli theorem, or the Rellich compactness theorem) to extract a subsequence of approximate solutions that converge (in a topology weaker than that associated to the available uniform bounds) to a limit. (Note that there is no reason
*a priori*to expect such limit points to be unique, or to have any regularity properties beyond that implied by the available uniform bounds..) - Show that this limit solves the original equation in a suitable weak sense.

The quality of these weak solutions is very much determined by the type of uniform bounds one can obtain on the approximate solution; the stronger these bounds are, the more properties one can obtain on these weak solutions. For instance, if the approximate solutions enjoy an energy identity leading to uniform energy bounds, then (by using tools such as Fatou’s lemma) one tends to obtain energy *inequalities* for the resulting weak solution; but if one somehow is able to obtain uniform bounds in a higher regularity norm than the energy then one can often recover the full energy *identity*. If the uniform bounds are at the regularity level needed to obtain well-posedness, then one generally expects to upgrade the weak solution to a strong solution. (This phenomenon is often formalised through *weak-strong uniqueness* theorems, which we will discuss later in these notes.) Thus we see that as far as attacking global regularity is concerned, both the theory of strong solutions and the theory of weak solutions encounter essentially the same obstacle, namely the inability to obtain uniform bounds on (exact or approximate) solutions at high regularities (and at arbitrary times).

For simplicity, we will focus our discussion in this notes on finite energy weak solutions on . There is a completely analogous theory for periodic weak solutions on (or equivalently, weak solutions on the torus which we will leave to the interested reader.

In recent years, a completely different way to construct weak solutions to the Navier-Stokes or Euler equations has been developed that are not based on the above compactness methods, but instead based on techniques of convex integration. These will be discussed in a later set of notes.

** — 1. A brief review of some aspects of distribution theory — **

We have already been using the concept of a distribution in previous notes, but we will rely more heavily on this theory in this set of notes, so we pause to review some key aspects of the theory. A more comprehensive discussion of distributions may be found in this previous blog post. To avoid some minor subtleties involving complex conjugation that are not relevant for this post, we will restrict attention to real-valued (scalar) distributions here. (One can then define vector-valued distributions (taking values in a finite-dimensional vector space) as a vector of scalar-valued distributions.)

Let us work in some non-empty open subset of a Euclidean space (which may eventually correspond to space, time, or spacetime). We recall that is the space of (real-valued) test functions . It has a rather subtle topological structure (see previous notes) which we will not detail here. A (real-valued) distribution on is a continuous linear functional from test functions to the reals . (This pairing may also be denoted or in other texts.) There are two basic examples of distributions to keep in mind:

- Any locally integrable function gives rise to a distribution (which by abuse of notation we also call ) by the formula .
- Any Radon measure gives rise to a distribution (which we will again call ) by the formula . For instance, if , the Dirac mass at is a distribution with .

Two distributions are equal in the sense of distributions of for all . For instance, it is not difficult to show that two locally integrable functions are equal in the sense of distributions if and only if they agree almost everywhere, and two Radon measures are equal in the sense of distributions if and only if they are identical.

As a general principle, any “linear” operation that makes sense for “nice” functions (such as test functions) can also be defined for distributions, but any “nonlinear” operation is unlikely to be usefully defined for arbitrary distributions (though it may still be a good concept to use for distributions with additional regularity). For instance, one can take a partial derivative (known as the weak derivative) of any distribution by the definition

for all . Note that this definition agrees with the “strong” or “classical” notion of a derivative when is a smooth function, thanks to integration by parts. Similarly, if is smooth, one can define the product distribution by the formula

for all . One can also take linear combinations of two distributions in the usual fashion, thus

for all and .

Exercise 1Let be a connected open subset of . Let be a distribution on such that in the sense of distributions for all . Show that is a constant, that is to say there exists such that in the sense of distributions.

A sequence of distributions is said to converge in the weak-* sense or *converge in the sense of distributions* to another distribution if one has

as for every test function ; in this case we write . This notion of convergence is sometimes referred to also as weak convergence (and one writes instead of ), although there is a subtle distinction between weak and weak-* convergence in non-reflexive spaces and so I will try to avoid this terminology (though in many cases one will be working in a reflexive space in which there is no distinction).

The linear operations alluded to above tend to be continuous in the distributional sense. For instance, it is easy to see that if , then for all , and for any smooth ; similarly, if , , and , are sequences of real numbers, then .

Suppose that one places a norm or seminorm on . Then one can define a subspace of the space of distributions, defined to be the space of all distributions for which the norm

is finite. For instance, if is the norm for some , then is just the dual space (with the (equivalence classes of) locally integrable functions in identified with distributions as above).

We have the following version of the Banach-Alaoglu theorem which allows us to easily create sequences that converge in the sense of distributions:

Proposition 2 (Variant of Banach-Alaoglu)Suppose that is a norm or seminorm on which makes the space separable. Let be a bounded sequence in . Then there is a subsequence of the which converges in the sense of distributions to a limit .

*Proof:* By hypothesis, there is a constant such that

for all . For each given , we may thus pass to a subsequence of such that converges to a limit. Passing to a subsequence a countably infinite number of times and using the Arzelá-Ascoli diagonalisation trick, we can thus find a dense subset of (using the metric) and a subsequence of the such that the limit exists for every , and hence for every by a limiting argument and (1). If one then defines to be the function

then one can verify that is a distribution, and by (1) we will have . By construction, converges in the sense of distributions to , and we are done.

It is important to note that there is no uniqueness claimed for ; while any given subsequence of the can have at most one limit , it is certainly possible for different subsequences to converge to different limits. Also, the proposition only applies for spaces that have preduals ; this covers many popular function spaces, such as spaces for , but omits endpoint spaces such as or . (For instance, approximations to the identity are uniformly bounded in , but converge weakly to a Dirac mass, which lies outside of .)

From definition we see that if , then we have the Fatou-type lemma

Thus, upper bounds on the approximating distributions are usually inherited by their limit . However, it is essential to be aware that the same is not true for lower bounds; there can be “loss of mass” in the limit. The following four examples illustrate some key ways in which this can occur:

- (Escape to spatial infinity) If is a non-zero test function, and is a sequence in going to infinity, then the translations of converge in the sense of distributions to zero, even though they will not go to zero in many function space norms (such as ).
- (Escape to frequency infinity) If is a non-zero test function, and is a sequence in going to infinity, then the modulations of converge in the sense of distributions to zero (cf. the Riemann-Lebesgue lemma), even though they will not go to zero in many function space norms (such as ).
- (Escape to infinitely fine scales) If , is a sequence of positive reals going to infinity, and , then the sequence converges in the sense of distributions to zero, but will not go to zero in several function space norms (e.g. with ).
- (Escape to infinitely coarse scales) If , is a sequence of positive reals going to zero, and , then the sequence converges in the sense of distributions to zero, but will not go to zero in several function space norms (e.g. with ).

Related to this loss of mass phenomenon is the important fact that the operation of pointwise multiplication is generally *not* continuous in the distributional topology: and does *not* necessarily imply in general (in fact in many cases the products or might not even be well-defined). For instance:

- Using the escape to frequency infinity example, the functions converge in the sense of distributions to zero, but their squares instead converge in the sense of distributions to , as can be seen from the double angle formula .
- Using the escape to infinitely fine scales example, the functions converge in the sense of distributions to zero, but their squares will not if .

This lack of continuity of multiplication means that one has to take a non-trivial amount of care when applying the theory of distributions to nonlinear PDE; a sufficiently careless regard for this issue (or more generally, treating distribution theory as some sort of “magic wand“) is likely to lead to serious errors in one’s arguments.

One way to recover continuity of pointwise multiplication is to somehow upgrade distributional convergence to stronger notions of convergence. For instance, from Hölder’s inequality one sees that if converges strongly to in (thus and both lie in , and goes to zero), and converges strongly to in , then will converge strongly in to , where .

One key way to obtain strong convergence in some norm is to obtain uniform bounds in an even stronger norm – so strong that the associated space embeds compactly in the space associated to the original norm. More precisely

Proposition 3 (Upgrading to strong convergence)Let be two norms on , with associated spaces of distributions. Suppose that embeds compactly into , that is to say the closed unit ball in is a compact subset of . If is a bounded sequence in that converges in the sense of distributions to a limit , then converges strongly in to as well.

*Proof:* By the Urysohn subsequence principle, it suffices to show that every subsequence of has a further subsequence that converges strongly in to . But by the compact embedding of into , every subsequence of has a further subsequence that converges strongly in to some limit , and hence also in the sense of distributions to by definition of the norm. But thus subsequence also converges in the sense of distributions to , and hence , and the claim follows.

** — 2. Simple examples of weak solutions — **

We now study weak solutions for some very simple equations, as a warmup for discussing weak solutions for Navier-Stokes.

We begin with an extremely simple initial value problem, the ODE

on a half-open time interval with , with initial condition , where and given and is the unknown. Of course, when are smooth, then the fundamental theorem of calculus gives the unique solution

for . If one integrates the identity against a test function (that is to say, one multiplies both sides of this identity by and then integrates) on , one obtains

which upon integration by parts and rearranging gives

where we extend by zero to the open set . Thus, we have

in the sense of distributions (on ). More generally, if are locally integrable functions on , we say that is a *weak solution* to the initial value problem if (4) holds in the sense of distributions on . Thanks to the fundamental theorem of calculus for locally integrable functions, we still recover the unique solution (16):

Exercise 4Let be locally integrable functions (extended by zero to all of ), and let . Show that the following are equivalent:

Now let be a finite dimensional vector space, let be a continuous function, let , and consider the initial value problem

on some forward time interval . The Picard existence theorem lets us construct such solutions when is Lipschitz continuous and is small enough, but now we are merely requiring to be continuous and not necessarily Lipschitz. As in the preceding case, we introduce the notion of a weak solution. If is locally bounded (and measurable) on , then will be locally integrable on ; we then extend by zero to be distributions on , and we say that is a *weak solution* to (5) if one has

in the sense of distributions on , or equivalently that one has the identity

for all test functions compactly supported in . In this simple ODE setting, the notion of a weak solution coincides with stronger notions of solutions:

Exercise 5Let be finite dimensional, let be continuous, let , and let be locally bounded and measurable. Show that the following are equivalent:

In particular, if the ODE initial value problem (5) exhibits finite time blowup for its (unique) classical solution, then it will also do so for weak solutions (with exactly the same blouwp time). This will be in contrast with the situation for PDE, in which it is possible for weak solutions to persist beyond the time in which classical solutions exist.

Now we give a compactness argument to produce weak solutions (which will then be classical solutions, by the above exercise):

Proposition 6 (Weak existence)Let be a finite dimensional vector space, let , let , and let be a continuous function. Let be the timeThen there exists a continuously differentiable solution to the initial value problem (5) on .

*Proof:* By construction, we have

Using the Weierstrass approximation theorem (or Stone-Weierstrass theorem), we can express on as the uniform limit of Lipschitz continuous functions , such that

for all ; we can then extend in a Lipschitz continuous fashion to all of . (The Lipschitz constant of is permitted to diverge to infinity as ). We can then apply the Picard existence theorem (Theorem 8 of Notes 1), for each we have a (continuously differentiable) maximal Cauchy development of the initial value problem

with as if is finite. (We could also solve the ODE backwards in time, but will not need to do so here.) We now claim that , and furthermore that one has the uniform bound

for all and all . Indeed, if this were not the case then by continuity (and the fact that ) there would be some and some such that , and for all . But then by the fundamental theorem of calculus and the triangle inequality (and (6)) we have

a contradiction. Thus we have (8) for all and , so takes values in on . Applying (7), (6) we conclude that

for all and all ; in particular, the are uniformly Lipschitz continuous and uniformly bounded on . Applying the Arzelá-Ascoli theorem, we can then pass to a subsequence in which the converge uniformly on to a limit , which then also takes values in . (Alternatively, one could use Proposition 2 to have converge in the sense of distributions, followed by Proposition 3 to upgrade to uniform convergence.) As converges uniformly to on , we conclude that converges uniformly to on . Since we have

in the sense of distributions (extending , by zero to ), we can take distributional limits and conclude that

in the sense of distributions, which by Exercise 5 shows that is a continuously differentiable solution to the initial value problem (5) as required.

In contrast to the Picard theory when is Lipschitz, Proposition 6 does not assert any uniqueness of the solution to the initial value problem (5). And in fact uniqueness often fails once the Lipschitz hypothesis is dropped! Consider the simple example of the scalar initial value problem

on , so the nonlinearity here is the continuous, but not Lipschitz continuous, function . Clearly the zero function is a solution to this ODE. But so is the function . In fact there are a continuum of solutions: for any , the function is a solution. Proposition 6 will select one of these solutions, but the precise solution selected will depend on the choice of approximating functions :

Exercise 7Let . For each , let denote the function

- (i) Show that each is Lipschitz continuous, and the converge uniformly to the function as .
- (ii) Show that the solution to the initial value problem is given by
for and

for .

- (iii) Show that as , converges uniformly to the function .

Now we give a simple example of a weak solution construction for a PDE, namely the linear transport equation

where the initial data and a position-dependent velocity field is given, and is the unknown field.

Suppose for the moment that are smooth, with bounded. Then one can solve this problem using the method of characteristics. For any , let denote the solution to the initial value problem

The Picard existence theorem gives us a smooth maximal Cauchy development for this problem; as is bounded, this development cannot go to infinity in finite time (either forward or backwards in time), and so the solution is global. Thus we have a well-defined map for each time . In fact we can say more:

Exercise 8Let the assumptions be as above.

- (i) Show the semigroup property for all .
- (ii) Show that is a homeomorphism for each .
- (iii) Show that for every , is differentiable, and the derivative obeys the linear initial value problem
(Hint: while this system formally can be obtained by differentiating (10) in , this formal differentiation requires rigorous justification. One can for instance proceed by first principles, showing that the Newton quotients approximately obey this equation, and then using a Gronwall inequality argument to compare this approximate solution to an exact solution.)

- (iv) Show that is a diffeomorphism for each ; that is to say, and its inverse are both continuously differentiable.
- (v) Show that is a smooth diffeomorphism (that is to say and its inverse are both smooth). (Caution: one may require a bit of planning to avoid the proof from becoming extremely long and tedious.)

From (10) and the chain rule we have the identity

for any smooth function (cf. the material derivative used in Notes 0). Thus, one can rewrite the initial value problem (9) as

at which point it is clear that the unique smooth solution to the initial value problem (10) is given by

Among other things, this shows that the sup norm is a conserved quantity:

Now we drop the hypothesis that is bounded. One can no longer assume that the trajectories are globally defined, or even that they are defined for a positive time independent of the starting point . Nevertheless, we have

Proposition 9 (Weak existence)Let be smooth, and let be smooth and bounded. Then there exists a bounded measurable function which weakly solves (10) in the sense thatin the sense of distributions on ) (extending by zero outside of ), or equivalently that

*Proof:* By multiplying by appropriate smooth cutoff functions, we can express as the locally uniform limit of smooth bounded functions with equal to on (say) . By the preceding discussion, for each we have a smooth global solution to the initial value problem

in the sense of distributions on . By (11), the are uniformly bounded with

Thus, by Proposition 2, we can pass to a subsequence and assume that converges in the sense of distributions to an element on ; by (2) we have

Since the are all supported on , is also. Taking weak limits in (13) (multiplying first by a cutoff function to localise to

This gives the required weak solution.

The following exercise shows that while one can construct global weak solutions, there is significant failure of uniqueness and persistence of regularity:

Exercise 10Set , thus we are solving the ODE

- (i) If are bounded measurable functions, show that the function defined by
for and

for is a weak solution to (14) with initial data

for and

for . (Note that one does not need to specify these functions at , since this describes a measure zero set.)

- (ii) Suppose further that , and that is smooth and compactly supported in . Show that the weak solution described in (i) is the solution constructed by Proposition 9.
- (iii) Show that there exist at least two bounded measurable weak solutions to (14) with initial data , thus showing that weak solutions are not unique. (Of course, at most one of these solutions could obey the inequality (12), so there are some weak solutions that are not constructible using Proposition 9.) Show that this lack of uniqueness persists even if one also demands that the weak solutions be smooth; conversely, show that there exist weak solutions with initial data that are discontinuous.

Remark 11As the above example illustrates, the loss of mass phenomenon for weak solutions arises because the approximants to those weak solutions “escape to infinity”in the limit, similarly, the loss of uniqueness phenomenon for weak solutions arises because the approximants “come from infinity” in the limit. In this particular case of a transport equation, the infinity is spatial infinity, but for other types of PDE it can be possible for approximate solutions to escape from, or come from, other types of infinity, such as frequency infinity, fine scale infinity, or coarse scale infinity. (In the former two cases, the loss of mass phenomenon will also be closely related to a loss of regularity in the weak solution.) Eliminating these types of “bad behaviour” for weak solutions is morally equivalent to obtaining uniform bounds for the approximating solutions that are strong enough to prevent such solutions from having a significant presence near infinity; in the case of Navier-Stokes, this basically corresponds to controlling such solutions uniformly in subcritical or critical norms.

** — 3. Leray-Hopf weak solutions — **

We now adapt the above formalism to construct weak solutions to the Navier-Stokes equations, following the fundamental work of Leray, who constructed such solutions on , (as before, we discard the case as being degenerate). The later work of Hopf extended this construction to other domains, but we will work solely with here for simplicity.

In the previous set of notes, several formulations of the Navier-Stokes equations were considered. For smooth solutions (with suitable decay at infinity, and in some cases a normalisation hypothesis on the pressure also), these formulations were shown to be essentially equivalent to each other. But at the very low level of regularity that weak solutions are known to have, these different formulations of Navier-Stokes are no longer obviously equivalent. As such, there is not a single notion of a “weak solution to the Navier-Stokes equations”; the notion depends on which formulation of these equations one chooses to work with. This leads to a number of rather technical subtleties when developing a theory of weak solutions. We will largely avoid these issues here, focusing on a specific type of weak solution that arises from our version of Leray’s construction.

It will be convenient to work with the formulation

of the initial value problem for the Navier-Stokes equations. Writing out the divergence as and interchanging with , we can rewrite this as

The point of this formulation is that it can be interpreted distributionally with fairly weak regularity hypotheses on . For Leray’s construction, it turns out that a natural regularity class is

basically because the norms associated to these function spaces are precisely the quantities that will be controlled by the important *energy identity* that we will discuss later. With this regularity, we have in particular that

by which we mean that

for all . Next, we need a special case of the Sobolev embedding theorem:

Exercise 12 (Non-endpoint Sobolev embedding theorem)Let be such that . Show that for any , one has with(

Hint:this non-endpoint case can be proven using the Littlewood-Paley projections from the previous set of notes.) The endpoint case of the Sobolev embedding theorem is also true (as long as ), but the proof requires the Hardy-Littlewood-Sobolev fractional integration inequality, which we will not cover here; see for instance these previous lecture notes.

We conclude that there is some for which

and hence by Hölder’s inequality

for all . (The precise value of is not terribly important for our arguments.)

Next, we invoke the following result from harmonic analysis:

Proposition 13 (Boundedness of the Leray projection)For any , one has the boundfor all . In particular, has a unique continuous extension to a linear map from to itself.

For , this proposition follows easily from Plancherel’s theorem. For , the proposition is more non-trivial, and is usually proven using the Calderón-Zygmund theory of singular integrals. A proof can be found for instance in Stein’s “Singular integrals“; we shall simply assume it as a black box here. We conclude that for in the regularity class (16), we have

In particular, is locally integrable in spacetime and thus can be interpreted as a distribution on (after extending by zero outside of . Thus also can be interpreted as a distribution. Similarly for the other two terms in (15). We then say that a function in the regularity class (16) is a *weak solution* to the initial value problem (15) for some distribution if one has

in the sense of spacetime distributions on (after extending by zero outside of . Unpacking the definitions of distributional derivative, this is equivalent to requiring that

for all spacetime test functions .

We can now state a form of Leray’s theorem:

Theorem 14 (Leray’s weak solutions)Let be divergence free (in the sense of distributions), and let . Then there exists a weak solution to the initial value problem (15). Furthermore, obeys the energy inequality

for almost every .

We now prove this theorem using the same sort of scheme that was used previously to construct weak solutions to other equations. We first need to set up some approximate solutions to (15). There are many ways to do this – the traditional way being to use some variant of the Galerkin method – but we will proceed using the Littlewood-Paley projections that were already introduced in the previous set of notes. Let be a sequence of dyadic integers going to infinity. We consider solutions to the initial value problem

this is (15) except with some additional factors of inserted in the initial data and in the nonlinear term. Formally, in the limit , the factors should converge to the identity and one should recover (15); but this requires rigorous justification. The number of factors of in the nonlinear term may seem excessive, but as we shall see, this turns out to be a convenient choice as it will lead to a favourable energy inequality for these solutions.

The Fujita-Kato theory of mild solutions for (15) from the previous set of notes can be easily adapted to the initial value problem (19), because the projections are bounded on all the function spaces of interest. Thus, for any , and any divergence-free , we can define an -mild solution to (15) on a time interval to be a function in the function space

such that

(in the sense of distributions) for all ; a mild solution on is a solution that is an mild solution when restricted to every compact subinterval . Note that the frequency-localised initial data lies in every space. By a modification of the theory of the previous set of notes, we thus see that there is a maximal Cauchy development that is a smooth solution to (19) (and an mild solution for every ), with if . Note that as is divergence-free, , and preserves the divergence-free property, and projects to divergence-free functions, is divergence-free for all . Similarly, as projects to functions with Fourier transform supported on the ball in , and this property is preserved by , , and we see that also has Fourier transform supported on the ball . This (non-uniformly) bounded frequency support is the key additional feature enjoyed by our approximate solutions that has no analogue for the actual solution , and effectively serves as a sort of “discretisation” of the problem (as per the uncertainty principle).

The next step is to ensure that the approximate solutions exist globally in time, that is to say that . We can do this by exploiting the energy conservation law for this equation. Indeed for any time , define the energy

(compare with Exercise 4 from Notes 0). From (19) we know that and lie in for any and any . This very high regularity allows us to easily justify operations such as integration by parts or differentiation under the integral sign in what follows. In particular, it is easy to establish the identity

for any . Inserting (19) (and suppressing explicit dependence on for brevity), we obtain

For the second term, we integrate by parts to obtain

For the first term

we use the self-adjointness of and , the skew-adjointness of , the fact that all three of these operators (being Fourier multipliers) commute with each other to write it as

Since is divergence-free, the Leray projection acts as the identity on it, so we may write the above expression as

Recalling the rules of thumb for the energy method from the previous set of notes, we locate a total derivative to rewrite the preceding expression as

(It is here that we begin to see how important it was to have so many factors of in our approximating equation.) We may now integrate by parts (easily justified using the high regularity of ) to obtain

But is divergence-free, so vanishes. To summarise, we conclude the (differential form of) the *energy identity*

by the fundamental theorem of calculus, we conclude in particular that

for all . Among other things, this gives a uniform bound

Ordinarily, this type bound would be too weak to combine with the blowup criterion mentioned earlier. But we know that has Fourier transform supported in , so in particular we have the reproducing formula . We may thus use the Bernstein inequality (Exercise 52 from Notes 1) and conclude that

This bound is not uniform in , but it is still finite, and so by combining with the blowup criterion we conclude that .

Now we need to start taking limits as . For this we need uniform bounds. Returning to the energy identity (20), we have the uniform bounds

so in particular for any finite one has

This is enough regularity for Proposition 2 to apply, and we can pass to a subsequence of which converges in the sense of spacetime distributions in (after extending by zero outside of to a limit , which is in for every .

Now we work on verifying the energy inequality (18). Let be a test function with which is non-increasing on . From (20) and integration by parts we have

Taking limit inferior and using the Fatou-type lemma (2), we conclude that

Now let , take to equal on and zero outside of for some small . Then we have

The function is supported on , is non-negative, and has total mass one. By the Lebesgue differentiation theorem applied to the bounded measurable function , we conclude that for almost every , we have

as . The claim (18) follows.

It remains to show that is a weak solution of (15), that is to say that (17) holds in the sense of spacetime distributions. Certainly the smooth solution of (19) will also be a weak solution, thus

in the sense of spacetime distributions on , where we extend by zero outside of .

At this point it is tempting to just take distributional limits of both sides of (22) to obtain (17). Certainly we have the expected convergence for the linear components of the equation:

However, it is not immediately clear that

mainly because of the previously mentioned problem that multiplication is not continuous with respect to weak notions of convergence. But if we can show (23), then we do indeed recover (17) as the limit of (22), which will complete the proof of Theorem 14.

Let’s try to simplify the task of proving (23). The partial derivative operator is continuous with respect to convergence in distributions, so it suffices to show that

where

We now try to get rid of the outer Littlewood-Paley projection. We claim that

Let be a fixed time. By Sobolev embedding and (21), is bounded in , uniformly in , for some . The same is then true for , hence by Hölder’s inequality and Proposition 13, is uniformly bounded in . On the other hand, for any spacetime test function , it is not difficult (using the rapid decrease of the Fourier transform of ) to show that goes to zero in the dual space . This gives (24).

It thus suffices to show that converges in the sense of distributions to , thus one wants

for any spacetime test function . One can easily calculate that lies in the dual space to the space that and are bounded in, so it will suffices to show that converges strongly in to for sufficiently close to . and any compact subset of spacetime (since the norm of outside of can be made arbitrarily small by making large enough.)

Let be a dyadic integer, then we can split

The functions are uniformly bounded in by some bound , hence by Plancherel’s theorem the functions , have an norm of (assuming is large enough so that ). Indeed, by Littlewood-Paley decomposition and Bernstein’s inequality we also see that these functions have an norm of if is close enough to that the exponent of is negative. It will therefore suffice to show that

strongly in for every fixed and .

We already know that goes to zero in the sense of distributions, so (as Proposition 3 indicates) the main difficulty is to obtain compactness of the sequence. The operator localises in spatial frequency, and the restriction to localises in both space and time, however there is still the possibility of escaping to temporal frequency. To prevent this, we need some sort of equicontinuity in time. For this, we may turn to the equation (19) obeyed by . Applying , we see that

when is large enough. We have already seen that is bounded in uniformly in , so by the Bernstein inequality is bounded in (we allow the bound to depend on ). Similarly for . We conclude that is bounded in uniformly in ; taking weak limits using (2), the same is true for , and hence is bounded in . Also, is bounded in by Bernstein’s inequality; thus is equicontinuous in . By the Arzelá-Ascoli theorem and Proposition 3, must therefore go to zero uniformly, and the claim follows. This completes the proof of Theorem 14.

Exercise 15 (Rellich compactness theorem)Let be such that .

- (i) Show that if is a bounded sequence in that converges in the sense of distributions to a limit , then there is a subsequence which converges strongly in to (thus, for any compact set , the restrictions of to converge strongly in to the restriction of to ).
- (ii) Show that for any compact set , the linear map defined by setting to be the restriction of to is a compact linear map.
- (iii) Show that the above two claims fail at the endpoint (which of course only occurs when ).

The weak solutions constructed by Theorem 14 have additional properties beyond the ones listed in the above theorem. For instance:

Exercise 16Let be as in Theorem 14, and let be a weak solution constructed using the proof of Theorem 14.

- (i) Show that is divergence-free in the sense of spacetime distributions.
- (ii) Show that there is a measure zero subset of such that one has the energy inequality
for all with . Furthermore, show that for all , the time-shifted function defined by is a weak solution to the initial value problem (15) with initial data .

- (iii) Show that after modifying on a set of measure zero, the function is continuous for any . (
Hint:first establish this when is a test function.)

We will discuss some further properties of the Leray weak solutions in later notes.

** — 4. Weak-strong uniqueness — **

If is a (non-zero) element in a Hilbert space , and is another element obeying the inequality

then this is very far from the assertion that is equal to , since the ball of elements of obeying (25) is far larger than the single point . However, if one also posseses the information that agrees with when tested against , in the sense that

then (25) and (26) combine to indeed be able to conclude that . Geometrically, this is because the above-mentioned ball is tangent to the hyperplane described by (26) at the point . Algebraically, one can establish this claim by the cosine rule computation

giving the claim.

This basic argument has many variants. Here are two of them:

Exercise 17 (Weak convergence plus norm bound equals strong convergence (Hilbert spaces))Let be an element of a Hilbert space , and let be a sequence in which weakly converges to , that is to say that for all . Show that the following are equivalent:

- (i) .
- (ii) .
- (iii) converges
stronglyto .

Exercise 18 (Weak convergence plus norm bound equals strong convergence ( norms))Let be a measure space, let be an absolutely integrable non-negative function, and let be a sequence of absolutely integrable non-negative functions that converge pointwise to . Show that the following are equivalent:

- (i) .
- (ii) .
- (iii) converges strongly in to .
(

Hint:express and in terms of the positive and negative parts of . The latter can be controlled using the dominated convergence theorem.)

Exercise 19Let be as in Theorem 14, and let be a weak solution constructed using the proof of Theorem 14. Show that (after modifying on a set of measure zero if necessary), converges strongly in to as . (Hint:use Exercise 16(iii) and Exercise 17.)

Now we give a variant relating to weak and strong solutions of the Navier-Stokes equations.

Proposition 20 (Weak-strong uniqueness)Let be an mild solution to the Navier-Stokes equations (15) for some , , and with . Let be a weak solution to the Navier-Stokes equation which obeys the energy inequality (18) for almost all . Then and agree almost everywhere on .

Roughly speaking, this proposition asserts that weak solutions obeying the energy inequality stay unique as long as a strong solution exists (in particular, it is unique whenever it is regular enough to be a strong solution). However, once a strong solution reaches the end of its maximal Cauchy development, there is no further guarantee of uniqueness for the rest of the weak solution. Also, there is no guarantee of uniqueness of weak solutions if the energy inequality is dropped, and indeed there is now increasing evidence that uniqueness is simply false in this case; see for instance this paper of Buckmaster and Vicol for recent work in this direction. The conditions on can be relaxed somewhat (in particular, it is possible to drop the condition ), though they still need to be “subcritical” or “critical” in nature; see for instance the classic papers of Prodi, of Serrin, and of Ladyzhenskaya, which show that weak solutions on obeying the energy inequality are necessarily unique and smooth (after time ) if they lie in the space for some exponents with and ; the endpoint case was worked out more recently by Escauriaza, Seregin, and Sverak. For a recent survey of weak-strong uniqueness results for fluid equations, see this paper of Wiedemann.

*Proof:* Before we give the formal proof, let us first give a non-rigorous proof in which we pretend that the weak solution can be manipulated like a strong solution. Then we have

and

As in the beginning of the section, the idea is to analyse the norm of the difference . Writing in the first equation and subtracting from the second equation, we obtain the *difference equation*

If we formally differentiate the energy using this equation, we obtain

(omitting the explicit dependence of the integrand on and ) which after some integration by parts (noting that is divergence-free and thus is the identity on formally becomes

The and terms formally cancel out by the usual trick of writing as a total derivative and integrating by parts, using the divergence-free nature of both and . For the term , we can cancel it against the term by the arithmetic mean-geometric mean inequality

to obtain

thanks to Hölder’s inequality. As is an mild solution, it lies in , which by Sobolev embedding and Hölder means that it is also in . Since , Gronwall’s inequality then should give for all , giving the claim.

Now we begin the rigorous proof, in which is only known to be a weak solution. Here, we do not directly manipulate the difference equation, but instead carefully use the equations for and as a substitute. Define and as before. From the cosine rule we have

where we drop the explicit dependence on in the integrand. From the energy inequality hypothesis (18), we have

for almost all , where we also drop explicit dependence on in the integrand. The strong solution also obeys the energy inequality; in fact we have the energy equality

as can be seen by first working with smooth solutions and taking limits using the local well-posedness theory. We conclude that

Now we work on the integral . Because we only know to solve the equation

in the sense of spacetime integrals, it is difficult to directly treat this spatial integral. Instead (similarly to the proof of the energy inequality for Leray solutions), we will first work with a proxy

where is a test function in time, which we normalise with ; eventually we will make an approximation to the indicator function of and apply the Lebesgue differentiation theorem to recover information about for almost every .

By hypothesis, we have

for any spacetime test function . We would like to apply this identity with replaced by (in order to obtain an identity involving the expression (28)). Now is not a test function; however, as is an mild solution, it has the regularity

also, using the equation (15), Sobolev embedding, Hölder’s inequality, and the hypotheses and we see that

(If one wishes, one can first obtain this bound for smooth solutions, and take limits using the local well-posedness theory.) As a consequence, one can find a sequence of test functions , such that converges to in and norm (so converges to in norm), and converges to in norm. Since lies in , lies in , and lies in by Hölder and Sobolev, we can take limits and conclude that

Since is divergence-free, and does not depend on the spatial variables, we can simplify this slightly as

and so we can write (28) as

Using the Lebesgue differentiation theorem as in the proof of Theorem 14, we conclude that for almost every , one has the identity

Applying (15), the right-hand side is

(Note that expressions such as are well defined because lie in .) We can integrate by parts (justified using the usual limiting argument and the bounds on ) and use the divergence-free nature of to write this as

Inserting this into (27), we conclude that

We write and write this as

noting from the regularity , on and Sobolev embedding that one can ensure that all integrals here are absolutely convergent.

The integral can be rewritten using integration by parts as (noting that there is enough regularity to justify the integration by parts by the usual limiting argument); expressing as a total derivative and integrating by parts again using the divergence-free nature of , we see that this expression vanishes. Similarly for the term. Now we eliminate the remaining terms which are linear in :

We may integrate by parts, and write the dot product in coordinates, to write this as

Applying the Leibniz rule and the divergence-free nature of , we see that this expression vanishes. We conclude that

Now we use the Leibniz rule, the divergence-free nature of , and the arithmetic mean-geometric mean inequality to write

to obtain

and hence by Sobolev embedding we have

for almost all . Applying Gronwall’s inequality (modifying on a set of measure zero) we conclude that for almost all , giving the claim.

One application of weak-strong uniqueness results is to give (in the case at least) *partial regularity* on the weak solutions constructed by Leray, in that the solutions agree with smooth solutions on large regions of spacetime – large enough, in fact, to cover all but a measure zero set of times . Unfortunately, the complement of this measure zero set could be disconnected, and so one could have different smooth solutions agreeing with at different epochs, so this is still quite far from an assertion of global regularity of the solution. Nevertheless it is still a non-trivial and interesting result:

Theorem 21 (Partial regularity)Let . Let be as in Theorem 14, and let be a weak solution constructed using the proof of Theorem 14.

- (i) (Eventual regularity) There exists a time such that (after modification on a set of measure zero), the weak solution on agrees with an mild solution on with initial data (where we time shift the notion of a mild solution to start at instead of ).
- (ii) (Epochs of regularity) There exists a compact exceptional set of measure zero, such that for any time , there is a time interval containing in its interior such that on agrees almost everywhere whtn an mild solution on with initial data .

*Proof:* (Sketch) We begin with (i). From (18), the norm of and the norm of are finite. Thus, for any , one can find a positive measure set of times such that

which by Plancherel and Cauchy-Schwarz implies that

In particular, by Exercise 16, one can find a time such that is a weak solution on with initial data obeying the energy inequality, with

By the small data global existence theory (Theorem 45 from Notes 1), if is chosen small enough, then there is then a global mild solution on to the Navier-Stokes equations with initial data , which must then agree with by weak-strong uniqueness. This proves (i).

Now we look at (ii). In view of (i) we can work in a fixed compact interval . Let be a time, and let be a sufficiently small constant. If there is a positive measure set of times for which

then by the same argument as above (but now using well-posedness theory instead of well-posedness theory), we will be able to equate (almost everywhere) with an mild solution on for some neighbourhood of . Thus the only times for which we cannot do this are those for which one has

for almost all . In particular, for any , one can cover such times by a collection of intervals of length , such that for almost every in that interval. On the other hand, as is bounded in , the number of disjoint time intervals of this form is at most (where we allow the implied constant to depend on and ). Thus the set of exceptional times can be covered by intervals of length , and thus its closure has Lebesgue measure . Sending we see that the exceptional times are contained in a closed measure zero subset of , and the claim follows.

The above argument in fact shows that the exceptional set in part (ii) of the above theorem will have upper Minkowski dimension at most (and hence also Hausdorff dimension at most ). There is a significant strengthening of this partial regularity result due to Caffarelli, Kohn, and Nirenberg, which we will discuss in later notes.

## 14 comments

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2 October, 2018 at 5:56 pm

Quanling DengReblogged this on gonewithmath.

3 October, 2018 at 3:44 am

Gabriel ApolinarioThese notes are a great resource.

Terry, there’s a missing word in the first paragraph: “are not high enough regularity”

[Reworded – T.]3 October, 2018 at 6:16 am

Dr. Anil PedgaonkarWhy maths people are left with old topics like fluid mechanics while topics like relativity theory string theory quantum theory are with physics

3 October, 2018 at 9:14 am

AnonymousTurbulence is still not well understood and since we live in turbulent times it may become a contemporary subject.

4 October, 2018 at 10:53 pm

AnonymousNo doubt you are aware of https://en.wikipedia.org/wiki/Straw_man in your post, but let us overlook that. The reason why fluid mechanics, in particular Navier-Stokes equations, is mathematically interesting is that its behavior is not well-understood, even though its formulation is extremely simple. Anyone with modest math education can “understand” the governing equations, but no-one knows whether they are well-posed for arbitrarily long time intervals. Certainly there are open problems and plenty of theory building to be done in string theory and other fields practiced by “big bad physicists” where the problem setting itself is more complicated, but one is simply less surprised to encounter unsolved problems in more complicated settings. Moreover, all the fields you mentioned belong to mathematical physics and are under active research by quite a few well-trained people who consider themselves primarily as mathematicians.

4 October, 2018 at 11:26 pm

Juha-Matti PerkkiöProf. Tao,

the recent polymath-projects discussed in this blog have been very inspiring and seem to be rather efficient when their goals are set properly. Could there be a case for some kind of mixed analysis/numerical analysis/computational project to gain some insight on Navier-Stokes? What comes to mind is for example to generate solutions with H^2- or H^1-norm growing as fast as possible/decaying as slow as possible or perhaps to generate solutions decaying as fast as possible in some relevant weaker norm like L^2. I am aware that even the existence of suitable quantitative criteria for the reliability of such numerical experiments is not self-evident, but perhaps there are some.

5 October, 2018 at 12:44 pm

SamApologies, I must be missing something basic, but why do the bounds on the norm of and -norm of imply an bound on ?

5 October, 2018 at 12:45 pm

SamSorry, I should specify that I’m talking about estimate (21).

5 October, 2018 at 2:23 pm

Terence TaoThis was a typo: it has been corrected to .

7 October, 2018 at 1:55 am

SamAh, yes, that makes sense. Thanks!

7 October, 2018 at 5:14 am

HuangTerry，I am Chinese. I have a new opinion about Callatz Conjecture. How can I send it to you?

8 October, 2018 at 9:29 am

AnonymousAre you Huang Deren mathematician?

8 October, 2018 at 4:18 pm

HuangNo

9 October, 2018 at 12:44 pm

254A, Notes 3: Local well-posedness for the Euler equations | What's new[…] weak compactness (Proposition 2 of Notes 2), one can pass to a subsequence such that converge weakly to some limits , such that and all […]