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 1 Let
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 4 Let
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 5 Let
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 time
Then 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 7 Let
. 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 locally 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 8 Let 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 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 that
in 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 a compact set) we have
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 10 Set
, 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 11 As 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 bound
for 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. Since
is uniformly bounded in
and
, we see from Hölder (and Proposition 13 that
is bounded in
uniformly in
for some
, 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
. By the fundamental theorem of calculus and Cauchy-Schwarz, this gives Hölder continuity in time (of order
). 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 on
, 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 16 Let
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) (Note: this exercise is tricky.) Assume
. Show that the weak solution
obeys a local energy inequality
for all
. (Hint: compute the time derivative of
, where
is a smooth cutoff supported on
that equals one in
, and use Sobolev inequalities and Hölder to control the various terms that arise from integration by parts; one will need to expand out the Leray projection and use the fact that
is bounded on every
space for
.) Using this inequality, 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
. (The arguments here can be extended to dimensions
, but it is open for
whether one can construct Leray-Hopf solutions obeying the strong energy inequality.)
- (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 strongly to
.
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 19 Let
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 with
and
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
for almost all .
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.
18 comments
Comments feed for this article
2 October, 2018 at 5:56 pm
Quanling Deng
Reblogged this on gonewithmath.
3 October, 2018 at 3:44 am
Gabriel Apolinario
These 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 Pedgaonkar
Why 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
Anonymous
Turbulence 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
Anonymous
No 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 Perkkiö
Prof. 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
Sam
Apologies, 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
Sam
Sorry, I should specify that I’m talking about estimate (21).
5 October, 2018 at 2:23 pm
Terence Tao
This was a typo: it has been corrected to
.
7 October, 2018 at 1:55 am
Sam
Ah, yes, that makes sense. Thanks!
7 October, 2018 at 5:14 am
Huang
Terry,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
Anonymous
Are you Huang Deren mathematician?
8 October, 2018 at 4:18 pm
Huang
No
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 […]
8 November, 2018 at 5:48 am
Stefan
As a double major in theoretical physics/theoretical computer science, I’m very intersted in this.
Does anybody know if this is related to “Homotopy Analysis Method in Nonlinear Differential Equations” and “Liao’s Method of Directly Defining the Inverse Mapping (MDDiM).”?
https://www.researchgate.net/publication/266832165_Homotopy_Analysis_Method_in_Nonlinear_Differential_Equations
https://link.springer.com/article/10.1007/s11075-015-0077-4
Anyway, since reading the post about ultrafilters and hierarchical infinitesimals, I feel that more boundries between physics and computer science could be broken and new solutions be found.
I think it is about time to solve the crisis in physics
8 November, 2018 at 3:39 pm
Anonymous
In physics, the subjective(!) description “more beautiful” for a desired feature of a better new theory should be replaced (or interpreted) by the objective(!) description “more symmetrical”. It is interesting to observe that each new physical theory is indeed “more symmetrical” (i.e. invariant under a larger group of transformations) – leading to (the already observed) microscopic “fearfull symmetry” for all known elementary particles interactions, and also macroscopically for gravitation.
9 December, 2018 at 2:32 pm
254A, Supplemental: Weak solutions from the perspective of nonstandard analysis (optional) | What's new
[…] a Notes 2, we reviewed the classical construction of Leray of global weak solutions to the Navier-Stokes […]
16 December, 2018 at 2:37 pm
255B, Notes 1: The Lagrangian formulation of the Euler equations | What's new
[…] 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: […]