The Navier–Stokes regularity problem

There is currently no proof guaranteeing that, given a smooth initial condition, the three-dimensional Navier–Stokes equations have a unique solution that exists for all positive times. This paper reviews the key rigorous results concerning the existence and uniqueness of solutions for this model. In particular, the link between the regularity of solutions and their uniqueness is highlighted. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.


The Clay Millennium Problem
The 3D Navier-Stokes equations (NSE) that model the flow of a viscous, incompressible fluid, are ∂ t u − ν∆u + (u · ∇)u + ∇p = 0, ∇ · u = 0, subject to an initial condition u(x, 0) = u 0 (x). Here u(x, t) denotes the time-dependent three-component fluid velocity, ν > 0 is the kinematic viscosity, and p is the scalar pressure. Throughout the rest of this paper, for notational simplicity ν is set to be equal to 1, which can be achieved by the rescaling uν (x, t) = νu(x, νt).
Usually the problem is posed on one of the following three domains: • a bounded domain Ω with Dirichlet boundary conditions u| ∂Ω = 0 (the 'most physical' case); • the whole space R 3 , requiring 'sufficient decay' at infinity (this makes the problem amenable to techniques from harmonic analysis, e.g. use of the Fourier transform); or • the three-dimensional torus T 3 , i.e. [0, 2π) 3 with periodic boundary conditions, and the convenient zero-momentum constraint T 3 u = 0 (an artificial but useful setting, since the domain here is both bounded and boundaryless).

The significance of rigorous existence and uniqueness results
The Navier-Stokes equations are well established as the canonical model of the flow of viscous incompressible fluids, and the starting point for countless theoretical and numerical investigations of fluid behaviour. As such, it is natural to ask what would be gained from a theorem guaranteeing the existence of unique solutions to these equations. There are a number of possible answers, some of which I hope are more convincing to those naturally sceptical about such things than the vague reply that it is unsatisfying not to have a rigorous proof.
• Lack of uniqueness would mean a lack of predictive power, so uniqueness of solutions is crucial for a 'good model'. However, it is also of interest to have a way to check that a particular solution is unique, even in the absence of a general uniqueness result, and there are ways to do this (see Sections 5(b) and 6).
• Without an existence result, what do numerical calculations approximate? Put more abstractly, how can one prove the convergence of a numerical scheme without being able to guarantee the existence of an underlying solution towards which to converge? (This is in fact possible under the assumption that a sufficiently smooth solution does exist, see Chernyshenko et al. [3].) • One could hope that a better mathematical understanding of the equations will lead to an enhanced physical understanding of the phenomena they model. (It is probably more reasonable to expect that a better mathematical understanding will require an enhanced physical understanding.) • Some very natural quantitative bounds on solutions are lacking, e.g. how to estimate the maximum value of the velocity in the future given the initial velocity profile. (In fact, bounds of this sort would be sufficient to solve the regularity problem, see Section 6.) • Theoretical methods to treat this problem should also apply to other PDE models. This is not entirely clear, however. There are artificial models that share many of the features of the NSE that have regular solutions (e.g. the dissipative Burgers equation [24]) and other such models that have solutions that blow up in finite time [22,36]. It may be that an eventual proof/disproof of the regularity of solutions of the NSE will involve such particular features of the equations that it says little about other models.
• Any such long-standing open mathematical question -particularly one so relevant to an important problem in physics -becomes something of a 'grand challenge', not to mention the financial incentive provided by the Clay Foundation.

Brief overview of the contents of this paper
The most important results on existence and uniqueness go back to Leray's work from 1934 [19], who treated the equations on the whole space (for a nice review of his argument in modern notation see Ożański & Pooley [23]). In 1951 Hopf [14] obtained similar results for the equations on a bounded domain. While we now know more about the properties of the class of solutions treated by Leray and Hopf, it seems not unfair to say that the resolution of the Regularity Problem is not significantly closer now than it was 85 years ago. Both Leray and Hopf proved the existence for all time of finite-energy 'weak solutions', i.e. solutions with Ω |u| 2 < ∞. However, these are not known to be unique. We discuss this class of solutions in Section 4. They also proved the existence for short times of smoother 'strong solutions' with finite 'enstrophy' (square integral of the vorticity: Ω |ω| 2 < ∞). While they exist these strong solutions are both smooth and unique; we cover these in Section 5.
Indeed, we will see that uniqueness of solutions is closely connected to their smoothness. It is therefore interesting to have results that guarantee the smoothness of a given solution when it satisfies some additional assumption. There are many such 'conditional regularity results'; one simple illustrative example is given in Section 6. Finally there is a class of (usually fairly sophisticated) results that, while not proving smoothness of solutions, serve to limit the size of the set of possible singularities of any weak solution (i.e. the Leray-Hopf solutions that are known to exist for all positive times). The first such results were proved by Scheffer [30,31], and then refined by a number of authors, in particular Caffarelli, Kohn, & Nirenberg [2]. These 'partial regularity' results probably give the most information that we currently have about solutions of the Navier-Stokes equations, and are briefly discussed in Section 7.

Weak solutions (finite kinetic energy): existence for all t ≥ 0
First we give an indication of why we should expect the existence of weak solutions, which are solutions with finite kinetic energy that satisfy the equations in an averaged sense.

(a) Evolution of the kinetic energy
Suppose that u is a smooth solution of the equations, posed on a bounded domain Ω with Dirichlet boundary conditions u| ∂Ω = 0, that exists for all t ≥ 0. If we want to follow the evolution of the kinetic energy then we can dot the equations with u and integrate in space: where here we have integrated by parts in the linear term and by |∇u| 2 we mean 3 i,j=1 |∂ i u j | 2 . Integrating by parts the nonlinear and pressure terms vanish 1 (again using the fact that u is divergence free and zero on the boundary). 1 In these calculations we use the Einstein summation convention and sum over repeated indices. Note that very similar Integrating in time from 0 to t yields which we can write more neatly as adopting the notation Such 'formal' computations as these (i.e. an argument in which we assume that the solutions are smooth, so that all the manipulations we make are rigorously allowed) lie at the basis of the proof of the existence of weak solutions.

(b) Definition of weak solutions and their existence
A function u is a weak solution of the Navier-Stokes equations if it satisfies 2)) and solves the Navier-Stokes equations in an averaged sense. To obtain this formulation we dot the equations with some smooth divergence-free function ϕ and integrate in space and time to obtain 2 For a weak solution we require this 'averaged equation' to hold for every smooth divergencefree function ϕ that is zero outside some compact subset of Ω. Since u satisfies the integrability conditions in (4.3) all the terms in (4.4) make sense. [There are many equivalent definitions of a weak solution, but this is probably the simplest. For example, one can take ϕ to be a function that also depends on time, or alternatively a finite linear combination of eigenfunctions of the linear Stokes operator, see [13] or Chapter 3 of [28], for details. In order to be meaningful all the various formulations require u to satisfy (4.3).] Note that, because we have integrated once by parts in the linear term, a weak solution only needs to be smooth enough to make sense of one derivative, and then only in an integrated sense (the required smoothness is precisely that in (4.3)). The fact that the formulation involves integration is significant: functions can be very bad and still integrable. For example, if (qn) ∞ n=1 is an enumeration of the rationals between 0 and 1 and while f is integrable (and so finite almost everywhere) it is unbounded in a neighbourhood of every rational.
To prove rigorously the existence of (weak) solutions of any partial differential equation there are two well established methods. One is to use an explicit representation formula (e.g. using the a Green's function to write down the solution of a linear PDE), and the other is to proceed via approximations, usually reducing the original PDE to a system of ODEs and then taking limits. For example, on a bounded domain one can make the formal estimates of Section 4(a) the basis of a rigorous existence proof as follows. First find the eigenfunctions {ψ j } ∞ j=1 and corresponding eigenvalues {λ j } ∞ j=1 of the Stokes equation, which form a complete orthonormal basis for L 2 (Ω). Then for each n look for an approximate solution expanded in terms of the {ψ j } n j=1 , where the coefficients {û n,k } n k=1 satisfy the truncated equations This is a finite system of locally Lipschitz ODEs (note that the c ijk are entirely determined by the Stokes eigenfunctions), so has a unique solution (at least while the solution remains bounded). The function un reconstructed from the coefficients in (4.6) satisfies similar estimates to those in Section 4(a): uniformly in n we have These uniform bounds (along with uniform bounds on dun/dt) allow us to use compactness results to find a subsequence un j that converges to a function u that one can then check really is a weak solution. All the more sophisticated mathematical argumentation is involved in what is quickly described in the previous sentence; with the details filled in, one can then guarantee the existence of at least one weak solution that exists for all t ≥ 0.

(c) The problem of uniqueness for weak solutions
Given the existence of a weak solution we now want to consider its uniqueness. Again, we give some 'formal' calculations to illustrate the problem that arises when trying to prove such uniqueness. Suppose that u and v are two weak solutions that arise from initial conditions u 0 and v 0 . The equation for their difference z : If we dot with z and integrate in space then we obtain  [17], If we split the terms on the right-hand side using the inequality 5 ab ≤ a 4 + b 4/3 and absorb the resulting term involving ∇z in the corresponding term on the left-hand side we obtain The key question for uniqueness of weak solutions (if we follow this analysis) is therefore whether if this quantity is finite then the estimate in (4.9) guarantees that z(t) = 0 whenever z 0 = 0. However, we do not know that this quantity is finite for an arbitrary weak solution: our requirement on a weak solution u in (4.3) is only that t 0 ∇u 2 L 2 < ∞, and it is easy to find a function whose square is integrable but not its fourth power, e.g. suppose that ∇u(s) 2 L 2 ∼ (t − s) −1/2 . So weak solutions do not have sufficient smoothness to guarantee their uniqueness.

(d) Evolution of the kinetic energy for weak solutions
We saw in Section 4(a) that smooth solutions of the NSE satisfy the energy equality (4.2). Leray showed that for any choice of (finite-energy) initial condition there exists at least one weak solution that satisfies the strong energy inequality for almost every s, including s = 0 (this is the class of 'Leray-Hopf weak solutions'). [The proof of this is quite involved when taking the equations on the whole space: in some ways Leray treated the hardest case first.] However, it is unknown whether all weak solutions must satisfy (4.10), even just for the choice s = 0. In other words, the existence of weak solutions in which the kinetic energy instantaneously increases cannot currently be ruled out. Of course, this is not a claim that a physical fluid system might spontaneously generate energy; rather, that with our current techniques it is not possible to rule this out for the usual mathematical model. A proof of the uniqueness of weak solutions would deal with this problem.

(e) Non-uniqueness of 'very weak' solutions
Non-uniqueness does occur for solutions of the Navier-Stokes equations, as Buckmaster & Vicol (2017) have recently shown. However, their proof treats 'very weak' solutions rather than 3 The proof is not difficult: the trick is to prove first a similar inequality in two variables, |f | 4 ≤ |f | 2 |∇f | 2 , starting from the identity |v(x1, x2)| 2 = 2 x k −∞ u∂ k u dx k for k = 1, 2, and then deduce the three-dimensional version. 4 In what follows c denotes an absolute constant that may change from line to line. 5 Young's inequality ab ≤ a p /p + b q /q is valid for any a, b ≥ 0 and 1 < p, q < ∞ with 1/p + 1/q = 1. Since x → e x is a convex function of x, we have ab = exp( 1 p log a p + 1 q log b q ) ≤ 1 p exp(log a p ) + 1 q exp(log b q ) = a p /p + b q /q.
In this formulation we no longer require any derivatives of u for the equation to make sense.
Buckmaster & Vicol's proof of non-uniqueness adapts the method of 'convex integration', first exploited in the context of fluid dynamics by De Lellis & Székelyhidi [5], to prove non-uniqueness of weak solutions for the Euler equations. Starting with an approximate solution the method is iterative, adding finer and finer oscillations that are controlled to produce a solution satisfying any given energy profile. (As such this method necessarily produces solutions with very little smoothness.) There is some suggestion of non-uniqueness for weak solutions (not only these 'very weak solutions') in work of Jia & Šverák: they show that non-uniqueness would follow from certain spectral properties of a particular linear operator, and present some numerical calculations that seem to indicate that these may indeed hold in [15].

Strong solutions (finite enstrophy)
We now show how a little smoothness of solutions is sufficient to guarantee their uniqueness. However, there is currently no proof of the long-time existence of solutions with this degree of regularity for large initial conditions.
(a) Evolution of the enstrophy 'Enstrophy' is the square integral of the vorticity ω := ∇ ∧ u, Strong solutions are (roughly speaking) weak solutions for which the enstrophy remains bounded. We again make some 'formal' calculations to give an idea of the expected evolution of the enstrophy for smooth solutions, just as we did for the kinetic energy in Section 4(a). We first note that controlling the enstrophy is enough is to control the square integral of all the first derivatives, since By taking the curl of the equations we obtain 6 an evolution equation for ω, If we dot with ω and integrate over space then proved above. Using the inequality ab ≤ a 4 + b 4/3 on the right-hand side we can absorb the term involving |∇ω| 2 in the corresponding term on the left-hand side, and so obtain Dropping the second term on the left-hand side it follows that the quantity X(t) := ω(t) 2 L 2 satisfies the differential inequalityẊ ≤ cX 3 . It is straightforward to integrate this to obtain the upper bound This shows that if the initial enstrophy is bounded (equivalently if ω 0 L 2 = ∇u 0 L 2 is bounded) then there is a time T (take any T < 1/2c ω 0 4 L 2 ) such that the enstrophy ω(t) 2 remains bounded for all t ∈ [0, T ]. We can therefore expect 'short time' (local in time) existence of finite-enstrophy solutions. While this argument uses the inequality in (5.3) to tell us something about the short-time existence of solutions, it also has implications for solutions whose existence cannot be extended to all positive times. Suppose that ω(t) L 2 → ∞ as t → T ; then starting the NSE from the initial condition ω 0 = ω(s) and using (5.3) yields letting t → T now yields the lower bound To prove rigorously the existence of strong solutions we proceed as with weak solutions, obtaining estimates for the solutions un of the approximating ODEs from Section 4(b) that parallel those in the previous section (5(a)) and then showing that these persist in the limit as n → ∞, i.e. for the resulting solution of the PDE. It follows that given ω 0 with ω 0 L 2 < ∞, there exists a time However, while strong solutions are unique, we cannot in general guarantee that they exist for all time, only for a short time that depends on the size of the initial enstrophy. Nevertheless, if the initial enstrophy is sufficiently small then we can prove the existence of strong solutions for all t ≥ 0. If we return to (5.2) and use the Poincaré inequality (cf. Rayleigh-Ritz principle) then we obtain 8 d dt Once more if we set X(t) = ω(t) 2 L 2 then this yields the differential inequalitẏ which guarantees that if cX 2 0 < λ 1 then cX(t) 2 < λ 1 for all t ≥ 0. This means that the enstrophy of solutions starting with ω 0 2 L 2 < λ 1 /c remains finite for all t ≥ 0, i.e. for 'small initial data' a unique strong solution will exist for all time.
[If u(x, t) is a solution of the equations on R 3 with u(x, 0) = u 0 (x) then so is the rescaled solution u α (x, t) := αu(αx, α 2 t), which arises from the rescaled initial condition u α 0 := αu 0 (αx). Since the solution u exists for all t ≥ 0 if and only if the solution u α does, if one is not to gain 'something for nothing', the measure of 'smallness' of the initial data should be in terms of some quantity that does not change under this rescaling. Spaces in which the natural norm have this scaling-invariance property are termed 'critical', and much of the recent progress in the analysis of the NSE has been in extending arguments similar to those in this section (short-time existence when the initial condition is 'large' and existence for all t ≥ 0 when the initial condition is 'small') to families of spaces of this type, see the book by Lemarié-Rieusset [20], for example.]

Conditional regularity
Strong solutions have many good properties in addition to their uniqueness. For example, if a solution is strong on [0, T ], i.e. once we know that the enstrophy is finite for all t ∈ [0, T ], then the solution is actually smooth for all t ∈ (0, T ], so for all 0 < t ≤ T the equations are satisfied in the classical sense (see Theorem 10.6 in [10], Chapter 7 of [28], or Section 6 in [13]). In fact, in the periodic case, such a solution is even analytic, see [12].
This motivates the search for other 'conditional' regularity results: given a weak solution u starting from smooth initial data, what additional properties will ensure that it is in fact a smooth solution? The statement that a strong solution is smooth can be interpreted in this light, where the 'additional property' of the weak solution is that its enstrophy is bounded. Here we give another simple example of a conditional regularity result: we show that the assumption that is sufficient to ensure that u is a strong solution on [0, T ], and hence smooth (and unique). Note that (6.1) certainly holds if for some As ever, we proceed with a formal calculation, taking the inner product of the equations with −∆u; then we obtain recalling that ω L 2 = ∇u L 2 this implies that Given our assumption in (6.1) the argument of the exponential is uniformly bounded over all t ∈ [0, T ] and so the solution is strong on [0, T ]. This is one 'endpoint' (r = 2, s = ∞) of a family of conditional regularity/uniqueness results, due severally to Ladzyhendskaya [18], Prodi [25], and Serrin [32,33]: in particular, Ladyzhenskaya showed that if u is a weak solution that in addition satisfies the Ladyzhenskaya-Prodi-Serrin condition u ∈ L r (0, T ; L s ), i.e.

Partial regularity
Finally, we discuss 'partial regularity results', which limit the size of the set of possible singularities of weak solutions (or a subclass of weak solutions).

(a) The set of singular times
Scheffer [30] showed, using a result of Leray (!), that the set T of singular times has Hausdorff dimension no larger than 1/2, dim H (T ) ≤ 1/2. We give the proof of a similar result for the boxcounting dimension here, since it is very simple but nevertheless provides a model of the proof of a much more powerful result about space-time singularities that we discuss shortly. It combines the fact that near any 'singular time' T the solution must blow up at the particular rate in (5.4) with the bound ∞ 0 ∇u(s) 2 < ∞, valid for any weak solution. The finiteness of the integral means that there cannot be too many times T near which the lower bound holds.
There are number of equivalent definitions of the (upper) box-counting dimension (see [9]), but the most convenient to use here is where N (A, ε) denotes the maximal number of ε-separated points in Let T be the set of singular times, and suppose that dim B (T ) = d > 1/2; then taking δ with this leads to a contradiction as j → ∞ since the integral on the left-hand side is finite.
(b) The set of space-time singularities Scheffer [31] also proved the first partial regularity result for the set S of all space-time singularities, i.e. the set of all 'singular points' (x, t) for which u(x, t) is not bounded in an open 9 For a solution u with T 0 Ω |u(x, t)| 4 < ∞, Prodi [25] proved uniqueness under this condition; in [33] Serrin showed that under this condition alone a solution u is unique in the class of all weak solutions that satisfy the energy inequality. In [32] Serrin proved a local conditional regularity result: if u ∈ L r (t1, t2; L s (U )) for some 2/r + 3/s < 1 then u is smooth in space in the space-time domain (t1, t2) × U ; the proof of this result is much harder (and that in the case 2/r + 3/s = 1, due to Fabes, Jones, & Rivière [8] and Struwe [34], harder still).  [2] improved this to show that dim H (S) ≤ 1. These results are valid for a weak solutions that also satisfy a localised version of the energy inequality, termed 'suitable weak solutions' in [2]. The argument there shows that if (x, t) is a singular point then the solution u must satisfy certain lower bounds on its rate of blow-up near (x, t); as in the previous section this can be coupled with the fact that certain integrals of the solution are finite to obtain bounds on the dimension of S. (For a simplified version of their argument, neglecting the pressure, see [29]. ) Robinson & Sadowksi showed in [27] that it is an easy corollary of the results in [2] that the boxcounting dimension of the singular set is no larger than 10 5/3, and as a consequence were able to prove that for any particular suitable weak solution there is a unique particle trajectory for almost every initial condition: even when the solution is only 'weak', the Lagrangian formulation of the dynamics still makes sense. (The proof combines an argument due to Foias, Guillopé, & Temam [11] that produces a volume-preserving particle flow for any weak solution, with a dynamical result that for almost any initial condition such a flow must avoid the singular set. The almosteverywhere uniqueness follows relatively easily.)

Conclusion
We end with a quick summary of the above discussion. The key fact to take away is that proving the uniqueness of solutions depends upon being able to show that they are sufficiently smooth. There is therefore a natural hierarchy of solutions, whose properties improve with their regularity.
• Very weak solutions are not unique.
• Weak (finite-energy) solutions may not be unique, but exist for all time. They enjoy various 'partial regularity' properties, which are, for example, sufficient to show that 'particle paths' exist for almost all initial conditions, so that a Lagrangian approach is valid even for weak solutions. This is an avenue that deserves to be explored further.
• Strong (finite-enstrophy) solutions are unique but can be guaranteed to exist for only a short time, unless the initial enstrophy is sufficiently small.
As mentioned in Section 2, given that there are models that share many features in common with the Navier-Stokes equations, some of which form singularities in a finite time and others for which there is global existence of regular solutions, one can expect that a solution of the Regularity Problem will require a better understanding of the fine structure of interactions between modes, as embodied in the equation in Fourier space, (cf. (4.7)); if this understanding reflects some physical insight so much the better. Note that this structure is encoded in the eigenvalues λ j and the coefficients c ijk , which are entirely determined by the spectral properties of the linear Stokes problem in (4.5).
Competing Interests. The author declares that he has no competing interests.