Sharp nonuniqueness for the Navier–Stokes equations

Abstract

In this paper, we prove a sharp nonuniqueness result for the incompressible Navier–Stokes equations in the periodic setting. In any dimension $d \ge 2$ and given any $p<2$, we show the nonuniqueness of weak solutions in the class $L^{p}_t L^\infty$, which is sharp in view of the classical Ladyzhenskaya–Prodi–Serrin criteria. The proof is based on the construction of a class of non-Leray–Hopf weak solutions. More specifically, for any $p<2$, $q<\infty$, and $\varepsilon >0$, we construct non-Leray–Hopf weak solutions $u \in L^{p}_t L^\infty \cap L^1_t W^{1,q}$ that are smooth outside a set of singular times with Hausdorff dimension less than $\varepsilon$. As a byproduct, examples of anomalous dissipation in the class $L^{ {3}/{2} - \varepsilon }_t C^{ {1}/{3}}$ are given in both the viscous and inviscid case.

Appendices

Appendix A: $X^{p,q}$ weak solutions on the torus

In this section, we show that sub-critical and critical weak solutions in $X^{p,q}([0,T] ; {\mathbb {T}}^d)$ are in fact Leray–Hopf. In particular, by the weak-strong uniqueness of Ladyzhenskaya–Prodi–Serrin, this implies the uniqueness part of Theorem 1.3. The content of Theorem A.1 is classical [32, 35, 46, 53] and we include a proof in the regime $q>2$ for the convenience of the readers. Note that the proof applies to the case $q=\infty$ which is most relevant to the results of this paper, but was omitted in [32].

Theorem A.1

Let $d \ge 2$ be the dimension and $u \in X^{p,q}([0,T] ; {\mathbb {T}}^d)$ be a weak solution of (1.1) with $\frac{2}{p} + \frac{d}{q} = 1$, $d \le q \le \infty$. Then u is a Leray–Hopf solution.

We prove Theorem A.1 for $q>2$ only. The case $d=q=2$, as discussed in [38], can be handled by the argument of [35] in 2D. The method we present here follows the duality approach in [53] and use only classical ingredients.

The first ingredient is an existence result for a linearized Navier–Stokes equation.

Theorem A.2

Let $u \in X^{p,q}([0,T] ; {\mathbb {T}}^d)$ be a weak solution of (1.1) with $\frac{2}{p} + \frac{d}{q} = 1$. For any divergence-free $v_0 \in L^2({\mathbb {T}}^d)$, there exists a weak solution $v \in C_w L^2 \cap L^2_t H^1$ to the linearized Navier–Stokes equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v -\Delta v + u \cdot \nabla v + \nabla p = 0 &{}\\ {{\,\mathrm{div}\,}}v =0, \end{array}\right. } \end{aligned}$$

(A.1)

satisfying the energy inequality

$$\begin{aligned} \frac{1}{2}\Vert v(t)\Vert _2^2+ \int _{t_0}^t \Vert \nabla v(s)\Vert _2^2 \, ds\le \frac{1}{2}\Vert v(t_0)\Vert _2^2 , \end{aligned}$$

for all $t\in [t_0,T]$, a.e. $t_0\in [0,T]$ (including $t_0=0$).

Proof

This follows by a standard Galerkin method and can be found in many textbooks. See [67, Chapter 4] or [36] for details. $\square$

Let v be the weak solution given by Theorem A.2 with initial data u(0). The goal is to show $u \equiv v$. Setting $w = u -v$, the equation for w reads

$$\begin{aligned} \partial _t w -\Delta w + u \cdot \nabla w + \nabla q = 0 , \end{aligned}$$

and its weak formulation

$$\begin{aligned} \int _0^T \int _{{\mathbb {T}}^d} w\cdot ( \partial _t \varphi + \Delta \varphi + u\cdot \nabla \varphi ) \, dx dt =0 \quad \text {for any }\varphi \in {\mathcal {D}}_T, \end{aligned}$$

(A.2)

where we recall that the test function class ${\mathcal {D}}_T$ consists of smooth divergence-free functions vanishing for $t \ge T$.

Fix $F \in C^\infty _c([0,T] \times {\mathbb {T}}^d)$. Let $\Phi :[0,T]\times {\mathbb {T}}^d \rightarrow {\mathbb {R}}^d$ and $\chi :[0,T]\times {\mathbb {T}}^d \rightarrow {\mathbb {R}}$ satisfy the system of equations

$$\begin{aligned} {\left\{ \begin{array}{ll} -\partial _t \Phi -\Delta \Phi - u \cdot \nabla \Phi + \nabla \chi = F&{}\\ {{\,\mathrm{div}\,}}\Phi =0 &{}\\ \Phi (T) =0. \end{array}\right. } \end{aligned}$$

(A.3)

Note that the equation of $\Phi$ is “backwards in time” and by a change of variable one can convert (A.3) into a more conventional form.

If we can use $\varphi =\Phi$ as the test function in the weak formulation (A.2), then immediately

$$\begin{aligned} \int _{[0,T] \times {\mathbb {T}}^d} w \cdot F \, dx dt = 0 . \end{aligned}$$

Since $F \in C^\infty _c([0,T] \times {\mathbb {T}}^d)$ is arbitrary, we have

$$\begin{aligned} w = 0 \quad \text {for }a.e. (t,x) \in [0,T] \times {\mathbb {T}}^d. \end{aligned}$$

So the question of whether $u \equiv v$ reduces to showing a certain regularity of $\Phi$. More specifically, we can prove the following theorem.

Theorem A.3

Let $d \ge 2$ be the dimension and $u \in X^{p,q}([0,T] ; {\mathbb {T}}^d)$ be a weak solution of (1.1) with $\frac{2}{p} + \frac{d}{q} = 1$ with $q>2$. For any $F \in C^\infty _c([0,T] \times {\mathbb {T}}^d)$, the system (A.3) has a weak solution $\Phi \in L^\infty _t L^2 \cap L^2_t H^1$ such that $\Phi$ can be used as a test function in (A.2).

Proof

We will prove the weak solution $\Phi$ satisfies the regularity

$$\begin{aligned} \partial _t \Phi ,\Delta \Phi ,u\cdot \nabla \Phi , \nabla \chi \in L^2_{t,x} , \end{aligned}$$

(A.4)

which implies $\Phi$ can be used in (A.2) since $w \in L^2_{t,x}$.

The solution $\Phi$ will be constructed by the Galerkin method of the following finite-dimensional approximation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\partial _t \Phi _n -\Delta \Phi _n - P_n\big [ u_n \cdot \nabla \Phi _n\big ] + \nabla \chi _n = F_n &{}\\ {{\,\mathrm{div}\,}}\Phi _n =0 &{}\\ \Phi _n (T) =0, \end{array}\right. } \end{aligned}$$

(A.5)

where $\Phi _n, u_n, \chi _n, F_n$ are restricted to the first n Fourier modes, and $P_n$ is the projection operator on those modes.

It suffices to verify the following a priori estimates as they are preserved in the limit as $n \rightarrow \infty$. We also only need to show the estimates for $\partial _t \Phi$, $\Delta \Phi$, and $u\cdot \nabla \Phi$ since the pressure $\chi$ satisfies the equation

$$\begin{aligned} \Delta \chi = {{\,\mathrm{div}\,}}( u \cdot \nabla \Phi )+ {{\,\mathrm{div}\,}}F. \end{aligned}$$

Step 1: Energy bounds $L^\infty _t L^2 \cap L^2_t H^1$.

The solutions to the Galerkin approximation (A.5) enjoy the energy estimate

$$\begin{aligned} -\frac{1}{2} \frac{d}{dt} \Vert \Phi \Vert _2^2 + \Vert \nabla \Phi \Vert _2^2 \le \int _{{\mathbb {T}}^d} |F \cdot \Phi |\,dx, \end{aligned}$$

which implies the desired energy bounds.

Step 2: Higher bounds $L^\infty _t H^1 \cap L^2_t H^2$.

We now take the $L^2$ inner product of (A.5) with $\Delta \Phi$ to obtain

$$\begin{aligned} -\frac{1}{2} \frac{d}{dt} \Vert \nabla \Phi \Vert _2^2 + \Vert \Delta \Phi \Vert _2^2 \le \int _{{\mathbb {T}}^d} |F \cdot \Delta \Phi |\,dx + \int _{{\mathbb {T}}^d} |u \cdot \nabla \Phi \cdot \Delta \Phi |\,dx. \end{aligned}$$

• Case 1: $p<\infty$ and $d<q\le \infty$.

  Thanks to Proposition A.4,

  $$\begin{aligned} \int _{{\mathbb {T}}^d} |u \cdot \nabla \Phi \cdot \Delta \Phi |\,dx \lesssim \Vert u \Vert _q \Vert \nabla \Phi \Vert _2^{ \frac{q-d}{q}} \Vert \Delta \Phi \Vert _2^{ \frac{q+d}{q}}. \end{aligned}$$

  (A.6)

  Then Hölder’s, Poincaré’s, and Young’s inequalities yield

  $$\begin{aligned} -\frac{1}{2} \frac{d}{dt} \Vert \nabla \Phi \Vert _2^2 + \frac{1}{2}\Vert \Delta \Phi \Vert _2^2 \lesssim \Vert u\Vert _q^{\frac{2q}{q-d}}\Vert \nabla \Phi \Vert _2^2 + \Vert F\Vert _2^2. \end{aligned}$$

  Thanks to the integrability of $\Vert u\Vert _q^{\frac{2q}{q-d}}$ ($\Vert u\Vert _\infty ^2$ when $q=\infty$), Gronwall’s inequality immediately implies that $\Phi \in L^\infty _t H^1 \cap L^2_t H^2$.

• Case 2: $p=\infty$ and $q=d > 2$.

  Since $u \in C_t L^{d}$, for any $\varepsilon >0$ there exists a decomposition $u = u_1 +u_2$ such that

  $$\begin{aligned} \Vert u_1 \Vert _{C_t L^{d} } \le \varepsilon \quad \text { and }\quad u_2 \in L^\infty _{t,x}, \end{aligned}$$

  and then the $u_1$ portion of the nonlinear term in (A.6) can be absorbed by $\Vert \Delta \Phi \Vert _2^2$, so we arrive at the same conclusion.

Step 3: Conclusion from maximal regularity of the heat equation.

Taking Leary’s projection ${\mathbb {P}}$ onto the divergence-free vector fields, the equation for $\Phi$ can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} -\partial _t \Phi -\Delta \Phi = {\mathbb {P}} (u \cdot \nabla \Phi ) + {\mathbb {P}} F&{}\\ \Phi (T) =0. \end{array}\right. } \end{aligned}$$

Therefore, by the maximal $L^p_t L^q$ regularity of the heat equation (see for instance [67, Theorem 5.4]), we only need to show the estimates (A.4) for $u \cdot \nabla \Phi$ to conclude the proof. By Sobolev interpolations we have $\nabla \Phi \in L^r L^s$ for any $\frac{2}{r} + \frac{d}{s} =\frac{d}{2}$ such that $2\le s \le \frac{2d}{d-2}$. We can find r, s in this regime that satisfy the Hölder relations $\frac{1}{p} + \frac{1}{r} = \frac{1}{2}$ and $\frac{1}{q} + \frac{1}{s} = \frac{1}{2}$, where recall $\frac{2}{p} + \frac{d}{q} =1$. Since $u \in L^{p}_t L^q$ with some $\frac{2}{p} + \frac{d}{q} =1$, this choice of r, s implies that

$$\begin{aligned} u \cdot \nabla \Phi \in L^2([0,T] \times {\mathbb {T}}^d ) . \end{aligned}$$

$\square$

The last result is a classical estimate of the nonlinear term, cf. [67, pp. 172]. Notice that one of the embedding fails when $q=2$ which is the reason we can only prove Theorem A.3 for $q > 2$.

Proposition A.4

Let $d \ge 2$ be the dimension and $d \le q \le \infty$ such that $q > 2$. For any smooth vector fields $u,v \in C^\infty _0({\mathbb {T}}^d)$,

$$\begin{aligned} \int _{{\mathbb {T}}^d} |u \cdot \nabla v \cdot \Delta v | \,dx \lesssim \Vert u \Vert _{q} \Vert \nabla v \Vert _{2}^{\frac{q-d}{q}} \Vert \Delta v \Vert _{2}^{\frac{q+d}{q}}. \end{aligned}$$

Proof

We apply Hölder’s inequality with exponents $\frac{1}{q} + \frac{1}{r} + \frac{1}{2} =1$, $r = \frac{2q}{q-2} \in [2 , \frac{2d}{d-2})$,

$$\begin{aligned} \int _{{\mathbb {T}}^d} |u \cdot \nabla v \cdot \Delta v | \,dx \lesssim \Vert u \Vert _{q} \Vert \nabla v \Vert _{r} \Vert \Delta v \Vert _{ 2} . \end{aligned}$$

Since $2 \le r< \infty$, by the Sobolev embedding $H^{s}({\mathbb {T}}^d ) \hookrightarrow L^{ r} ({\mathbb {T}}^d )$, $s = d( \frac{1}{2} - \frac{1}{r})$,

$$\begin{aligned} \Vert \nabla v \Vert _{r} \lesssim \Vert \nabla v \Vert _{H^{s}} . \end{aligned}$$

Finally, by a standard Sobolev interpolation and the $L^2$-boundedness of Riesz transform, we have

$$\begin{aligned} \Vert \nabla v \Vert _{H^{s}} \lesssim \Vert \nabla v \Vert _{2}^{\frac{q- d}{q}}\Vert \Delta v \Vert _{2}^{\frac{d}{q} }, \end{aligned}$$

which concludes the proof. $\square$

Appendix B: Some technical tools

1.1 Improved Hölder’s inequality on ${\mathbb {T}}^d$

We recall the following result due to Modena and Székelyhidi [60], which was inspired by [11, Lemma 3.7]. This lemma allows us to quantify the decorrelation in the usual Hölder’s inequality when we increase the oscillation of one function.

Lemma B.1

Let $p \in [1,\infty ]$ and $a,f :{\mathbb {T}}^d \rightarrow {\mathbb {R}}$ be smooth functions. Then for any $\sigma \in {\mathbb {N}}$ ,

$$\begin{aligned} \Big | \Vert a f(\sigma \cdot ) \Vert _{p } - \Vert a \Vert _{p} \Vert f \Vert _{p } \Big |\lesssim \sigma ^{-\frac{1}{p}} \Vert a\Vert _{C^1} \Vert f \Vert _{ p }. \end{aligned}$$

(B.1)

The proof is based on the interplay between the Poincare’s inequality and the fast oscillation of $f (\sigma \cdot )$ and can be found in [60, Lemma 2.1].

1.2 Tensor-valued antidivergence ${\mathcal {R}}$

For any $f\in C^\infty ({\mathbb {T}}^d)$, there exists a $v \in C^\infty _0({\mathbb {T}}^d)$ such that

And we denote v by $\Delta ^{-1}f$. Note that if $f\in C^\infty _0({\mathbb {T}}^d)$, then by rescaling we have

$$\begin{aligned} \Delta ^{-1} \big ( f(\sigma \cdot ) \big ) = \sigma ^{-2} v(\sigma \cdot ) \quad \text { for }\sigma \in {\mathbb {N}}. \end{aligned}$$

We recall the following antidivergence operator ${\mathcal {R}}$ introduced in [26].

Definition B.2

${\mathcal {R}} : C^\infty ({\mathbb {T}}^d ,{\mathbb {R}}^d) \rightarrow C^\infty ({\mathbb {T}}^d, {\mathcal {S}}^{d \times d }_0)$ is defined by

$$\begin{aligned} ({\mathcal {R}} v )_{ij} = {\mathcal {R}}_{ijk} v_k \end{aligned}$$

(B.2)

where

$$\begin{aligned} {\mathcal {R}}_{ijk} = \frac{2-d}{d-1}\Delta ^{-2} \partial _i \partial _j \partial _k - \frac{1}{d-1}\Delta ^{-1} \partial _k \delta _{ij} + \Delta ^{-1} \partial _i \delta _{jk} + \Delta ^{-1} \partial _j \delta _{ik} . \end{aligned}$$

It is clear that ${\mathcal {R}}$ is well-defined since ${\mathcal {R}}_{ijk}$ is symmetric in i, j and taking the trace gives

$$\begin{aligned} {{\,\mathrm{Tr}\,}}{\mathcal {R}} v&= \frac{2-d}{ d -1} \Delta ^{-1} \partial _k v_k + \frac{-d}{d-1} \Delta ^{-1} \partial _k v_k + \Delta ^{-1} \partial _k v_k + \Delta ^{-1} \partial _k v_k\\&=(\frac{2-d}{ d -1} + \frac{-d}{d-1}+ 2) \Delta ^{-1} \partial _k v_k=0. \end{aligned}$$

By a direct computation, one can also show that

and

$$\begin{aligned} {\mathcal {R}} \Delta v = \nabla v + \nabla v ^T \quad \text {for any divergence-free }v \in C^\infty ({\mathbb {T}}^d ,{\mathbb {R}}^d). \end{aligned}$$

(B.3)

We can show that ${\mathcal {R}}$ is bounded on $L^{p} ({\mathbb {T}}^d)$ for any $1\le p \le \infty$.

Theorem B.3

Let $1 \le p \le \infty$. For any vector field $f \in C^\infty ({\mathbb {T}}^d,{\mathbb {R}}^d)$, there holds

$$\begin{aligned} \Vert {\mathcal {R}} f \Vert _{L^{p}({\mathbb {T}}^d )} \lesssim \Vert f \Vert _{L^{p}({\mathbb {T}}^d )}. \end{aligned}$$

In particular, if $f \in C^\infty _0({\mathbb {T}}^d,{\mathbb {R}}^d)$, then

$$\begin{aligned} \Vert {\mathcal {R}} f(\sigma \cdot ) \Vert _{L^{p}({\mathbb {T}}^d )} \lesssim \sigma ^{-1}\Vert f \Vert _{L^{p}({\mathbb {T}}^d )} \quad \text {for any }\sigma \in {\mathbb {N}}. \end{aligned}$$

Proof

Once the first bound is established, the second bound follows from the definition of ${\mathcal {R}}$. It suffices to only consider f with zero mean since ${\mathcal {R}} (C) =0$ for any constant C. Then we only need to show that the operator

$$\begin{aligned} \Delta ^{-1} \Delta ^{-1} \partial _i \partial _j \partial _k \end{aligned}$$

is bounded on $L^p({\mathbb {T}}^d)$ for $1\le p \le \infty$ since the argument applies also to $\Delta ^{-1} \partial _i$.

When $1< p < \infty$, this follows from the boundedness of the Riesz transforms and the Poincare inequality.

When $p= \infty$, the Sobolev embedding $W^{1,d+1}({\mathbb {T}}^d) \hookrightarrow L^{\infty }({\mathbb {T}}^d )$ implies that

$$\begin{aligned} \Vert \Delta ^{-1} \Delta ^{-1} \partial _i \partial _j \partial _k f \Vert _{L^\infty ({\mathbb {T}}^d) }\lesssim & {} \Vert \Delta ^{-1} \Delta ^{-1} \partial _i \partial _j \partial _k f \Vert _{ W^{1,d+1} ({\mathbb {T}}^d) } \\\lesssim & {} \Vert f \Vert _{L^{d + 1}({\mathbb {T}}^d) }\le \Vert f \Vert _{L^{\infty } ({\mathbb {T}}^d) } \end{aligned}$$

where we have used the boundedness of the Riesz transforms once again.

When $p= 1$, one can use a duality approach and use the boundedness in $L^\infty$ since integrating by parts yields

$$\begin{aligned} \langle \Delta ^{-1} \Delta ^{-1} \partial _i \partial _j \partial _k f ,\varphi \rangle = -\langle f ,\Delta ^{-1} \Delta ^{-1} \partial _i \partial _j \partial _k \varphi \rangle \quad \text { if } \varphi \in C^\infty _0({\mathbb {T}}^d). \end{aligned}$$

$\square$

1.3 Bilinear antidivergence ${\mathcal {B}}$

We can also introduce the bilinear version ${\mathcal {B}} : C^\infty ({\mathbb {T}}^d, {\mathbb {R}}^d) \times C^\infty ({\mathbb {T}}^d ,{\mathbb {R}}^{d\times d} ) \rightarrow C^\infty ({\mathbb {T}}^d, {\mathcal {S}}^{d \times d}_0)$ of ${\mathcal {R}}$. This bilinear antidivergence ${\mathcal {B}}$ allows us to gain derivative when the later argument has zero mean and a small period.

Let

$$\begin{aligned} ( {\mathcal {B}}( v,A ))_{i j} = v_l {\mathcal {R}}_{ijk}A_{lk} - {\mathcal {R}}( \partial _i v_l {\mathcal {R}}_{ijk}A_{lk} ) \end{aligned}$$

or by a slight abuse of notations

$$\begin{aligned} {\mathcal {B}}( v,A ) = v {\mathcal {R}} A - {\mathcal {R}}( \nabla v {\mathcal {R}} A ) . \end{aligned}$$

Theorem B.4

Let $1 \le p \le \infty$. For any $v \in C^\infty ({\mathbb {T}}^d,{\mathbb {R}}^d)$ and $A \in C^\infty _0({\mathbb {T}}^d,{\mathbb {R}}^{d\times d})$,

(B.4)

and

$$\begin{aligned} \Vert {\mathcal {B}} (v ,A) \Vert _{L^{p}({\mathbb {T}}^d )} \lesssim \Vert v \Vert _{C^{1}({\mathbb {T}}^d )} \Vert {\mathcal {R}} A \Vert _{L^{p}({\mathbb {T}}^d )}. \end{aligned}$$

Proof

A direct compuation gives

where we have used the fact that A has zero mean and ${\mathcal {R}}$ is symmetric.

Integrating by parts, we have

which implies that

The second estimate follows immediately from the definition of ${\mathcal {B}}$ and Theorem B.3. $\square$