Non-uniqueness for the Navier–Stokes
initial value problem

velocity field

Julien Guillod

Université Paris-Diderot

August 23, 2017

Joint work with
Vladimír Šverák

vorticity

1. Introduction

Navier–Stokes equations

$$\begin{cases} \partial_{t}\bu+\bu\bcdot\bnabla\bu=\Delta\bu-\bnabla p\\ \bnabla\bcdot\bu=0\\ \bu(0)=\bu_{0} \end{cases} \quad \text{in} \quad \mathbb{R}^3\tag{NS}$$
Main question
Well-posedness of the Cauchy problem (existence, uniqueness, regularity)
Main two methods
  • energy method (a priori bounds)
  • perturbation method (Banach fixed point)

Energy method

  • A priori bound on the energy:$$\left\Vert \bu(t)\right\Vert _{L^{2}}^{2}+2\int_{0}^{t}\left\Vert \bnabla\bu(\tau)\right\Vert _{L^{2}}^{2}\rd\tau\leq\left\Vert \bu_0\right\Vert _{L^{2}}^{2}$$
  • Leray–Hopf solution on $(0,T)$: $$\bu \in L^{\infty}(0,T;L^{2})\cap L^{2}(0,T;\dot{H}^{1})$$ satisfying (NS) in the sense of distribution and the energy inequality.
Theorem (Leray, 1934)
For any $\bu_0\in L^2$, existence of a global Leray–Hopf solution ($T=\infty$). The regularity and uniqueness of these weak solutions are open.

Perturbation method

  • Inversion of the linearity on the nonlinearity: $$\bu(t)=\e^{\Delta t}\bu_0+\int_{0}^{t}\e^{\Delta(t-\tau)}\mathbb{P}\bigl(\bu(\tau)\bcdot\bnabla\bu(\tau)\bigr)\,\rd\tau$$
Theorem (Kato, 1984)
For $q\geq3$ and $\bu_0\in L^q$, existence of a local solution $\bu\in L^\infty(0,T;L^q)$, which is global if $\bu_0$ is small.
Theorem (Prodi, 1959; Serrin, 1963;
Ladyzhenskaya, 1967; Escauriaza et al., 2003)
If $\bu\in L^{p}(0,T;L^{q})$ with $\frac{2}{p}+\frac{3}{q}\leq1$, then $\bu$ is smooth and unique on $(0,T)$.

Results in $L^{p}(0,T;L^{q})$

LpLq critical spaces
Scaling symmetry
$$\begin{aligned}\bu_{\lambda}(t,\bx) & =\lambda\bu(\lambda^{2}t,\lambda\bx)\\ p_{\lambda}(t,\bx) & =\lambda^{2}p(\lambda^{2}t,\lambda\bx)\\ \bu_{0\lambda}(\bx) & =\lambda\bu_{0}(\lambda\bx)\end{aligned}$$
Critical space
Banach space $X$ of functions $(0,\infty)\times\mathbb{R}^3\to\mathbb{R}^3$ such that $$\Vert\bu_\lambda\Vert_X = \Vert\bu\Vert_X$$

Scale-invariant solutions

  • Scale-invariant initial data: $$u_{0\lambda}=u_0 \quad\textit{i.e.}\quad u_0(\bx)=\frac{\varphi(\hat{\bx})}{|\bx|}$$
  • Scale-invariant solutions: $$u_{\lambda}=u \quad\textit{i.e.}\quad u(t,\bx)=\frac{1}{t^{1/2}}U\Bigl(\frac{\bx}{t^{1/2}}\Bigr)$$
  • Ansatz into (NS):
    $$\begin{cases} \Delta\bU+\frac{\bx}{2}\bcdot\bnabla\bU+\frac{1}{2}\bU-\bU\bcdot\bnabla\bU-\bnabla P=\bzero\\ \bnabla\bcdot\bU=0\\ \bU(\bx)=\bu_{0}(\bx)+o(\left|\bx\right|^{-1})\quad\text{as}\quad\left|\bx\right|\to\infty \end{cases}\tag{NS-self}$$
  • Short form: $$\text{(NS-self)} \quad\Longleftrightarrow\quad F(\bU)=\bzero$$

Results on scale-invariant solutions

Theorem (Koch & Tataru, 2001)
For $\bu_0\in BMO^{-1}$ small enough, local existence of a unique solution. In particular, for $\bu_0$ small and scale-invariant, existence of a unique global self-similar solution.
Theorem (Jia & Šverák, 2013)
For $\bu_0\in C^\infty(\mathbb{R}^3\setminus\{\bzero\})$ scale-invariant, there exists at least one scale-invariant solution $\bu\in C^\infty((0,\infty)\times\mathbb{R}^3)$.
Intuition behind the proofs
  1. small data ⇝ perturbation method ⇝ uniqueness
  2. large data ⇝ a priori bound ⇝ uniqueness unknown

2. Results

Aim and Strategy

Aim
Find a self-similar $\bu_0$ such that the Navier–Stokes equations admit at least two different self-similar solutions.
Strategy
  1. Work within the class of axisymmetric solutions (with swirl)
  2. Choose a self-similar initial data with pure swirl: $$\bu_0(r,z) = \sigma(r^{2}+z^{2})^{-1/2}f(z/r)\,\be_{\theta}$$
  3. Find a numerical solution $\bU_\sigma$ by continuation in $\sigma\geq0$
  4. Numerically compute the spectrum of $DF(\bU_\sigma)$ for each $\sigma\geq0$
  5. Resolve the solutions possibly bifurcating form $\bU_\sigma$

Numerical solution $\bU_\sigma$

solution Usigma

Spectrum of $DF(\bU_\sigma)$

spectrum

Bifurcation

$\sigma=\,$0

Bifurcated solution

bifurcated solution

Nonuniqueness results

Nonuniqueness numerical results (G. & Šverák, 2017)
There exists two different smooth solutions in $L^\infty(0,T;L^{3,\infty})$ with the same scale-invariant initial data $\bu_0\in L^{3,\infty}\subset BMO^{-1}$.
Theorem (G. & Šverák, 2017, inspired by Jia & Šverák, 2015)
Under a spectral assumption verified numerically, there exists two different smooth Leray–Hopf solutions with the same compactly supported initial data $\bu_{0}\in C^{\infty}(\mathbb{R}^{3}\setminus\{\bzero\})$ with $\bu_{0}(\bx)=O(\left|\bx\right|^{-1})$ near the origin. Moreover these two solutions belongs to $L^{p}(0,T;L^{q})$ for any $$\frac{2}{p}+\frac{3}{q}>1\qquad\text{and}\qquad q\geq2\,.$$

Supercritical implies nonuniqueness

LpLq critical spaces
Corollary
The known perturbation results are essentially optimal:
the Navier–Stokes equations are locally ill-posed in supercritical spaces.
Remark
Not the Millennium problem, which is about the regularity of global solutions.

3. Speculations

Philosophy

Hypothesis
Maybe what is not forbidden is allowed?
Examples on positive side
  • Double exponential growth for 2D Euler (Kiselev & Šverák, 2014)
  • Onsager conjecture for 3D Euler (Isett, 2016)
Examples on negative side
  • Global well-posedness for Burgers (Kiselev & Ladyzhenskaya, 1957)
  • Global well-posedness for Template matching (Plecháč & Šverák, 2003)

Plausible scenario

speculation