On the nonuniqueness
of Leray solutions

velocity field

Julien Guillod

logo Sorbonne Université

April 17, 2019

Joint work with
Vladimír Šverák

velocity field

1. Introduction

Navier–Stokes equations

$$\begin{cases} \partial_{t}\bu+\bu\bcdot\bnabla\bu=\Delta\bu-\bnabla p\\ \bnabla\bcdot\bu=0\\ \bu(0,\cdot)=\bu_{0} \end{cases} \quad \text{in} \quad \mathbb{R}^3\tag{NS}$$
Main question
Well-posedness of the Cauchy problem (existence, uniqueness, regularity)
Main properties
  • energy conservation ⇒ compactness
  • scaling invariance ⇒ Banach fixed point
  • rotational invariance ⇒ keep in mind

Energy conservation

  • A priori bound on the energy:$$\left\Vert \bu(t)\right\Vert _{L^{2}(\mathbb{R}^{3})}^{2}+2\int_{0}^{t}\left\Vert \bnabla\bu(\tau)\right\Vert _{L^{2}(\mathbb{R}^{3})}^{2}\rd\tau\leq\left\Vert \bu_0\right\Vert _{L^{2}(\mathbb{R}^{3})}^{2}$$
  • Leray–Hopf solution: $$\bu \in L^{\infty}(0,T;L^{2}(\mathbb{R}^{3}))\cap L^{2}(0,T;\dot{H}^{1}(\mathbb{R}^{3}))$$ satisfying (NS) as a distribution and the local energy inequality.
Theorem (Leray, 1934)
For any $\bu_0\in L^2(\mathbb{R}^3)$, existence of a global Leray–Hopf solution ($T=\infty$). The regularity and uniqueness of these weak solutions are open.

Scaling invariance

  • Scaling symmetry: $$\begin{aligned}\bu_{\lambda}(t,\bx) & =\e^\lambda\bu(\e^{2\lambda}t,\e^\lambda\bx)\\ \bu_{0\lambda}(\bx) & =\e^\lambda\bu_{0}(\e^\lambda\bx)\end{aligned}$$
  • Mild solution: $$\bu(t)=\e^{\Delta t}\bu_0+\int_{0}^{t}\e^{\Delta(t-\tau)}\mathbb{P}\bigl(\bu(\tau)\bcdot\bnabla\bu(\tau)\bigr)\,\rd\tau$$
  • Critical space $X$ of functions $(0,\infty)\times\mathbb{R}^3\to\mathbb{R}^3$: $$\Vert\bu_\lambda\Vert_X = \Vert\bu\Vert_X$$
  • Critical spaces are a priori the largest ones where a fixed point argument is expected to close.

Perturbative results

Theorem (Kato, 1984)
For $\bu_0\in L^3(\mathbb{R}^3)$, existence of a local mild solution $\bu\in L^\infty(0,T;L^3(\mathbb{R}^3))$, which is global if $\bu_0$ is small enough.
Theorem (Escauriaza, Seregin & Šverák, 2003)
If $\bu\in L^{\infty}(0,T;L^{3}(\mathbb{R}^{3}))$ is a Leray–Hopf solution, then $\bu$ is smooth and unique on $(0,T)$.
Theorem (Koch & Tataru, 2001)
For $\bu_0\in BMO^{-1}(\mathbb{R}^3)$, existence of a unique global solution provided $\bu_0$ is small enough in $BMO^{-1}(\mathbb{R}^3)$.

Conclusions

  • Global existence for large data in $L^{2}(\mathbb{R}^3)$ but no regularity nor uniqueness.
  • Local well-posedess for large data in $L^{3}(\mathbb{R}^3)$ but not in $L^{3,\infty}(\mathbb{R}^3)$.
  • Fundamental difference between the two spaces: $$\begin{align*}\frac{1}{|\bx|} & \notin L^{3}(\mathbb{R}^{3}) \,, & \frac{1}{|\bx|} & \in L^{3,\infty}(\mathbb{R}^{3}) \subset BMO^{-1}(\mathbb{R}^3) \end{align*}$$
  • Hidden smallness condition: $$\bu_{0\lambda}\big|_{B_R}\to0 \quad\text{as}\quad \lambda\to -\infty \quad\text{for any}\quad \bu_0\in L^{3}(\mathbb{R}^{3})$$
Question on the nonperturbative regime
What is happening for large data in $L^{3,\infty}(\mathbb{R}^3)$?

2. Scale-invariant
solutions

Scale-invariant solutions

  • Scale-invariant initial data: $$\bu_{0\lambda}=\bu_0 \quad\textit{i.e.}\quad \bu_0(\bx)=\frac{\varphi(\hat{\bx})}{|\bx|}$$
  • Scale-invariant solutions: $$\bu_{\lambda}=\bu \quad\textit{i.e.}\quad \bu(t,\bx)=\frac{1}{\sqrt{2\kappa t}}\bU\Bigl(\frac{\bx}{\sqrt{2\kappa t}}\Bigr)$$
  • Forward self-similar if $\kappa>0$, backward self-similar if $\kappa < 0$.
  • Ansatz into (NS):
    $$\begin{cases} \Delta\bU+\kappa\bigl(\bx\bcdot\bnabla\bU+\bU\bigr)-\bU\bcdot\bnabla\bU-\bnabla P=\bzero\\ \bnabla\bcdot\bU=0\\ \bU(\bx)=\bu_{0}(\bx)+o(\left|\bx\right|^{-1})\quad\text{as}\quad\left|\bx\right|\to\infty \end{cases}\tag{NS-self}$$

Forward self-similar solutions

existence

Theorem (Jia & Šverák, 2013)
For $\bu_0\in C^\infty(\mathbb{R}^3\setminus\{\bzero\})$ scale-invariant, existence of a forward self-similar solution $\bu\in C^\infty((0,\infty)\times\mathbb{R}^3)$.
Intuition
Since (NS-self) is elliptic, use Leray–Schauder as for proving the existence of steady solutions to the Navier–Stokes equations. Therefore, the solution is a priori unique only for small $\bu_0$.

Forward self-similar solutions

non-uniqueness

Numerical result (Guillod & Šverák, 2017)
Two forward self-similar solutions with the same scale-invariant initial data $\bu_0 \in C^\infty(\mathbb{R}^3\setminus\{\bzero\})$ can be constructed numerically.
Intuition
The two solutions are obtained through a pitchfork bifurcation happening when the size of the initial data $\bu_0$ is increased.
Corollary (adapted from Jia & Šverák, 2015)
Under a spectral assumption verified numerically, there exists two different Leray–Hopf solutions with the same compactly supported initial data $\bu_{0}\in C^{\infty}(\mathbb{R}^{3}\setminus\{\bzero\}) \cap L^{3,\infty}(\mathbb{R}^3)$.

Backward self-similar solutions

Theorem (Tsai, 1998)
Any backward self-similar solution satisfying
the local energy inequality is identically zero.
Intuition
Crucial use the structure of the equation: maximum principle satisfied by the total head pressure $\Phi=\frac{1}{2}|\bu|^2+p$.
Ruled-out scenario
Backward self-similar blow-up followed by nonunique forward self-similar solutions.
speculation

3. Generalized
scale-invariant
solutions

Generalized scale-invariant solutions

Main idea
Combine the rotational and scaling symmetries: $$\bu(t,\bx) = \e^{\lambda}\e^{\lambda\bL}\bu(\e^{2\lambda}t,\e^{\lambda}\e^{-\lambda\bL}\bx)$$ where $\bL\in \mathfrak{so}(3)$ i.e. $$\bu(t,\bx) = \frac{\e^{-\frac{1}{2}\log(2\kappa t)\bL} }{\sqrt{2\kappa t}} \bU\Bigl(\frac{\e^{\frac{1}{2}\log(2\kappa t)\bL}\bx}{\sqrt{2 \kappa t}}\Bigr)$$
Remark
The existence of generalized backward self-similar solutions ($\kappa < 0$ and $\bL\neq\bzero$) cannot be ruled out by Tsai argument.
generalized scale-invariant generalized scale-invariant
steady solution in $\mathbb{R}^2\setminus\{\bzero\}$
(Guillod & Wittwer, 2015)

Complex Ginzburg–Landau equation

$$\begin{cases} \partial_{t}u - i\bigl(\Delta u + |u|^2 u\bigr) = \nu\Delta u\\ u(0,\cdot)=u_{0} \end{cases} \quad \text{in} \quad \mathbb{R}^3\tag{CGL}$$
Main properties
  • same energy conservation as (NS)
  • same scaling symmetry as (NS)
  • invariance under $U(1)$ instead of $SO(3)$
For $\nu>0$, same results as for (NS)
  • global Leray solutions
  • local well-posedness
  • existence of forward self-similar solutions
  • no radial backward self-similar solutions

Generalized scale-invariant solutions

  • Combine the scaling and the $U(1)$-invariance: $$u_{0\lambda}(\bx)=\e^{\lambda}\e^{i\omega\lambda}u_{0}(\e^{\lambda}\bx)\,, \quad u_{\lambda}(t,\bx) = \e^{\lambda}\e^{i\omega\lambda}\bu(\e^{2\lambda}t,\e^{\lambda}\bx)$$
  • Generalized scale-invariant initial data: $$u_{0\lambda}=u_{0} \quad\textit{i.e.}\quad u_0(\bx)=\frac{\varphi(\hat{\bx})}{|\bx|^{1+i\omega}}$$
  • Generalized scale-invariant solutions: $$u_{\lambda} = u \quad\textit{i.e.}\quad u(t,\bx) = \frac{\e^{-\frac{1}{2}i\omega\log(2\kappa t)}}{\sqrt{2\kappa t}} U\Bigl(\frac{\bx}{\sqrt{2\kappa t}}\Bigr)$$
  • Generalized forward self-similar if $\kappa>0$, backward if $\kappa < 0$.
  • Concept introduced by Zakharov for the cubic nonlinear Schrödinger equation:
    $$i\partial_{t}u + \Delta u + |u|^2 u = 0 \tag{NLS}$$

Backward generalized self-similar

Numerical result (Plecháč & Šverák, 2001)
For $\nu\geq0$ small enough, there exist a radial backward generalized self-similar solution.
Generalized backward self-similar solutions for complex Ginzburg-Landau

Forward generalized self-similar

Existence and non-uniqueness

Numerical result (Guillod & Šverák)
Let $A(\kappa)\in\mathbb{C}$ denote the amplitude of the previous backward generalized self-similar branch of solution: $$u_{0}(\bx)=\frac{A(\kappa)}{|\bx|^{1+i\omega}}$$ For $\kappa\leq-0.15$ (i.e. $\nu\leq0.17$), there exists more than one solution to (CGL) with the same initial data $u_{0}$.

Scenario for complex Ginzburg–Landau

speculation

same scenario for Navier–Stokes?