1. Introduction

Nonlinear parabolic equations

• Cauchy problem for a scalar or vector field $\bu$ on $\mathbb{R}$ or $\mathbb{C}$:
  $$\begin{cases} \partial_{t} \bu-\nu\Delta \bu = N(\bu)\\ \bu(0,\cdot)=\bu_{0} \end{cases} \quad \text{in} \quad \mathbb{R}^3\tag{$*$}$$
• Assumptions on the nonlinearity:
  • energy conservation
  • scaling invariance
  • rotational invariance
• Examples to keep in mind:
  • incompressible Navier–Stokes: $$\bu:(0,\infty)\times\mathbb{R}^3\to\mathbb{R}^3 \quad\text{and}\quad N(\bu) = \mathbb{P}(\bu\cdot\bnabla\bu)$$
  • complex Ginzburg–Landau: $$\bu:(0,\infty)\times\mathbb{R}^3\to\mathbb{C} \quad\text{and}\quad N(\bu) = i (\Delta \bu + |\bu|^2 \bu)$$

Energy conservation

• A priori bound on the energy:$$\left\Vert \bu(t)\right\Vert _{L^{2}(\mathbb{R}^{3})}^{2}+2\nu\int_{0}^{t}\left\Vert \bnabla\bu(\tau)\right\Vert _{L^{2}(\mathbb{R}^{3})}^{2}\rd\tau\leq\left\Vert \bu_0\right\Vert _{L^{2}(\mathbb{R}^{3})}^{2}$$
• 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 ($*$) as a distribution and the local energy inequality.

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: $$\bu(t)=\e^{\Delta t}\bu_0+\int_{0}^{t}\e^{\Delta(t-\tau)}N(\bu)(\tau)\,\rd\tau$$
• 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.

Critical spaces $L^{p}(0,T;L^{q}(\mathbb{R}^{3}))$

Theorem (Kato, 1984 / Doering-Gibbon-Levermore, 1994)

For $p\geq3$ and $\bu_0\in L^p(\mathbb{R}^3)$, existence of a local mild solution $\bu\in L^\infty(0,T;L^p(\mathbb{R}^3))$.

Theorem (Prodi-Serrin-Ladyzhenskaya / Doering et al.)

If $\bu\in L^{p}(0,T;L^{q}(\mathbb{R}^{3}))$ with $\frac{2}{p}+\frac{3}{q}\leq1$, then $\bu$ is smooth and unique on $(0,T)$.

Ultra critical spaces

• 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: $$\begin{align*}\frac{1}{|\bx|} & \notin L^{3}(\mathbb{R}^{3}) \,, & \frac{1}{|\bx|} & \in L^{3,\infty}(\mathbb{R}^{3})\end{align*}$$
• Hidden smallness condition: $$\bu_{0\lambda}|_{B_R}\to0 \quad\text{as}\quad \lambda\to -\infty \quad\text{for any}\quad \bu_0\in L^{3}(\mathbb{R}^{3})$$

2. Scale-invariant
solutions

Scale-invariant solutions

Forward self-similar solutions

existence for Navier–Stokes

Numerical strategy

to construct multiple solutions

Numerical results

$\sigma=\,$0

Forward self-similar solutions

non-uniqueness

Ill-posedness

of the Navier–Stokes equations

Summary

• The Navier–Stokes equations are locally ill-posed for $$\bu_0\in L^2(\mathbb{R}^3) \cap L^{3,\infty}(\mathbb{R}^3)$$
• The Leray–Hopf solutions are not unique in $$\bu \in L^{p}(0,T;L^{q}(\mathbb{R}^{3}))$$ with $$\tfrac{2}{p}+\tfrac{3}{q}>1$$ even with the energy equality.

Backward self-similar solutions

Ruled-out scenario

Backward self-similar blow-up followed by nonunique forward self-similar solutions.

3. Generalized
scale-invariant
solutions

Generalized scale-invariant solutions

for Navier–Stokes

steady solution in $\mathbb{R}^2\setminus\{\bzero\}$
(Guillod & Wittwer, 2015)

Generalized scale-invariant solutions

for complex Ginzburg–Landau

Backward generalized self-similar

Forward generalized self-similar

Existence and non-uniqueness

Scenario for complex Ginzburg–Landau

Appendix

Numerical solution $\bU_\sigma$

Spectrum of the linearization around $\bU_\sigma$