Nonuniqueness in fluids

through bifurcations

Julien Guillod

18 May 2026

joint work with Dallas Albritton, Mikhail Korobkov, Xiao Ren & Vladimír Šverák

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Steady Navier–Stokes equations

$$\begin{cases} -\Delta\bu + \Ren \bu\bcdot\bnabla\bu + \bnabla p = \bff\\ \bnabla\bcdot\bu=0\\ \end{cases} \quad \text{in} \quad \mathbb{R}^{n=2,3} \tag{NS-steady}$$
Nonuniqueness for large $\Ren$
Physics/Numerics
  • generically solution not unique at large Reynolds
  • pitchfork or Hopf bifurcation
Mathematics
  • existence by compactness
    using Leray-Schauder
  • nonuniqueness expected
  • not much nonuniqueness results

Time-dependent problem

Main question
What about the uniqueness of the Cauchy problem? $$\begin{cases} \partial_t \bu -\Delta\bu + \Ren\bu\bcdot\bnabla\bu + \bnabla p = \bzero\\ \bnabla\bcdot\bu=0\\ \bu|_{t=0} = \bu_0 \end{cases} \quad \text{in} \quad \mathbb{R}^{n=2,3} \tag{NS}$$ Different solutions emerging from same initial data $\bu_0$?

Existence by energy bound

A priori bound on the energy
$$\frac{1}{2}\left\Vert \bu(t)\right\Vert _{L^{2}}^{2}+\int_{0}^{t}\left\Vert \bnabla\bu(\tau)\right\Vert _{L^{2}}^{2}\ud\tau \leq \frac{1}{2}\left\Vert \bu_0\right\Vert _{L^{2}}^{2}$$
Physics
  • solutions with finite kinetic energy
Mathematics
  • existence by compactness
    using Leray-Schauder
  • for any $\bu_0 \in L^2(\mathbb{R}^n)$
  • nothing about uniqueness

Existence by perturbation

Scaling symmetry
$$\bu_{\lambda}(t,\bx) :=\lambda\bu(\lambda^2t,\lambda\bx) \qquad \bu_{0\lambda}(\bx) :=\lambda\bu_{0}(\lambda\bx) \qquad \lambda >0$$
Physics
Linear part dominant over the nonlinearity when:
  • $\bu_0$ regular or singular like $|\bx|^{-\alpha}$ for $\alpha<1$
  • $\bu_0$ like $|\bx|^{-1}$ and small $\Ren$
  • small make sense only w.r.t scale-invariant quantities
Mathematics
Existence by Banach fixed-point under smallness assumptions:
  • $\bu_0 \in L^{n}(\mathbb{R}^n)$: hidden smallness (local by rescaling)
  • $\bu_0 \in L^{n,\infty}(\mathbb{R}^n)$ and small $\Ren$
  • uniqueness as by-product

Scale-invariant solutions

  • Scale-invariant initial data: $$\bu_{0\lambda}=\bu_0 \quad\textit{i.e.}\quad \bu_0(\bx)={|\bx|}^{-1}\varphi(\hat{\bx})$$
  • 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):
    $$\begin{cases} -\Delta\bU-\kappa\bigl(\bU+\bx\bcdot\bnabla\bU\bigr)+\Ren\bU\bcdot\bnabla\bU+\bnabla P=\bzero\\ \bnabla\bcdot\bU=0 \qquad\qquad\qquad\qquad\qquad \text{in} \quad \mathbb{R}^{n=2,3}\\ \bU(\bx)=\bu_{0}(\bx)+o(\left|\bx\right|^{-1})\quad\text{as}\quad\left|\bx\right|\to\infty \end{cases} \tag{NS-self}$$

Existence

of forward self-similar solutions

Theorem
$n=3$ (Jia & Šverák, 2013)
$n=2$ (Albritton, Guillod, Korobkov & Ren, 2026)
For $\bu_0\in C^\infty(\mathbb{R}^n\setminus\{\bzero\})$ scale-invariant, existence of a forward self-similar solution $\bu\in C^\infty((0,\infty)\times\mathbb{R}^n)$.
Remark: nonuniqueness expected
Physics
(NS-self) like (NS-steady)
Mathematics
existence by Leray-Schauder

Sketch of proof in $\mathbb{R}^3$

  • Equation on the remainer $\bU = \bu_0 + \bV$:
    $$-\Delta\bV-\kappa\bigl(\bV+\bx\bcdot\bnabla\bV\bigr)+\Ren\bigl(\bV\bcdot\bnabla\bu_0+\bU\bcdot\bnabla\bV\bigr)+\bnabla Q=\bF_0$$
  • Formal a priori bound on the enstrophy: $$\|\bnabla\bV\|_{L^2}^2 + \kappa \tfrac{n-2}{2}\|\bV\|_{L^2}^2 \leq \Ren \int |\bnabla\bu_0||\bV|^2 + \int |\bF_0||\bV|$$
  • Since $\bnabla\bu_0 \sim |\bx|^{-2}$, truncating on a large ball leads to smalleness
  • Work only for $n=3$
  • For $n=2$, like the steady case ($\kappa=0$) so pretty difficult

Sketch of proof in $\mathbb{R}^2$

Difficulites
  • Enstrophy bound $\|\bnabla\bV\|_{L^2}^2$ not sufficient to get decay
  • Initial data like $|\bx|^{-1}$ has infinite energy, even locally
  • Voriticity like $|\bx|^{-2}$ so not locally integrable
Ideas of the proof
  • compensated compactness to bound the pressure
  • maximum principle on a modified Bernoulli pressure
  • operator $\kappa(1+\bx\bcdot\bnabla)$ dominant at large distances
lead to a spectific a priori estimate: $$\|(1+|\bx|) \bU\| \leq C(\Ren)$$

Nonuniqueness in $\mathbb{R}^3$

Axisymmetric pure-swirl initial data: $$\bu_0=\frac{\e^{-4\cot^2\theta}}{r}\be_\varphi$$ in spherical coordinates

Nonuniqueness in $\mathbb{R}^2$

Initial data: $$\bu_0=-\frac{\cos\theta}{r}\be_r$$ in polar coordinates

Conclusions

Numerical results
$n=3$ (Guillod & Šverák, 2023)
$n=2$ (Albritton, Guillod, Korobkov & Ren, 2026)
For an explicit scale-invariant initial data $\bu_0 \sim |\bx|^{-1}$, numerical construction of three different forward self-similar solutions for large enough Reynolds number.
Theorem (Hou, Wang & Yang, 2026)
Computer-assisted proof of the nonuniqueness of self-similar and Leray-Hopf solutions for $n=3$.

Physical consequences

speculation