A roadmap to nonuniqueness of L^p weak solutions to Euler

In this post, I discuss a rather classical roadmap to obtain the non-uniqueness of ${L^p}$ weak solutions to the classical incompressible Euler equations; namely, focusing on the two-dimensional case, which reads in the vorticity formulation for vorticity function ${\omega}$ :

$\displaystyle \partial_t \omega + v \cdot \nabla \omega = 0, \qquad v = \nabla^\perp \Delta^{-1}\omega \ \ \ \ \ (1)$

on ${\mathbb{R}^2}$ , with initial vorticity in ${L^p}$ (hence, vorticity remains in ${L^p}$ for all times). It’s known, going far back to Yudovich ’63, that weak solutions with bounded vorticity are unique, leaving open the question of uniqueness of solutions whose vorticity is only in ${L^p}$ for ${p\in [1,\infty)}$ . This blog post is to discuss the possible quick roadmap to proving nonuniqueness arising from the instability nature of fluid models, focusing on the Euler equations (1).

1. Euler / Navier-Stokes invariant

The key observation is that the Euler equations are invariant under the following scaling

$\displaystyle \omega = \epsilon^{-\alpha} \widetilde \omega (x/\epsilon, t/\epsilon^\alpha) , \qquad v= \epsilon^{1-\alpha} \widetilde v (x/\epsilon, t/\epsilon^\alpha) \ \ \ \ \ (2)$

for any positive ${\alpha}$ . That is, ${(v,\omega)}$ solve the Euler equation in variables ${(x,t)}$ if and only if ${(\widetilde v, \widetilde \omega)}$ solve the same equation in variables ${(y,\tau) = (x/\epsilon, t/\epsilon^\alpha)}$ :

$\displaystyle \partial_\tau \widetilde \omega + \widetilde v \cdot \nabla_y \widetilde \omega = 0, \qquad \widetilde v = \nabla^\perp_y \Delta_y^{-1} \widetilde \omega.\ \ \ \ \ (3)$

Namely, the flow is identical when zooming into a point in the physical space. For Navier-Stokes equations (namely, Euler with ${\Delta \omega}$ on the right hand side), only ${\alpha =2}$ works.

The next key observation is that Euler equations are very unstable (e.g., Kelvin-Helmhotz instability or flows where vorticity changes sign). It is then tempting to construct growth mode solutions, say ${e^{\lambda_0 \tau}}$ , concentrated at a point (due to the invariant scaling), and obtain weak solutions to Euler equations as a weak limit of ${\epsilon \rightarrow 0}$ . This however does not work, since the instability time is of order ${\tau_* \sim \log \frac 1\epsilon}$ , which is vanishing in the original variable ${t_* \sim \epsilon^\alpha \log \frac 1\epsilon\rightarrow 0}$ . The instability is lost in the limit (noting that this is due to the limitation of our available nonlinear analyses which are very far from being enough to go beyond this instability time, not to mention to ever reaching ${t \sim \epsilon^{-\alpha}}$ nonlinearly!).

2. Modulation

To capture the instability within the time that nonlinearity remains neglected, we introduce the following modulation Ansatz:

$\displaystyle \omega = \epsilon(t)^{-\alpha} \widetilde \omega (x/\epsilon(t), \tau(t)) , \qquad v= \epsilon(t)^{1-\alpha} \widetilde v (x/\epsilon(t), \tau(t)) \ \ \ \ \ (4)$

for ${\epsilon = \epsilon(t)}$ and for ${\tau = \tau(t)}$ satisfying

$\displaystyle \epsilon'(t) = \epsilon(t)^{1-\alpha}, \qquad \tau'(t) = \epsilon^{-\alpha}(t).\ \ \ \ \ (5)$

This choice conveniently reduces the Euler equations to the following modified Euler equations:

$\displaystyle \partial_\tau \widetilde \omega + \widetilde v \cdot \nabla_y \widetilde \omega - (\alpha + y \cdot \nabla_y )\widetilde \omega =0, \qquad \widetilde v = \nabla^\perp_y \Delta^{-1}_y \widetilde \omega \ \ \ \ \ (6)$

which looks temptingly close to the original Euler equations and one then hopes that this modified Euler shares a similar instability structure, which would lead to a nonuniqueness of solutions in the limit. Note that taking initial value ${\epsilon(0) = \epsilon_0\ll1}$ , one gets from (5) that

$\displaystyle \epsilon(t) = \Big( \epsilon_0^\alpha + \alpha t\Big)^{1/\alpha}, \qquad \tau(t) = \frac{1}{\alpha} \log \frac{\epsilon_0^\alpha + \alpha t}{\epsilon_0^\alpha} .\ \ \ \ \ (7)$

It turns out that with this Ansatz, the instability time can now be of order one ${t_* \sim 1}$ in the limit ${\epsilon_0 \rightarrow 0}$ . That is, this growth mode indeed survives in the limit, giving the non-uniqueness, which I shall quickly discuss in the next section.

3. Non-uniqueness

Suppose for the moment that the modified Euler equations (6) indeed have an unstable background solution ${\omega_*(y) }$ : namely, there is an exact solution to the nonlinear problem (6), satisfying

$\displaystyle \widetilde \omega(y,\tau) \sim \omega_*(y) + \delta e^{\lambda_0 \tau}\omega_c(y) \ \ \ \ \ (8)$

for some eigenfunction ${\omega_c(y)}$ and for ${\Re \lambda_0>0}$ , as long as ${\delta e^{\lambda_0 \tau} \ll 1}$ (namely as long as the nonlinearity remains neglected via the standard linear to nonlinear iteration, see e.g., the classical argument by Bardos-Guo-Strauss). Equivalently, solution (8) is constructed for time ${\tau \le \tau_* = \frac{1}{\Re \lambda_0} \log \frac1\delta}$ . In view of (7), this implies that

$\displaystyle 1 + \alpha \epsilon_0^{-\alpha} t_* = \delta^{-\alpha / \Re \lambda_0} = \epsilon_0^{-\alpha}$

upon taking ${\delta = \epsilon_0^{\Re\lambda_0}}$ . That is, the original instability time is indeed of order ${t_* \sim 1}$ . Here, only some minimal regularity is assumed on the background solution as well as on the growth mode ${\omega_c(y)}$ so that ${\| \omega_c\|_{L^p}=1}$ for some ${p\ge 1}$ .

The non-uniqueness of ${L^p}$ weak solutions now follows trivially. Indeed, rescaling back to the original Euler equations, we obtain a family of exact solutions of the form

$\displaystyle \begin{aligned} \omega_\epsilon &= \omega_{*,\epsilon}(t,x) + \omega_{c,\epsilon}(t,x) \\&\sim \epsilon(t)^{-\alpha} \omega_*(x/\epsilon(t)) + \epsilon(t)^{-\alpha} \epsilon_0^{\Re \lambda_0} e^{ \lambda_0 \tau(t)} \omega_c(x/\epsilon(t)), \end{aligned}\ \ \ \ \ (9)$

recalling ${\epsilon(t) =( \epsilon_0^\alpha + \alpha t)^{1/\alpha}}$ and ${\delta = \epsilon_0^{\Re\lambda_0}}$ . It is now direct to see that in the limit of ${\epsilon_0\rightarrow0}$ , these are two distinct weak solutions ${ \omega_{*,0}(t,x) }$ and ${ \omega_{*,0}(t,x) + \omega_{c,0}(t,x) }$ to Euler equations starting from the same initial data. Indeed, at ${t=0}$ ,

$\displaystyle \omega_{c,\epsilon_0}(0,x) =\epsilon_0^{-\alpha + \Re \lambda_0} \omega_c(x/\epsilon_0) , \ \ \ \ \ (10)$

and so ${\| \omega_{c,\epsilon_0}(0,\cdot) \|_{L^p} \lesssim \epsilon_0^{-\alpha + \Re \lambda_0+d/p} \|\omega_c(\cdot)\|_{L^p} \rightarrow 0}$ , provided that ${p\le p_* = d/\alpha}$ . That is, ${\omega_{c,\epsilon_0}(0,\cdot) \rightharpoonup 0}$ , weakly in ${L^p}$ as ${\epsilon_0 \rightarrow 0}$ , while for ${t>0}$ ,

$\displaystyle \begin{aligned} \| \omega_{c,\epsilon_0}(t) \|_{L^p} &\approx \epsilon_0^{\Re \lambda_0} \epsilon(t)^{-\alpha + d/p} e^{\Re \lambda_0\tau(t)} \| \omega_c(\cdot)\|_{L^{p}} \end{aligned}$

for any ${p\ge 1}$ . By definition, we have ${\epsilon(t) \rightarrow (\alpha t)^{1/\alpha}}$ as ${\epsilon_0\rightarrow 0}$ . In addition, we compute

$\displaystyle \epsilon_0^{\Re \lambda_0} e^{\Re \lambda_0 \tau(t)} = (\epsilon_0^\alpha + \alpha t)^{\Re \lambda_0 /\alpha} \rightarrow (\alpha t)^{\Re \lambda_0/\alpha}$

as ${\epsilon_0 \rightarrow 0}$ . That is, there holds in fact the strong convergence

$\displaystyle \omega_{c,\epsilon_0}(t,x) \rightarrow (\alpha t)^{-1 + d/(\alpha p) + \Re \lambda_0/\alpha} \omega_c(x/(\alpha t)^{1/\alpha}),$

as ${\epsilon_0 \rightarrow 0}$ , strongly in ${L^p}$ for any ${p\ge 1}$ . This solution is thus different from the background solution ${ \omega_{*,0}(t,x) = t^{-1} \omega_*(x/t^{1/\alpha})}$ , starting from the same initial data in ${L^p}$ for any ${p\in [1,p_*]}$ with ${p_* = d/\alpha}$ , with $d=2$ being the dimension. Taking ${\alpha}$ to zero, this covers the whole range of $L^p$ spaces, however the self-similar solution decays too slowly for such a small value of ${\alpha}$ (see below). It’s thus natural to conjecture that there is a critical value of $p_c$ , for which the uniqueness would still hold for vorticity in $L^p$ for $p> p_c$ .

4. Vishik’s and other results

The previous approach is rather simple and robust, however faces a fundamental difficulty, which remains notoriously open: namely, to construct an unstable background solution to the modified Euler equations (6), which I recall (dropping titles):

$\displaystyle \partial_\tau \omega + v \cdot \nabla_y \omega - (\alpha + y \cdot \nabla_y ) \omega =0, \qquad v = \nabla^\perp_y \Delta^{-1}_y \omega . \ \ \ \ \ (11)$

A natural idea is to search for radial solutions, which turn out to be uniquely determined: ${\omega(r) = r^{-\alpha}}$ . Since the vorticity is monotone, this natural background solution is spectrally stable (i.e. Rayleigh’s inflexion point criterium). That is, nonuniqueness of ${L^p}$ weak solutions to Euler must come from symmetry breaking profiles. There is a very interesting and promising approach by my colleagues here at Penn State, who attempt, with help from numerics, to construct unstable background solutions from patching two radial solutions. Instability or spectrum for such a non-radial background solution however remains open.

Recently, M. Vishik, see Part 1 and Part 2, was able to construct unstable background solutions that are radial (i.e. by force, literally!) and as a consequence only solves the Euler equations with a forcing. When background is radial, there is no unstable continuous spectrum and the linear problem reduces to a study of Rayleigh equations (namely, an ODE describes the spectrum of the linearized Euler near shear flows or radial background). A similar idea is also recently exploited to give nonuniqueness of Leray’s weak solutions to Navier-Stokes equations, with a forcing as in Vishik’s. Once an unstable background profile is constructed, the non-uniqueness follows directly as discussed, up to obvious modifications to Navier-Stokes equations. The original problem however remains widely open. See also Jia-Sverak’s program for further discussion on the subject.

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A roadmap to nonuniqueness of L^p weak solutions to Euler

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