Systematic search for singularities in 3D Euler flows

Abstract

We consider the question whether starting from a smooth initial condition 3D inviscid Euler flows on a periodic domain $\mathbb{T}^3$ may develop singularities in a finite time. Our point of departure is the well-known result by Kato (1972), which asserts the local existence of classical solutions to the Euler system in the Sobolev space $H^m(\mathbb{T}^3)$ for $m > 5/2$. Thus, potential formation of a singularity must be accompanied by an unbounded growth of the $H^m$ norm of the velocity field as the singularity time is approached. We perform a systematic search for "extreme" Euler flows that may realize such a scenario by formulating and solving a PDE-constrained optimization problem where the $H^3$ norm of the solution at a certain fixed time $T > 0$ is maximized with respect to the initial data subject to suitable normalization constraints. This problem is solved using a state-of-the-art Riemannian conjugate gradient method where the gradient is obtained from solutions of an adjoint system. Computations performed with increasing numerical resolutions demonstrate that, as asserted by the theorem of Kato (1972), when the optimization time window $[0, T]$ is sufficiently short, the $H^3$ norm remains bounded in the extreme flows found by solving the optimization problem, which indicates that the Euler system is well-posed on this "short" time interval. On the other hand, when the window $[0, T]$ is long, possibly longer than the time of the local existence asserted by Kato's theorem, then the $H^3$ norm of the extreme flows diverges upon resolution refinement, which indicates a possible singularity formulation on this "long" time interval. The extreme flow obtained on the long time window has the form of two colliding vortex rings and is characterized by certain symmetries. In particular, the region of the flow in which a singularity might occur is nearly axisymmetric.

Figures (16)

• Figure 1: Schematic illustration of the Riemannian conjugate gradient method \ref{['eq:RCG']}.
• Figure 2: [Short time window, $T = 25$] Dependence of the objective functional $\Phi_{25}\left(\boldsymbol{\eta}^{(n)}\right)$ on the iteration index $n$ for different values of $\sigma$ when (a) $\boldsymbol{\eta}_{{\text{TG}}}$ and (b) $\boldsymbol{\eta}_{{\text{rand}}}$ are used as initial guesses in \ref{['eq:RCG']}.
• Figure 3: [Short time window, $T = 25$] (a) Dependence of the objective functional $\Phi_{25}\left(\boldsymbol{\eta}^{(n)}\right)$ on the iteration index $n$ and (b) energy spectra \ref{['eq:Ek']} of the optimal initial conditions $\widetilde{\boldsymbol{\eta}}_{25}$ corresponding to different initial guesses $\boldsymbol{\eta}_{{\text{TG}}}$, $\boldsymbol{\eta}_{{\text{rand}}}$, $\boldsymbol{\eta}_{{\text{K}}}$ and $\boldsymbol{\eta}_{{\text{H}}}$ used in iterations \ref{['eq:RCG']}. In panel (b) the solid line represents the Gaussian filter we use hl07.
• Figure 4: [Short time window, $T = 25$] Dependence of (a) the objective functional $\Phi_{25}^N\left(\boldsymbol{\eta}^{(n)}\right)$ on the iteration index $n$ for different resolutions $N^3$ and (b) of the corresponding maximum attained values $\widetilde{\Phi}_{25}^N$ of the objective functional on $N$.
• Figure 5: [Short time window, $T = 25$] The energy spectra of (a) the optimal initial conditions $\widetilde{\boldsymbol{\eta}}_{25}^N$ and (b) of the corresponding terminal states $\boldsymbol{u}^N\left(25; \widetilde{\boldsymbol{\eta}}_{25}^N\right)$ obtained for different resolutions $N^3$. The solid lines represent the Gaussian filters we use hl07 whereas the dashed lines mark the threshold wavenumber $k_0$ above which aliasing errors occur.

Theorems & Definitions (2)

• Theorem 1.1
• Remark 2.1