Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows

Abstract

We consider long-time behavior of the zero-homogeneous solutions with 3-fold symmetry to the two-dimensional Euler equation. This is the remaining case in the relaxation theory of Said, Elgindi, and Murray [Ann. Sci. Éc. Norm. Supér. (4) \textbf{58} (2025), no.~4, 943--970], which treats $m$-fold symmetry solutions with $m\ge 4$. We prove that every nonconstant $W^{1,p}$ solution satisfies $\Vert g_{\theta }(t){\Vert }_{L^p}\to \infty$ as $t\to \pm \infty$ for $1<p\le \infty$. For $p=1$, the total variation is conserved, but the $L\log L$ modular tends to infinity whenever it is initially finite. If $D_{\theta }{g}_0$ is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in $W^{\alpha ,r}$ for $\alpha r<1$. For such data, every weak $L^2$ infinite-time limit generates a complete $L^2$-precompact orbit. This structural assumption on $D_{\theta }{g}_0$ is automatic for $C^1$ data. In particular, these results answer the question concerning small-scale creation and compact orbit raised by Drivas and Elgindi [EMS Surv. Math. Sci. \textbf{10} (2023), no.~1, 1--100, Problem~5] for all nonconstant smooth 3-fold symmetric scale-invariant flows. Together with the known theory for $m\ge 4$, they cover the full well-posed scale-invariant range $m\ge 3$.