Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows
We consider the zero-homogeneous reduction of the two-dimensional Euler equation in the 3-fold symmetric case $m=3$, which is left unsolved in the $m\geq4$ relaxation theory of Said, Elgindi, and Murray \cite{EMS}. We prove that every nonconstant $W^{1,p}$ solution satisfies $\|g_\theta(t)\|_{L^p}\to\infty$ as $t\to\pm\infty$ for $1<p\leq\infty$. 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 $\alpha r<1$. The structural assumption on $D_\theta g_0$ is automatic for $C^1$ data. Moreover, every weak $L^2$ infinite-time limit generates a complete $L^2$-precompact orbit.