平流-扩散方程的逆能量级联与非唯一性

中文摘要 AI 辅助

我们证明了在对数近临界类中低于$L_t^2L_x^\infty$的漂移下，平流-扩散方程的非唯一性。第一个构造是扩散辅助的标量能量逆级联，其能量随$t \to 0$发散。第二个构造使用压缩逆混合，其中扩散起微扰作用，并产生一个在初始时刻具有能量跳跃的有界抛物解。在这两种构造中，漂移和标量在正时间内都是光滑的。我们还考虑了$L^{2,\infty}_tL^\infty_x$中的漂移，以及从几乎所有时间开始满足能量不等式的解。在这种情况下，我们证明了有界能量解的唯一性，并构造了随时间趋于零能量爆发的非唯一解。

英文摘要

We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2L_x^\infty$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t \to 0$. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in $L^{2,\infty}_tL^\infty_x$ and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.

发表机构

• Westlake University（西湖大学）

机构由 AI 辅助整理，请以论文原文为准。