Quantitative Parabolic Decay With Bounded Drift

Abstract

A bounded time-dependent drift together with an arbitrarily small real potential can produce double exponential decay of the full spatial $L^2$ norm. We prove matching quantitative two-time lower bounds for energy solutions of $\partial_tu-Δu=vu+w\cdot\nabla u$ on $\mathbb{R}^n$, for $n\geq3$, when $w\in L^\infty_{t,x}$ and either $v$ is sufficiently small in $L^\infty_tL^{n/2}_x$ or $v\in L^\infty_tL^p_x$ for some $n/2<p\leq\infty$, without a smallness assumption. In particular, a nonzero global solution satisfies a double exponential lower bound at large times. This decay scale is optimal for these coefficient classes: In every dimension $n\geq3$, we construct a real bounded drift, a real potential arbitrarily small in the critical space and in any prescribed subcritical $L^p$ space, and a nonzero real energy solution with double exponential decay. The construction localizes Rowan's torus example by absorbing the localization error.