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 L2 norm. We prove matching quantitative two-time lower bounds for energy solutions of ∂tu−Δu=vu+w·∇ u on ℝn, for n≥3, when w∈ L∞t,x and either v is sufficiently small in L∞tLn/2x or v∈ L∞tLpx for some n/2<p≤∞, 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≥3, we construct a real bounded drift, a real potential arbitrarily small in the critical space and in any prescribed subcritical Lp space, and a nonzero real energy solution with double exponential decay. The construction localizes Rowan's torus example by absorbing the localization error.
Related
- Quantitative Decay for Linear Parabolic Equations
- Heat equation with singular drift which has bounded solution discontinuous at an interior point
- Exponential Decay for a Boundary-Controlled Nonlinear Parabolic Reactor Model
- Geometry property of stochastic parabolic equations and Prékopa-Leindler inequality on bounded domains
- Stochastic Itô Equations and Parabolic Second-Order Equations with singular Drift
BibTeX
@misc{kukavica2026quantitative,
title = {{Quantitative Parabolic Decay With Bounded Drift}},
author = {Igor Kukavica and Qi Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02685},
eprint = {2610.02685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02685},
}
RIS
TY - UNPB
TI - Quantitative Parabolic Decay With Bounded Drift
AU - Igor Kukavica
AU - Qi Xu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02685
UR - https://arxiv.org/abs/2610.02685
AB - 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 ∂_tu−Δu=vu+w·∇ u on ℝ^n, for n≥3, when w∈ L^∞_t,x and either v is sufficiently small in L^∞_tL^n/2_x or v∈ L^∞_tL^p_x for some n/2<p≤∞, 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≥3, 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.
ER -