Optimal decay rates and space-time analyticity of solutions to the Patlak-Keller-Segel equations

TL;DR

This paper establishes optimal decay rates and space-time analyticity of solutions to the Patlak-Keller-Segel equations using heat kernel estimates and bootstrapping, with implications for understanding solution regularity and extending to other PDEs.

Contribution

It introduces a novel bootstrapping method to prove optimal decay rates and space-time analyticity for solutions with initial data in L^1, extending to more general equations like Navier-Stokes.

Findings

Optimal decay rates for space-time derivatives established.

Solutions exhibit joint space-time analyticity.

Results extend to solutions bounded in space and time.

Abstract

Based on some elementary estimates for the space-time derivatives of the heat kernel, we use a bootstrapping approach to establish the optimal decay rates for the $L^q(\mathbb{R}^d)$ ($1\leq q\leq\infty$, $d\in\mathbb{N}$) norm of the space-time derivatives of solutions to the (modified) Patlak-Keller-Segel equations with initial data in $L^1(\mathbb{R}^d)$, which implies the joint space-time analyticity of solutions. When the $L^1(\mathbb{R}^d)$ norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space-time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in $L^1(\mathbb{R}^d)$. The decay estimates and space-time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations, including the Navier-Stokes equations.