Decay of higher order derivatives for $L^p$ solutions to the compressible fluid model of Korteweg type

TL;DR

This paper introduces a novel method using Gevrey estimates to determine the optimal decay rates of higher order derivatives in $L^p$ solutions for the compressible Korteweg fluid model, applicable to various dissipative systems.

Contribution

It develops a new approach based on Gevrey estimates and Besov norms to analyze decay of derivatives, extending techniques to a broad class of dissipative PDEs.

Findings

Established optimal decay rates for higher order derivatives.

Developed uniform bounds on the analyticity radius in Besov norms.

Applicable to a wide range of dissipative systems.

Abstract

We present a new derivation for the optimal decay of \textit{arbitrary} higher order derivatives for $L^p$ solutions to the compressible fluid model of Korteweg type. This approach, based on Gevrey estimates, is to establish uniform bounds on the growth of the radius of analyticity of the solution in negative Besov norms. For that end, the maximal regularity property involving Gevrey multiplier of heat kernel and non standard product Besov estimates are well developed. Our approach is partly inspired by Oliver-Titi's work and is applicable to a wide range of dissipative systems.