Gaussian bounds of fundamental matrix and maximal $L^1$ regularity for Lam\'{e} system with rough coefficients

TL;DR

This paper establishes Gaussian bounds for the fundamental matrix of a generalized parabolic Lamé system with rough coefficients and proves maximal L^1 regularity results, leading to well-posedness of certain viscous fluid models.

Contribution

It introduces a novel approach to derive Gaussian bounds and maximal L^1 regularity for Lamé systems with minimal regularity assumptions on coefficients.

Findings

Gaussian bounds for fundamental matrix established

Maximal L^1 regularity for the Lamé system derived

Global-in-time well-posedness for viscous pressureless flow proved

Abstract

The purpose of this paper is twofold. First, we use a classical method to establish Gaussian bounds of the fundamental matrix of a generalized parabolic Lam\'{e} system with only bounded and measurable coefficients. Second, we derive a maximal $L^1$ regularity result for the abstract Cauchy problem associated with a composite operator. In a concrete example, we also obtain maximal $L^1$ regularity for the Lam\'{e} system, from which it follows that the Lipschitz seminorm of the solutions to the Lam\'{e} system is globally $L^1$-in-time integrable. As an application, we use a Lagrangian approach to prove a global-in-time well-posedness result for a viscous pressureless flow provided that the initial velocity satisfies a scaling-invariant smallness condition. The method established in this paper might be a powerful tool for studying many issues arising from viscous fluids with truly variable densities.