Decay of solutions to anisotropic conservation laws with large initial data

TL;DR

This paper investigates the long-term decay behavior of solutions to two-dimensional anisotropic conservation laws with large initial data, establishing decay rates in L^2 and Sobolev spaces without smallness restrictions.

Contribution

It provides the first decay rate results for large initial data solutions to anisotropic conservation laws using time-frequency decomposition and energy methods.

Findings

Decay rates in L^2 space established

Decay rates in homogeneous Sobolev space obtained

No smallness assumption on initial data required

Abstract

In this paper, we study the large time behavior of solutions to the Cauchy problem for the anisotropic conservation laws in two dimensional space. Without any smallness assumption on the initial data, the decay rates of solutions in $L^2$ space and homogeneous Sobolev space $\dot{H}^\gamma$ are obtained by using the method of time-frequency decomposition and the classical energy method.