On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: global well-posedness and decay property

TL;DR

This paper investigates the well-posedness and decay properties of entropy solutions to multidimensional scalar conservation laws with continuous flux in Besicovitch almost periodic functions, establishing existence, uniqueness, and decay conditions.

Contribution

It introduces a framework for analyzing scalar conservation laws with Besicovitch almost periodic flux, proving well-posedness and decay criteria for solutions.

Findings

Existence and uniqueness of entropy solutions are proven.

A necessary and sufficient condition for decay of solutions is established.

The results extend the theory to a broader class of almost periodic functions.

Abstract

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We propose also the necessary and sufficient condition for the decay of almost periodic entropy solutions as time $t\to+\infty$.