# On the long time behavior of almost periodic entropy solutions to scalar conservation laws

## TL;DR

This paper characterizes the long-term decay and asymptotic behavior of almost periodic entropy solutions to scalar conservation laws, identifying conditions for convergence to traveling waves and describing the flux function's properties.

## Contribution

It establishes the precise decay conditions for multidimensional solutions and proves convergence to traveling waves in one dimension, linking flux function properties to the limit profile.

## Key findings

- Decay conditions for multidimensional solutions
- Asymptotic convergence to traveling waves in 1D
- Flux function is affine on the minimal segment of the limit profile

## Abstract

We found the precise condition for the decay as $t\to\infty$ of Besicovitch almost periodic entropy solutions of multidimensional scalar conservation laws. Moreover, in the case of one space variable we establish asymptotic convergence of the entropy solution to a traveling wave (in the Besicovitch norm). Besides, the flux function turns out to be affine on the minimal segment containing the essential range of the limit profile while the speed of the traveling wave coincides with the slope of the flux function on this segment.