# On Riemann solutions under different initial periodic perturbations at two infinities for 1-d scalar convex conservation laws

## TL;DR

This paper studies the long-term behavior of entropy solutions to 1D scalar convex conservation laws with different periodic initial conditions at infinities, revealing algebraic convergence to Riemann solutions and a novel shift effect caused by differing perturbations.

## Contribution

It demonstrates that differing periodic perturbations at infinities can cause a constant shift in the background shock, extending previous results where perturbations were identical.

## Key findings

- Solutions approach Riemann solutions algebraically over time.
- Different periodic perturbations can induce a constant shift in the shock.
- The shift effect is a new discovery contrasting previous work.

## Abstract

This paper is concerned with the large time behaviors of the entropy solutions to one-dimensional scalar convex conservation laws, of which the initial data are assumed to approach two arbitrary $ L^\infty $ periodic functions as $ x\rightarrow-\infty $ and $ x\rightarrow+\infty, $ respectively. We show that the solutions approach the Riemann solutions at algebraic rates as time increases. Moreover, a new discovery in this paper is that the difference between the two periodic perturbations at two infinities may generate a constant shift on the background shock wave, which is different from the result in [11], where the two periodic perturbations are the same.