Nested discontinuous asymptotic profiles for the viscous Burgers equation with infinite mass

TL;DR

This paper investigates the long-time behavior of solutions to the viscous Burgers equation with infinite mass initial data, revealing a hierarchy of discontinuous asymptotic profiles through iterative rescaling.

Contribution

It introduces a novel multi-scale analysis showing the existence of infinitely many discontinuous profiles for the viscous Burgers equation with infinite mass initial conditions.

Findings

Solutions converge to a bounded discontinuous profile over time

Multiple rescalings reveal new discontinuous profiles near the initial discontinuity

The process of profile discovery can be repeated infinitely many times

Abstract

We study the viscous Burgers equation with a family of initial data having infinite mass. After rescaling, the solution converges toward a bounded discontinuous profile in the long-time limit. Moreover, by changing the scale near the discontinuity point again, we find a new profile that is also discontinuous. This process can be repeated an arbitrary number of times.