Singular, finite-time $L^2$ attractors for odd, smooth solutions of Burgers equation on the torus

TL;DR

This paper demonstrates that certain singular functions serve as finite-time attractors for odd, smooth solutions of the 1D inviscid Burgers equation on the torus, and provides an alternative proof of blowup in the fractal case.

Contribution

It introduces a novel class of finite-time attractors for Burgers solutions and offers a new proof technique for blowup in the supercritical fractal regime.

Findings

Positive multiples of a singular function are finite-time attractors.

Provides an alternative proof of blowup for fractal Burgers equation.

Attractor property extends to broader class of odd functions.

Abstract

In this paper, we show that the positive multiples of a particular function $F$ -- which is singular with a jump discontinuity at the origin -- are finite-time global attractors in $L^2$ for generic odd, smooth solutions of the one dimensional inviscid Burgers equation. Furthermore, the identity that leads to this result provides to an alternative proof of finite-time blowup for the fractal Burgers equation in the supercritical range $0<\alpha<\frac{1}{2}$. This proof is based on lower bounds on a Lyapunov functional given by the inner product of the solution with the global attractor $F$. We will also show that this property holds for a broader class of odd functions that are strictly increasing on $(0,\pi)$.