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

TL;DR

The paper investigates the finite-time dynamics of 1D Burgers-type equations on the torus, showing that for odd solutions the singular jump function $F$ generates finite-time $L^2$ attractors through a key inner-product identity. This leads to a precise evolution law for the $L^2$ distance to $rF$ and a fastest-approaching attractor given by $\|u^0\|_{L^2}\,\Phi$, where $\Phi=F/\|F\|_{L^2}$. The authors then use a Lyapunov functional approach to prove finite-time blowup for the fractal Burgers equation with fractional dissipation $0<\alpha<\tfrac{1}{2}$, providing explicit thresholds and an example initial data. Finally, they extend these attractor considerations to a broader class of odd, increasing functions and connect the attractor structure to Fourier-space dynamics, highlighting the broader relevance of energy-based methods for related dissipative PDE models.

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 $0<α<\frac{1}{2}$. 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 $(0,π)$.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Corollary 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10