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arXiv · 2511.09528

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

Evan Miller
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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 $(0,\pi)$.

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Evan Miller. 2025-11-12. Singular, finite-time $L^2$ attractors for odd, smooth solutions of Burgers equation on the torus. https://arxiv.org/abs/2511.09528

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