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

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 $(0,\pi )$.