Singularity formation for Burgers equation with transverse viscosity

Abstract

We consider Burgers equation with transverse viscosity $${\partial }_{t}u+u{\partial }_{x}u-{\partial }_{yy}u=0,(x,y)\in {\mathbb{R}}^2,u:[0,T)\times {\mathbb{R}}^2\to \mathbb{R}.$$ We construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the $x$ variable, whose scaling parameters evolve according to parabolic equations along the $y$ variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.

79 pages (in version 2 minor corrections have been performed)