Singularity formation for Burgers' equation with transverse viscosity
Charles Collot, Tej‐Eddine Ghoul, Nader Masmoudi
Abstract
We consider Burgers equation with transverse viscosity\n$$\\partial_tu+u\\partial_xu-\\partial_{yy}u=0, \\ \\ (x,y)\\in \\mathbb R^2, \\ \\\nu:[0,T)\\times \\mathbb R^2\\rightarrow \\mathbb R.$$ We construct and describe\nprecisely a family of solutions which become singular in finite time by having\ntheir gradient becoming unbounded. To leading order, the solution is given by a\nbackward self-similar solution of Burgers equation along the $x$ variable,\nwhose scaling parameters evolve according to parabolic equations along the $y$\nvariable, one of them being the quadratic semi-linear heat equation. We develop\na new framework adapted to this mixed hyperbolic/parabolic blow-up problem,\nrevisit the construction of flat blow-up profiles for the semi-linear heat\nequation, and the self-similarity in the shocks of Burgers equation.\n