Singularity formation for Burgers equation with transverse viscosity

We consider Burgers equation with transverse viscosity ∂ₜu+u∂ₓu-∂yyu=0, \ \ (x,y)∈ R², \ \ u:[0,T)× R²→ 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.