Singularity formation for Burgers equation with transverse viscosity

2018/03/21 by Collot, Charles, Ghoul, Tej-Eddine, Masmoudi, Nader · 3 citations
#35A20 #35B35 #35B40 #35B44 #35K58 #35L67 #35M10 #35Q35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.07826

Abstract

We consider Burgers equation with transverse viscosity ∂_tu+u∂_xu-∂_yyu=0, (x,y)∈ \mathbb R², u:[0,T)× \mathbb R²→ \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.

Cited by

• Dynamics of Ideal Fluid Flows
• Stable Type I blow-up for the one-dimensional wave equation with time-derivative nonlinearity
• Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations

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