## Abstract

We consider Burgers’ equation with transverse viscosity (Equation presented). 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 singularities of the inviscid Burgers’ equation.

Original language | English (US) |
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Pages (from-to) | 1047-1133 |

Number of pages | 87 |

Journal | Bulletin de la Societe Mathematique de France |

Volume | 55 |

DOIs | |

State | Published - 2022 |

## ASJC Scopus subject areas

- Mathematics(all)