We consider Burgers equation with transverse viscosityWe 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.

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We consider the two dimensional unsteady Prandtl's system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative along the transversal axis solves a closed one dimensional equation. We give a precise description of singular solutions for this reduced problem. A stable blow-up pattern and a countable family of other unstable solutions are found. The blow-up point is ejected to infinity in finite time, and the solutions form a plateau with growing length. The proof uses modulation techniques and different energy estimates in the various zones of interest.

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