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Separation for the stationary Prandtl equation. (English) Zbl 1427.35201

Summary: In this paper, we prove that separation occurs for the stationary Prandtl equation, in the case of adverse pressure gradient, for a large class of boundary data at \(x=0\). We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at \(x=0\), there exists \(x^*>0\) such that \(\partial_y u_{|y=0}(x) \sim C \sqrt{x^* -x}\) as \(x\to x^*\) for some positive constant \(C\), where \(u\) is the solution of the stationary Prandtl equation in the domain \(\{0< x< x^*, y> 0\}\). Our proof relies on three main ingredients: the computation of a “stable” approximate solution, using modulation theory arguments; a new formulation of the Prandtl equation, for which we derive energy estimates, relying heavily on the structure of the equation; and maximum principle and comparison principle techniques to handle some of the nonlinear terms.

MSC:

35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
35B50 Maximum principles in context of PDEs
35B44 Blow-up in context of PDEs
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