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Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations. (English) Zbl 1084.76020

Summary: We use the method of characteristics to prove the short-time existence of smooth solutions of the unsteady inviscid Prandtl equations, and present a simple explicit solution that forms a singularity in finite time. We give numerical and asymptotic solutions which indicate that this singularity persists for nonmonotone of viscous Prandtl equations. We also solve the linearization of inviscid Prandtl equations about a shear flow. We show that the resulting problem is weakly, but not strongly, well-posed, and that it has an unstable continuous spectrum when the shear flow has a critical point, in contrast with the behavior of linearized Euler equations.

MSC:

76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
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