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Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points. (English) Zbl 1442.35305

Summary: We study the well-posedness of the Prandtl system without monotonicity and analyticity assumption. Precisely, for any index \(\sigma\in[3/2, 2],\) we obtain the local in time well-posedness in the space of Gevrey class \(G^\sigma\) in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised by D. Gérard-Varet and N. Masmoudi [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 6, 1273–1325 (2015; Zbl 1347.35201)], who solved the case \(\sigma=7/4\).

MSC:

35Q30 Navier-Stokes equations
35Q31 Euler equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Citations:

Zbl 1347.35201
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