Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points

Abstract

In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. 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 in the paper of D. Gérard-Varet and N. Masmoudi [{\it Ann. Sci. Éc. Norm. Supér}. (4) 48 (2015), no. 6, 1273-1325], in which the case $\sigma =7/4$ is solved.

50pages

References in corpus (2)

• Almost global existence for the Prandtl boundary layer equations
• Long time well-posdness of the Prandtl equations in Sobolev space

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• The inviscid limit for the Navier-Stokes equations with data analytic only near the boundary
• Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow
• Blow-up criterion for the 2-D Prandtl equation
• Well-posedness and blowup of the geophysical boundary layer problem
• Global-in-time Stability of 2D MHD boundary Layer in the Prandtl-Hartmann Regime
• Local-in-time well-posedness for Compressible MHD boundary layer
• Real analytic local well-posedness for the Triple Deck