Well-posedness in Gevrey function space for the three-dimensional Prandtl equations

Abstract

In the paper, we study the three-dimensional Prandtl equations without any monotonicity condition on the velocity field. We prove that when one tangential component of the velocity field has a single curve of non-degenerate critical points with respect to the normal variable, the system is locally well-posed in the Gevrey function space with Gevrey index in $]1,2].$ The proof is based on some new observation of cancellation mechanism in the three space dimensional system in addition to those in the two-dimensional setting obtained in [1,7,19,22].

Enlarged version without any monotonicity assumption

References in corpus (5)

• Almost global existence for the Prandtl boundary layer equations
• On the ill-posedness of the Prandtl equations in three space dimensions
• Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points
• Long time well-posdness of Prandtl system with small and analytic initial data
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Cited by in corpus (4)

• Well-posedness in Gevrey function space for 3D Prandtl equations without Structural Assumption
• Justification of Prandtl Ansatz for MHD boundary layer
• Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption
• Separation of the two-dimensional unsteady Prandtl boundary layers under an adverse pressure gradient