Long time well-posdness of the Prandtl equations in Sobolev space

Abstract

In this paper, we study the long time well-posedness for the nonlinear Prandtl boundary layer equation on the half plane. While the initial data are small perturbations of some monotonic shear profile, we prove the existence, uniqueness and stability of solutions in weighted Sobolev space by energy methods. The key point is that the life span of the solution could be any large $T$ as long as its initial date is a perturbation around the monotonic shear profile of small size like $e^{-T}$. The nonlinear cancellation properties of Prandtl equations under the monotonic assumption are the main ingredients to establish a new energy estimate.

In this version, reviser some typos, 43 pages

References in corpus (4)

• Almost global existence for the Prandtl boundary layer equations
• Long time well-posdness of Prandtl system with small and analytic initial data
• A well-posedness theory for the Prandtl equations in three space variables
• Global existence of weak solutions to the three-dimensional Prandtl equations with A special structure

Cited by in corpus (3)

• Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points
• Well-posedness in Gevrey function space for the three-dimensional Prandtl equations
• The van Dommelen and Shen singularity in the Prandtl equations