Almost global existence for the Prandtl boundary layer equations

Abstract

We consider the Prandtl boundary layer equations on the half plane, with initial datum that lies in a weighted $H^1$ space with respect to the normal variable, and is real-analytic with respect to the tangential variable. The boundary trace of the horizontal Euler flow is taken to be a constant. We prove that if the Prandtl datum lies within $\epsilon$ of a stable profile, then the unique solution of the Cauchy problem can be extended at least up to time $T_{\epsilon }\ge \exp ({\epsilon }^{-1}/\log ({\epsilon }^{-1}))$.

Fixed a number of typos

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Cited by in corpus (31)

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• Improved well-posedness for the Triple-Deck and related models via concavity
• Global existence and decay of solutions to Prandtl system with small analytic data
• Justification of Prandtl Ansatz for MHD boundary layer
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• On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class
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• Long time well-posdness of the Prandtl equations in Sobolev space
• Global small analytic solutions of MHD boundary layer equations
• Regularity and Expansion for Steady Prandtl Equations
• Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force
• Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption
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• Separation of the two-dimensional unsteady Prandtl boundary layers under an adverse pressure gradient
• Blow-up criterion for the 2-D Prandtl equation
• Local-in-time Well-posedness of Boundary Layer System for the Full Incompressible MHD Equations by Energy Methods
• Global Solvability of 2D MHD Boundary Layer Equations in Analytic Function Spaces
• Well-posedness and blowup of the geophysical boundary layer problem
• Well-posedness in Sobolev spaces of the two-dimensional MHD Boundary layer equations without viscosity
• Symmetrical Prandtl boundary layer expansions of steady Navier-Stokes equations on bounded domain
• Local-in-time well-posedness for Compressible MHD boundary layer
• Real analytic local well-posedness for the Triple Deck