Almost Global Existence for the Prandtl Boundary Layer Equations

By Mihaela Ignatova, Vlad Vicol. 2015

130 citations recorded by the source corpus.

We consider the Prandtl boundary layer equations on the half plane, with initial datum that lies in a weighted H1 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 $${\varepsilon}$$ε of a stable profile, then the unique solution of the Cauchy problem can be extended at least up to time $${T_{\varepsilon} \geqq {\rm exp}(\varepsilon^{-1} / {\rm log}(\varepsilon^{-1}))}$$Tε≧exp(ε-1/log(ε-1)).

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/392449ce62132730c1cdf0ad793b4947a2afd62f

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Global existence and decay of solutions to Prandtl system with small analytic data. (2019, unknown): “3 of [15], here we do not need any boundary condition for u on the boundary y = 0.”
The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary (2019, methodology): “We refer the reader to [1, 10, 17, 28, 35, 36, 37, 39, 47, 51, 52] for the well-posedness theory for the Prandtl equations, to [14, 26, 18, 38] for the identification of ill-posed regimes, and to [19, 20, 21, 22, 23] for recent works which show the invalidity of the Prandtl expansion at the level of”
Well-posedness of the hydrostatic Navier–Stokes equations (2018, methodology): “We refer to [10, 42, 13, 20, 26, 9] and references therein for more results on the Prandtl system such as singularities, long time behavior, and Gevrey-class stability.”
The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space (2017, unknown): “Notably, solutions to the Prandtl equations have been constructed for monotonic data ([31, 32, 1, 29]) or data with Gevrey or analytic regularity ([35, 11, 22], among others).”