Local-in-Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods

Nader Masmoudi, Tak Kwong Wong

Research output: Contribution to journalArticle

Abstract

We prove local existence and uniqueness for the two-dimensional Prandtl system in weighted Sobolev spaces under Oleinik's monotonicity assumption. In particular we do not use the Crocco transform or any change of variables. Our proof is based on a new nonlinear energy estimate for the Prandtl system. This new energy estimate is based on a cancellation property that is valid under the monotonicity assumption. To construct the solution, we use a regularization of the system that preserves this nonlinear structure. This new nonlinear structure may give some insight into the convergence properties from the Navier-Stokes system to the Euler system when the viscosity goes to 0.

Original languageEnglish (US)
Pages (from-to)1683-1741
Number of pages59
JournalCommunications on Pure and Applied Mathematics
Volume68
Issue number10
DOIs
StatePublished - Oct 1 2015

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Sobolev spaces
Energy Method
Existence and Uniqueness of Solutions
Energy Estimates
Viscosity
Monotonicity
Euler System
Navier-Stokes System
Weighted Sobolev Spaces
Change of Variables
Local Existence
Cancellation
Convergence Properties
Regularization
Existence and Uniqueness
Valid
Transform

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Local-in-Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods. / Masmoudi, Nader; Wong, Tak Kwong.

In: Communications on Pure and Applied Mathematics, Vol. 68, No. 10, 01.10.2015, p. 1683-1741.

Research output: Contribution to journalArticle

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