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Local‐in‐Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods
Author(s) -
Masmoudi Nader,
Wong Tak Kwong
Publication year - 2015
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21595
Subject(s) - uniqueness , mathematics , prandtl number , nonlinear system , sobolev space , monotonic function , mathematical analysis , regularization (linguistics) , euler's formula , euler equations , physics , mechanics , heat transfer , quantum mechanics , artificial intelligence , computer science
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. © 2015 Wiley Periodicals, Inc.