Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure

Cheng-Jie  Liu; Ya-Guang  Wang; Tong Yang

doi:10.3934/dcdss.2016082

Article Contents

2016, Volume 9, Issue 6: 2011-2029. Doi: 10.3934/dcdss.2016082

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Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure

1.
Department of Mathematics, City University of Hong Kong, Hong Kong
2.
School of Mathematical Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240
3.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received: March 31, 2015

Revised: August 31, 2016

Published: November 2016

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Abstract

The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang [14] on the two-dimensional Prandtl equations to the three-dimensional setting.

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Mathematics Subject Classification: 35M13, 35Q35, 76D03, 76D10, 76N20.

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