Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow

Feimin Huang; Dehua Wang; Difan Yuan

doi:10.3934/dcds.2019146

Article Contents

2019, Volume 39, Issue 6: 3535-3575. Doi: 10.3934/dcds.2019146

This issue Previous Article On the periodic Zakharov-Kuznetsov equation Next Article Well-posedness of general 1D initial boundary value problems for scalar balance laws

Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow

1.
Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100190, China
2.
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
3.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
4.
Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, China

* Corresponding author: Difan Yuan
* Corresponding author: Difan Yuan

Received: August 31, 2018

Published: February 2019

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We are concerned with the vortex sheet solutions for the inviscid two-phase flow in two dimensions. In particular, the nonlinear stability and existence of compressible vortex sheet solutions under small perturbations are established by using a modification of the Nash-Moser iteration technique, where a priori estimates for the linearized equations have a loss of derivatives. Due to the jump of the normal derivatives of densities of liquid and gas, we obtain the normal estimates in the anisotropic Sobolev space, instead of the usual Sobolev space. New ideas and techniques are developed to close the energy estimates and derive the tame estimates for the two-phase flows.

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Mathematics Subject Classification: Primary: 76T10; Secondary: 34B05, 35L60, 35L65.

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