Communications in Mathematical Sciences

Volume 14 (2016)

Number 3

Rectilinear vortex sheets of inviscid liquid-gas two-phase flow: Linear stability

Pages: 735 – 776



Lizhi Ruan (The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan, China)

Dehua Wang (Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania, U.S.A.)

Shangkun Weng (Pohang Mathematics Institute, Pohang University of Science and Technology, Nam-Gu, Pohang, Gyungbuk, Korea)

Changjiang Zhu (School of Mathematics, South China University of Technology, Guangzhou, China)


The vortex sheet solutions are considered for the inviscid liquid-gas two-phase flow. In particular, the linear stability of rectilinear vortex sheets in two spatial dimensions is established for both constant and variable coefficients. The linearized problem of vortex sheet solutions with constant coefficients is studied by means of Fourier analysis, normal mode analysis, and Kreiss symmetrizer, while the linear stability with variable coefficients is obtained by Bony–Meyer paradifferential calculus theory. The linear stability is crucial to the existence of vortex sheet solutions of the nonlinear problem. A novel symmetrization and some weighted Sobolev norms are introduced to study the hyperbolic linearized problem with characteristic boundary.


inviscid liquid-gas two-phase flow, vortex sheet, linear stability

2010 Mathematics Subject Classification

34B05, 35L50, 35L65, 76T10

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Published 26 February 2016