Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

• Published in: Archive for rational mechanics and analysis Vol. 222; no. 2; pp. 603 - 730
• Main Author: Stevens, Ben
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2016; Springer Nature B.V
• Subjects: Classical Mechanics; Complex Systems; Compressibility; Computational fluid dynamics; Density; Entropy; Equations of state; Euler-Lagrange equation; Fluid- and Aerodynamics; Fluids; Linear equations; Mathematical analysis; Mathematical and Computational Physics; Mathematical models; Nonlinear equations; Physics; Physics and Astronomy; Pressure jump; Regularization; Structural stability; Surface stability; Surface tension; Theoretical; Time compression; Viscosity; Vortex sheets; Vortices
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-016-1009-8

Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We model the fluids by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density such that the sound speed is positive. We prove that, for a short time, there exists a unique solution of the equations with the same structure. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity which becomes fairly standard after rotating the velocity according to the interface normal. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al. in the setting of a compressible liquid-vacuum interface. Although already considered by Coutand et al. [ 10 ] and Lindblad [ 17 ], we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.