Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams R.A., Fournier J.J.F.: Sobolev spaces. Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edn. Elsevier/Academic Press, Amsterdam (2003)
Alinhac, S.: Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Commun. Partial Differ. Equ. 14(2), 173–230 (1989). doi:10.1080/03605308908820595
Benzoni-Gavage, S., Serre, D.: Multidimensional hyperbolic partial differential equations. First-order systems and applications. In: Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford (2007)
Chen, G.Q., Wang, Y.G.: Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics. Arch. Rat. Mech. Anal. 187(3), 369–408 (2008). doi:10.1007/s00205-007-0070-8
Chen, G.Q., Wang, Y.G.: Characteristic discontinuities and free boundary problems for hyperbolic conservation laws. Nonlinear Partial Differential Equations—The Abel Symposium 2010, vol. 7. Abel Symposia, Vol. 7 (Eds. Holden H. and Karlsen K.). Springer, Berlin, 2012
Cheng, C.H.A., Coutand, D., Shkoller, S.: On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity. Commun. Pure Appl. Math. 61(12), 1715–1752 (2008). doi:10.1002/cpa.20240
Coulombel, J.F., Morando, A., Secchi, P., Trebeschi, P.: A priori estimates for 3D incompressible current-vortex sheets. Commun. Math. Phys. 311(1), 247–275 (2012). doi:10.1007/s00220-011-1340-8
Coulombel, J.F., Secchi, P.: The stability of compressible vortex sheets in two space dimensions. Indiana Univ. Math. J. 53(4), 941–1012 (2004). doi:10.1512/iumj.2004.53.2526
Coulombel, J.F., Secchi, P.: Nonlinear compressible vortex sheets in two space dimensions. Ann. Sci. Éc. Norm. Supér. (4) 41(1), 85–139 (2008)
Coutand, D., Hole, J., Shkoller, S.: Well-posedness of the free-boundary compressible 3-d euler equations with surface tension and the zero surface tension limit. SIAM J. Math. Anal. 45(6), 3690–3767 (2013). doi:10.1137/120888697
Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007). doi:10.1090/S0894-0347-07-00556-5
Coutand, D., Shkoller, S.: A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete Contin. Dyn. Syst. Ser. S 3(3), 429–449 (2010). doi:10.3934/dcdss.2010.3.429
Coutand, D., Shkoller, S.: Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Rat. Mech. Anal. 206(2), 515–616 (2012). doi:10.1007/s00205-012-0536-1
Evans L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)
Fejer J.A., Miles J.W.: On the stability of a plane vortex sheet with respect to three-dimensional disturbances. J. Fluid Mech. 15, 335–336 (1963)
Jang J., Masmoudi N.: Well-posedness of compressible euler equations in a physical vacuum. Commun. Pure Appl. Math. 68(1), 61–111 (2015)
Lindblad H.: Well posedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 260(2), 319–392 (2005
Lindblad H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. 162(1), 109–194 (2005)
Miles J.W.: On the disturbed motion of a plane vortex sheet. J. Fluid Mech. 4, 538–552 (1958)
Trakhinin, Y.: The existence of current-vortex sheets in ideal compressible magnetohydrodynamics. Arch. Rat. Mech. Anal. 191(2), 245–310 (2009). doi:10.1007/s00205-008-0124-6
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Stevens, B. Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension. Arch Rational Mech Anal 222, 603–730 (2016). https://doi.org/10.1007/s00205-016-1009-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-016-1009-8