Abstract
We establish the local existence and uniqueness of solutions to the free-boundary ideal compressible magnetohydrodynamic equations with surface tension in three spatial dimensions by a suitable modification of the Nash–Moser iteration scheme. The main ingredients in proving the convergence of the scheme are the tame estimates and unique solvability of the linearized problem in the anisotropic Sobolev spaces \(H_*^m\) for m large enough. In order to derive the tame estimates, we make full use of the boundary regularity enhanced from the surface tension. The unique solution of the linearized problem is constructed by designing some suitable \(\varepsilon \)–regularization and passing to the limit \(\varepsilon \rightarrow 0\).
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Alinhac, S.: Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Commun. Partial Differ. Eqs. 14(2), 173–230 (1989)
Alinhac, S., Gérard, P.: Pseudo-differential Operators and the Nash–Moser Theorem. Translated from the 1991 French original by Stephen S. Wilson. American Mathematical Society, Providence (2007)
Chazarain, J., Piriou, A.: Introduction to the Theory of Linear Partial Differential Equations. North-Holland Publishing Co., Amsterdam-New York (1982)
Chen, G.-Q., Wang, Y.-G.: Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics. Arch. Rational Mech. Anal. 187(3), 369–408 (2008)
Chen, G.-Q., Secchi, P., Wang, T.: Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime. Arch. Rational Mech. Anal. 232(2), 591–695 (2019)
Chen, S.: Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary. Front. Math. China 2(1), 87–102 (2007). Translated from Chinese Ann. Math. 3(2), 222–232 (1982)
Coulombel, J.-F., Secchi, P.: Nonlinear compressible vortex sheets in two space dimensions. Annales scientifiques de l’École Normale Supérieure, Quatrième Série 41(1), 85–139 (2008)
Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Amer. Math. Soc. 20(3), 829–930 (2007)
Coutand, D., Shkoller, S.: Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Rational Mech. Anal. 206(2), 515–616 (2012)
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)
Delhaye, J.M.: Jump conditions and entropy sources in two-phase systems. Local instant formulation. Int. J. Multiphase Flow 1(3), 395–409 (1974)
Ebin, D.G.: The equations of motion of a perfect fluid with free boundary are not well posed. Commun. Partial Differ. Eqs. 12(10), 1175–1201 (1987)
Gu, X., Wang, Y.: On the construction of solutions to the free-surface incompressible ideal magnetohydrodynamic equations. J. de Mathématiques Pures et Appliquées, Neuvième Série 128, 1–41 (2019)
Hao, C., Luo, T.: A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows. Arch. Rational Mech. Anal. 212(3), 805–847 (2014)
Hao, C., Luo, T.: Ill-posedness of free boundary problem of the incompressible ideal MHD. Commun. Math. Phys. 376(1), 259–286 (2020)
Hörmander, L.: The boundary problems of physical geodesy. Arch. Rational Mech. Anal. 62(1), 1–52 (1976)
Jang, J., Masmoudi, N.: Well-posedness of compressible Euler equations in a physical vacuum. Commun. Pure Appl. Math. 68(1), 61–111 (2015)
Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, 2nd edn. Pergamon Press, Oxford (1984)
Lax, P.D., Phillips, R.S.: Local boundary conditions for dissipative symmetric linear differential operators. Commun. Pure Appl. Math. 13, 427–455 (1960)
Lindblad, H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. 162(1), 109–194 (2005)
Lindblad, H.: Well posedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 260(2), 319–392 (2005)
Métivier, G.: Stability of multidimensional shocks. In: Freistühler, H., Szepessy, A. (eds.) Advances in the Theory of Shock Waves, pp. 25–103. Birkhäuser, Boston (2001)
Mishkov, R.L.: Generalization of the formula of Faa di Bruno for a composite function with a vector argument. Int. J. Math. Math. Sci. 24, 481–491 (2000)
Morando, A., Trakhinin, Y., Trebeschi, P.: Local existence of MHD contact discontinuities. Arch. Rational Mech. Anal. 228(2), 691–742 (2018)
Samulyak, R., Du, J., Glimm, J., Xu, Z.: A numerical algorithm for MHD of free surface flows at low magnetic Reynolds numbers. J. Comput. Phys. 226(2), 1532–1549 (2007)
Secchi, P.: Well-posedness of characteristic symmetric hyperbolic systems. Arch. Rational Mech. Anal. 134, 155–197 (1996)
Secchi, P.: On the Nash-Moser iteration technique. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds.) Recent developments of mathematical fluid mechanics, pp. 443–457. Birkhäuser, Basel (2016)
Secchi, P., Trakhinin, Y.: Well-posedness of the plasma-vacuum interface problem. Nonlinearity 27(1), 105–169 (2014)
Shatah, J., Zeng, C.: Geometry and a priori estimates for free boundary problems of the Euler equation. Commun. Pure Appl. Math. 61(5), 698–744 (2008)
Shatah, J., Zeng, C.: Local well-posedness for fluid interface problems. Arch. Rational Mech. Anal. 199(2), 653–705 (2011)
Stone, J.M., Gardiner, T.: Nonlinear evolution of the magnetohydrodynamic Rayleigh–Taylor instability. Phys. Fluids 19(9), 094104 (2007)
Stone, J.M., Gardiner, T.: The magnetic Rayleigh-Taylor instability in three dimensions. The Astrophysical Journal 671(2), 1726–1735 (2007)
Trakhinin, Y.: The existence of current-vortex sheets in ideal compressible magnetohydrodynamics. Arch. Rational Mech. Anal. 191(2), 245–310 (2009)
Trakhinin, Y.: Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition. Commun. Pure Appl. Math. 62(11), 1551–1594 (2009)
Trakhinin, Y., Wang, T.: Well-posedness of free boundary problem in non-relativistic and relativistic ideal compressible magnetohydrodynamics. Arch. Rational Mech. Anal. 239(2), 1131–1176 (2021)
Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc. 12(2), 445–495 (1999)
Yang, F., Khodak, A., Stone, H.A.: The effects of a horizontal magnetic field on the Rayleigh–Taylor instability. Nuclear Materials and Energy 18, 175–181 (2019)
Zhang, P., Zhang, Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61(7), 877–940 (2008)
Zorich, V.: Mathematical analysis. I, 2nd edn. Universitext. Springer-Verlag, Berlin (2015)
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The research of Yuri Trakhinin was supported by Mathematical Center in Akademgorodok under Agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation. The research of Tao Wang was partially supported by the National Natural Science Foundation of China under Grants 11971359 and 11731008.
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Trakhinin, Y., Wang, T. Well-posedness for the free-boundary ideal compressible magnetohydrodynamic equations with surface tension. Math. Ann. 383, 761–808 (2022). https://doi.org/10.1007/s00208-021-02180-z
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DOI: https://doi.org/10.1007/s00208-021-02180-z
Keywords
- Free boundary problem
- Ideal compressible magnetohydrodynamics
- Surface tension
- Well-posedness
- Nash–Moser iteration