Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD
Authors:
Alessandro Morando, Yuri Trakhinin and Paola Trebeschi
Journal:
Quart. Appl. Math. 72 (2014), 549-587
MSC (2000):
Primary 76W05; Secondary 35Q35, 35L50, 76E17, 76E25, 35R35, 76B03
DOI:
https://doi.org/10.1090/S0033-569X-2014-01346-7
Published electronically:
April 22, 2014
MathSciNet review:
3237563
Full-text PDF Free Access
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Additional Information
Abstract: We study the free boundary problem for the plasma-vacuum interface in ideal incompressible magnetohydrodynamics (MHD). In the vacuum region the magnetic field is described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic field is tangent to the boundary. Under a suitable stability condition satisfied at each point of the plasma-vacuum interface, we prove the well-posedness of the linearized problem in Sobolev spaces.
References
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- Yuri Trakhinin, Stability of relativistic plasma-vacuum interfaces, J. Hyperbolic Differ. Equ. 9 (2012), no. 3, 469–509. MR 2974767, DOI https://doi.org/10.1142/S0219891612500154
- Ping Zhang and Zhifei Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math. 61 (2008), no. 7, 877–940. MR 2410409, DOI https://doi.org/10.1002/cpa.20226
References
- S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 14 (1989), no. 2, 173–230 (French, with English summary). MR 976971 (90h:35147b), DOI https://doi.org/10.1080/03605308908820595
- Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations: First-order systems and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. MR 2284507 (2008k:35002)
- I. B. Bernstein, E. A. Frieman, M. D. Kruskal, and R. M. Kulsrud, An energy principle for hydromagnetic stability problems, Proc. Roy. Soc. London. Ser. A. 244 (1958), 17–40. MR 0091737 (19,1009e)
- J.-F. Coulombel, A. Morando, P. Secchi, and P. Trebeschi, A priori estimates for 3D incompressible current-vortex sheets, Comm. Math. Phys. 311 (2012), no. 1, 247–275. MR 2892470, DOI https://doi.org/10.1007/s00220-011-1340-8
- Jean-François Coulombel and Paolo Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 85–139 (English, with English and French summaries). MR 2423311 (2010a:76075)
- Daniel Coutand and Steve Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc. 20 (2007), no. 3, 829–930. MR 2291920 (2008c:35242), DOI https://doi.org/10.1090/S0894-0347-07-00556-5
- Daniel Coutand and Steve Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S 3 (2010), no. 3, 429–449. MR 2660719 (2011h:35211), DOI https://doi.org/10.3934/dcdss.2010.3.429
- J. P. Goedbloed, S. Poedts, Principles of magnetohydrodynamics with applications to laboratory and astrophysical plasmas, Cambridge University Press, Cambridge, 2004.
- David Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18 (2005), no. 3, 605–654 (electronic). MR 2138139 (2005m:76021), DOI https://doi.org/10.1090/S0894-0347-05-00484-4
- Hans Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2) 162 (2005), no. 1, 109–194. MR 2178961 (2006g:35293), DOI https://doi.org/10.4007/annals.2005.162.109
- J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243 (40 \#512)
- Alessandro Morando, Yuri Trakhinin, and Paola Trebeschi, Stability of incompressible current-vortex sheets, J. Math. Anal. Appl. 347 (2008), no. 2, 502–520. MR 2440346 (2009d:76142), DOI https://doi.org/10.1016/j.jmaa.2008.06.002
- P. Secchi, Y. Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interfaces Free Bound. 15 (2013), no. 3, 323–357, DOI 10.4171/IFB/30. MR 3148595
- P. Secchi, Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlineariy 27 (2014), no. 1, 105–169. DOI 10.1088/0951-7715/27/1/105. MR 3151094
- Yuri Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal. 191 (2009), no. 2, 245–310. MR 2481071 (2011c:35471), DOI https://doi.org/10.1007/s00205-008-0124-6
- Yuri Trakhinin, On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD, J. Differential Equations 249 (2010), no. 10, 2577–2599. MR 2718711 (2011k:35257), DOI https://doi.org/10.1016/j.jde.2010.06.007
- Yuri Trakhinin, Stability of relativistic plasma-vacuum interfaces, J. Hyperbolic Differ. Equ. 9 (2012), no. 3, 469–509. MR 2974767, DOI https://doi.org/10.1142/S0219891612500154
- Ping Zhang and Zhifei Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math. 61 (2008), no. 7, 877–940. MR 2410409 (2009h:35461), DOI https://doi.org/10.1002/cpa.20226
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Additional Information
Alessandro Morando
Affiliation:
Dipartimento di Matematica, Facoltà di Ingegneria, Università di Brescia, Via Valotti, 9, 25133 Brescia, Italy
Email:
alessandro.morando@unibs.it
Yuri Trakhinin
Affiliation:
Sobolev Institute of Mathematics, Koptyug av. 4, 630090 Novosibirsk, Russia
Email:
trakhin@math.nsc.ru
Paola Trebeschi
Affiliation:
Dipartimento di Matematica, Facoltà di Ingegneria, Università di Brescia, Via Valotti, 9, 25133 Brescia, Italy
Email:
paola.trebeschi@unibs.it
Received by editor(s):
July 6, 2012
Received by editor(s) in revised form:
August 30, 2012
Published electronically:
April 22, 2014
Additional Notes:
Part of this work was done during the fellowship of Y.T. at the Landau Network–Centro Volta–Cariplo Foundation spent at the Department of Mathematics of the University of Brescia in Italy. Y.T. would like to warmly thank the Department of Mathematics of the University of Brescia for its kind hospitality during the visiting period. Research of Y.T. was also partially supported under RFBR (Russian Foundation for Basic Research) grant number 10-01-00320-a
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© Copyright 2014
Brown University
