Nonlinear surface waves on the plasma-vacuum interface
Author:
Paolo Secchi
Journal:
Quart. Appl. Math. 73 (2015), 711-737
MSC (2010):
Primary 76W05; Secondary 35Q35, 35L50, 76E17, 76E25, 35R35, 76B03
DOI:
https://doi.org/10.1090/qam/1405
Published electronically:
September 15, 2015
MathSciNet review:
3432280
Full-text PDF Free Access
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Additional Information
Abstract:
In this paper we study the propagation of weakly nonlinear surface waves on a plasma-vacuum interface. In the plasma region we consider the equations of incompressible magnetohydrodynamics, while in vacuum the magnetic and electric fields are governed by the Maxwell equations. A surface wave propagates along the plasma-vacuum interface when it is linearly weakly stable.
Following the approach of Ali and Hunter (2003), we measure the amplitude of the surface wave by the normalized displacement of the interface in a reference frame moving with the linearized phase velocity of the wave, and obtain that it satisfies an asymptotic nonlocal, Hamiltonian evolution equation. We show the local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables, and we derive a blow up criterion.
References
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- Sylvie Benzoni-Gavage, Local well-posedness of nonlocal Burgers equations, Differential Integral Equations 22 (2009), no. 3-4, 303–320. MR 2492823
- Sylvie Benzoni-Gavage and Jean-François Coulombel, On the amplitude equations for weakly nonlinear surface waves, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 871–925. MR 2960035, DOI https://doi.org/10.1007/s00205-012-0522-7
- Sylvie Benzoni-Gavage, Jean-François Coulombel, and Nikolay Tzvetkov, Ill-posedness of nonlocal Burgers equations, Adv. Math. 227 (2011), no. 6, 2220–2240. MR 2807088, DOI https://doi.org/10.1016/j.aim.2011.04.017
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- Davide Catania, Marcello D’Abbicco, and Paolo Secchi, Stability of the linearized MHD-Maxwell free interface problem, Commun. Pure Appl. Anal. 13 (2014), no. 6, 2407–2443. MR 3248397, DOI https://doi.org/10.3934/cpaa.2014.13.2407
- J. P. Goedbloed and S. Poedts, Principles of magnetohydrodynamics with applications to laboratory and astrophysical plasmas, Cambridge University Press, Cambridge, 2004.
- M. F. Hamilton, Yu. A. Il’insky, and E. A. Zabolotskaya, Evolution equations for nonlinear Rayleigh waves, J. Acoust. Soc. Am. 97 (1995), 891–897.
- John K. Hunter, Nonlinear surface waves, Current progress in hyperbolic systems: Riemann problems and computations (Brunswick, ME, 1988) Contemp. Math., vol. 100, Amer. Math. Soc., Providence, RI, 1989, pp. 185–202. MR 1033516, DOI https://doi.org/10.1090/conm/100/1033516
- John K. Hunter, Short-time existence for scale-invariant Hamiltonian waves, J. Hyperbolic Differ. Equ. 3 (2006), no. 2, 247–267. MR 2229856, DOI https://doi.org/10.1142/S0219891606000781
- John K. Hunter, Nonlinear hyperbolic surface waves, Nonlinear conservation laws and applications, IMA Vol. Math. Appl., vol. 153, Springer, New York, 2011, pp. 303–314. MR 2857003, DOI https://doi.org/10.1007/978-1-4419-9554-4_16
- John K. Hunter and J. B. Thoo, On the weakly nonlinear Kelvin-Helmholtz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equ. 8 (2011), no. 4, 691–726. MR 2864545, DOI https://doi.org/10.1142/S0219891611002548
- Tosio Kato, Nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 9–23. MR 843591
- Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 437941, DOI https://doi.org/10.1002/cpa.3160230304
- Nikita Mandrik and Yuri Trakhinin, Influence of vacuum electric field on the stability of a plasma-vacuum interface, Commun. Math. Sci. 12 (2014), no. 6, 1065–1100. MR 3194371, DOI https://doi.org/10.4310/CMS.2014.v12.n6.a4
- Alice Marcou, Rigorous weakly nonlinear geometric optics for surface waves, Asymptot. Anal. 69 (2010), no. 3-4, 125–174. MR 2760337
- Alessandro Morando, Yuri Trakhinin, and Paola Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD, Quart. Appl. Math. 72 (2014), no. 3, 549–587. MR 3237563, DOI https://doi.org/10.1090/S0033-569X-2014-01346-7
- Paolo Secchi and Yuri Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interfaces Free Bound. 15 (2013), no. 3, 323–357. MR 3148595, DOI https://doi.org/10.4171/IFB/305
- Paolo Secchi and Yuri Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlinearity 27 (2014), no. 1, 105–169. MR 3151094, DOI https://doi.org/10.1088/0951-7715/27/1/105
- Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. MR 1477408
- 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, 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
References
- Giuseppe Alì and John K. Hunter, Nonlinear surface waves on a tangential discontinuity in magnetohydrodynamics, Quart. Appl. Math. 61 (2003), no. 3, 451–474. MR 1999831 (2004f:35111)
- G. Alì, J. K. Hunter, and D. F. Parker, Hamiltonian equations for scale-invariant waves, Stud. Appl. Math. 108 (2002), no. 3, 305–321. MR 1895286 (2003i:37062), DOI https://doi.org/10.1111/1467-9590.01416
- Sylvie Benzoni-Gavage, Local well-posedness of nonlocal Burgers equations, Differential Integral Equations 22 (2009), no. 3-4, 303–320. MR 2492823 (2010m:35421)
- Sylvie Benzoni-Gavage and Jean-François Coulombel, On the amplitude equations for weakly nonlinear surface waves, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 871–925. MR 2960035, DOI https://doi.org/10.1007/s00205-012-0522-7
- Sylvie Benzoni-Gavage, Jean-François Coulombel, and Nikolay Tzvetkov, Ill-posedness of nonlocal Burgers equations, Adv. Math. 227 (2011), no. 6, 2220–2240. MR 2807088 (2012f:35455), DOI https://doi.org/10.1016/j.aim.2011.04.017
- S. Benzoni-Gavage and M. D. Rosini, Weakly nonlinear surface waves and subsonic phase boundaries, Comput. Math. Appl. 57 (2009), no. 9, 1463–1484. MR 2509960 (2010e:76075), DOI https://doi.org/10.1016/j.camwa.2008.12.001
- 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)
- Davide Catania, Marcello D’Abbicco, and Paolo Secchi, Stability of the linearized MHD-Maxwell free interface problem, Commun. Pure Appl. Anal. 13 (2014), no. 6, 2407–2443. MR 3248397, DOI https://doi.org/10.3934/cpaa.2014.13.2407
- J. P. Goedbloed and S. Poedts, Principles of magnetohydrodynamics with applications to laboratory and astrophysical plasmas, Cambridge University Press, Cambridge, 2004.
- M. F. Hamilton, Yu. A. Il’insky, and E. A. Zabolotskaya, Evolution equations for nonlinear Rayleigh waves, J. Acoust. Soc. Am. 97 (1995), 891–897.
- John K. Hunter, Nonlinear surface waves, Current progress in hyperbolic systems: Riemann problems and computations (Brunswick, ME, 1988) Contemp. Math., vol. 100, Amer. Math. Soc., Providence, RI, 1989, pp. 185–202. MR 1033516 (91a:35105), DOI https://doi.org/10.1090/conm/100/1033516
- John K. Hunter, Short-time existence for scale-invariant Hamiltonian waves, J. Hyperbolic Differ. Equ. 3 (2006), no. 2, 247–267. MR 2229856 (2007k:35414), DOI https://doi.org/10.1142/S0219891606000781
- John K. Hunter, Nonlinear hyperbolic surface waves, Nonlinear conservation laws and applications, IMA Vol. Math. Appl., vol. 153, Springer, New York, 2011, pp. 303–314. MR 2857003 (2012i:35229), DOI https://doi.org/10.1007/978-1-4419-9554-4_16
- John K. Hunter and J. B. Thoo, On the weakly nonlinear Kelvin-Helmholtz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equ. 8 (2011), no. 4, 691–726. MR 2864545 (2012j:35237), DOI https://doi.org/10.1142/S0219891611002548
- Tosio Kato, Nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 9–23. MR 843591 (87h:34091)
- Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 0437941 (55 \#10862)
- Nikita Mandrik and Yuri Trakhinin, Influence of vacuum electric field on the stability of a plasma-vacuum interface, Commun. Math. Sci. 12 (2014), no. 6, 1065–1100. MR 3194371, DOI https://doi.org/10.4310/CMS.2014.v12.n6.a4
- Alice Marcou, Rigorous weakly nonlinear geometric optics for surface waves, Asymptot. Anal. 69 (2010), no. 3-4, 125–174. MR 2760337 (2011m:35370)
- A. Morando, Y. Trakhinin, and P. Trebeschi, Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible mhd, Quart. Appl. Math. 72 (2014), no. 3, 549–587. MR 3237563
- Paolo Secchi and Yuri Trakhinin, Well-posedness of the linearized plasma-vacuum interface problem, Interfaces Free Bound. 15 (2013), no. 3, 323–357. MR 3148595, DOI https://doi.org/10.4171/IFB/305
- Paolo Secchi and Yuri Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlinearity 27 (2014), no. 1, 105–169. MR 3151094, DOI https://doi.org/10.1088/0951-7715/27/1/105
- Michael E. Taylor, Partial differential equations. III, Nonlinear equations, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Corrected reprint of the 1996 original. MR 1477408 (98k:35001)
- 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
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Additional Information
Paolo Secchi
Affiliation:
DICATAM, Mathematical Division, University of Brescia, Via Branze 43, 25123 Brescia, Italy
MR Author ID:
157900
Email:
paolo.secchi@unibs.it
Received by editor(s):
March 25, 2014
Published electronically:
September 15, 2015
Additional Notes:
The author is supported by the national research project PRIN 2012 “Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: Theoretical and applicative aspects”.
Article copyright:
© Copyright 2015
Brown University
