# American Institute of Mathematical Sciences

January  2014, 34(1): 121-143. doi: 10.3934/dcds.2014.34.121

## Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains

 1 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1 2 IMATH, EA 2134, Université du Sud Toulon-Var, BP 132, 83957 La Garde, France 3 Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China

Received  August 2012 Published  June 2013

We consider the compressible Navier-Stokes system coupled with the Maxwell equations governing the time evolution of the magnetic field. We introduce a relative entropy functional along with the related concept of dissipative solution. As an application of the theory, we show that for small values of the Mach number and large Reynolds number, the global in time weak (dissipative) solutions converge to the ideal MHD system describing the motion of an incompressible, inviscid, and electrically conducting fluid. The proof is based on frequency localized Strichartz estimates for the Neumann Laplacean on unbounded domains.
Citation: Eduard Feireisl, Antonin Novotny, Yongzhong Sun. Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 121-143. doi: 10.3934/dcds.2014.34.121
##### References:
 [1] F. Berthelin and A. Vasseur, From kinetic equations to multidimensional isentropic gas dynamics before shocks,, SIAM J. Math. Anal., 36 (2005), 1807. doi: 10.1137/S0036141003431554. Google Scholar [2] N. Burq, Global Strichartz estimates for nontrapping geometries: About an article by H. F. Smith and C. D. Sogge: "Global Strichartz estimates for nontrapping perturbations of the Laplacian,'', Comm. Partial Differential Equations, 28 (2003), 1675. doi: 10.1081/PDE-120024528. Google Scholar [3] M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Funct. Anal., 179 (2001), 409. doi: 10.1006/jfan.2000.3687. Google Scholar [4] C. M. Dafermos, The second law of thermodynamics and stability,, Arch. Rational Mech. Anal., 70 (1979), 167. doi: 10.1007/BF00250353. Google Scholar [5] R. Danchin, Low Mach number limit for viscous compressible flows,, M2AN Math. Model Numer. Anal., 39 (2005), 459. doi: 10.1051/m2an:2005019. Google Scholar [6] Y. Dermenjian and J.-C. Guillot, Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle,, Math. Methods Appl. Sci., 10 (1988), 87. doi: 10.1002/mma.1670100202. Google Scholar [7] B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations,, Commun. Partial Differential Equations, 22 (1997), 977. doi: 10.1080/03605309708821291. Google Scholar [8] B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars,, Commun. Math. Phys., 266 (2006), 595. doi: 10.1007/s00220-006-0052-y. Google Scholar [9] J. Edward and D. Pravica, Bounds on resonances for the Laplacian on perturbations of half-space,, SIAM J. Math. Anal., 30 (1999), 1175. doi: 10.1137/S003614109733172X. Google Scholar [10] D. M. Èĭdus, The principle of limiting amplitude,, (in Russian) Usp. Mat. Nauk, 24 (1969), 91. Google Scholar [11] E. Feireisl, Incompressible limits and propagation of acoustic waves in large domains with boundaries,, Commun. Math. Phys., 294 (2010), 73. doi: 10.1007/s00220-009-0954-6. Google Scholar [12] E. Feireisl, Local decay of acoustic waves in the low Mach number limits on general unbounded domains under slip boundary conditions,, Commun. Partial Differential Equations, 36 (2011), 1778. doi: 10.1080/03605302.2011.602168. Google Scholar [13] E. Feireisl, Low Mach number limits of compressible rotating fluids,, J. Math. Fluid Mechanics, 14 (2012), 61. doi: 10.1007/s00021-010-0043-9. Google Scholar [14] E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system,, Arch. Rational Mech. Anal., 204 (2012), 683. doi: 10.1007/s00205-011-0490-3. Google Scholar [15] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids,, J. Math. Fluid Mech., 3 (2001), 358. doi: 10.1007/PL00000976. Google Scholar [16] E. Feireisl, A. Novotný and Y. Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids,, Indiana Univ. Math. J., 60 (2011), 611. doi: 10.1512/iumj.2011.60.4406. Google Scholar [17] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I,", Springer-Verlag, (1994). Google Scholar [18] I. Gallagher, Résultats récents sur la limite incompressible,, Séminaire Bourbaki. Vol. 2003/2004, (2005), 29. Google Scholar [19] P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system,, J. Math. Fluid Mech., 13 (2011), 137. doi: 10.1007/s00021-009-0006-1. Google Scholar [20] X. P. Hu and D. H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows,, SIAM J. Math. Anal., 41 (2009), 1272. doi: 10.1137/080723983. Google Scholar [21] S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions,, Comm. Math. Phys., 297 (2010), 371. doi: 10.1007/s00220-010-0992-0. Google Scholar [22] S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients,, SIAM J. Math. Anal., 42 (2010), 2539. doi: 10.1137/100785168. Google Scholar [23] T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation,, Arch. Rational Mech. Anal., 25 (1967), 188. Google Scholar [24] T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow,, J. Funct. Anal., 56 (1984), 15. doi: 10.1016/0022-1236(84)90024-7. Google Scholar [25] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. Google Scholar [26] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids,, Comm. Pure Appl. Math., 34 (1981), 481. doi: 10.1002/cpa.3160340405. Google Scholar [27] H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations,, Tohoku Math. J. (2), 41 (1989), 471. doi: 10.2748/tmj/1178227774. Google Scholar [28] P. Kukučka, Singular limits of the equations of magnetohydrodynamics,, J. Math. Fluid Mech., 13 (2011), 173. doi: 10.1007/s00021-009-0007-0. Google Scholar [29] Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows,, J. Differential Equations, 251 (2011), 1990. doi: 10.1016/j.jde.2011.04.016. Google Scholar [30] P.-L. Lions, "Mathematical Topics in Fluid Dynamics. Vol. 2. Compressible Models,", Oxford Lecture Series in Mathematics and its Applications, 10 (1998). Google Scholar [31] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, J. Math. Pures Appl. (9), 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6. Google Scholar [32] N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199. doi: 10.1016/S0294-1449(00)00123-2. Google Scholar [33] N. Masmoudi, Examples of singular limits in hydrodynamics,, in, (2007). doi: 10.1016/S1874-5717(07)80006-5. Google Scholar [34] S. Schochet, The mathematical theory of low Mach number flows,, M2ANMath. Model Numer. Anal., 39 (2005), 441. doi: 10.1051/m2an:2005017. Google Scholar [35] P. Secchi, On the equations of ideal incompressible magnetohydrodynamics,, Rend. Sem. Univ. Padova, 90 (1993), 103. Google Scholar [36] H. F. Smith and C. D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian,, Comm. Partial Differential Equations, 25 (2000), 2171. doi: 10.1080/03605300008821581. Google Scholar [37] R. S. Strichartz, A priori estimates for the wave equation and some applications,, J. Functional Analysis, 5 (1970), 218. Google Scholar

show all references

##### References:
 [1] F. Berthelin and A. Vasseur, From kinetic equations to multidimensional isentropic gas dynamics before shocks,, SIAM J. Math. Anal., 36 (2005), 1807. doi: 10.1137/S0036141003431554. Google Scholar [2] N. Burq, Global Strichartz estimates for nontrapping geometries: About an article by H. F. Smith and C. D. Sogge: "Global Strichartz estimates for nontrapping perturbations of the Laplacian,'', Comm. Partial Differential Equations, 28 (2003), 1675. doi: 10.1081/PDE-120024528. Google Scholar [3] M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Funct. Anal., 179 (2001), 409. doi: 10.1006/jfan.2000.3687. Google Scholar [4] C. M. Dafermos, The second law of thermodynamics and stability,, Arch. Rational Mech. Anal., 70 (1979), 167. doi: 10.1007/BF00250353. Google Scholar [5] R. Danchin, Low Mach number limit for viscous compressible flows,, M2AN Math. Model Numer. Anal., 39 (2005), 459. doi: 10.1051/m2an:2005019. Google Scholar [6] Y. Dermenjian and J.-C. Guillot, Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle,, Math. Methods Appl. Sci., 10 (1988), 87. doi: 10.1002/mma.1670100202. Google Scholar [7] B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations,, Commun. Partial Differential Equations, 22 (1997), 977. doi: 10.1080/03605309708821291. Google Scholar [8] B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars,, Commun. Math. Phys., 266 (2006), 595. doi: 10.1007/s00220-006-0052-y. Google Scholar [9] J. Edward and D. Pravica, Bounds on resonances for the Laplacian on perturbations of half-space,, SIAM J. Math. Anal., 30 (1999), 1175. doi: 10.1137/S003614109733172X. Google Scholar [10] D. M. Èĭdus, The principle of limiting amplitude,, (in Russian) Usp. Mat. Nauk, 24 (1969), 91. Google Scholar [11] E. Feireisl, Incompressible limits and propagation of acoustic waves in large domains with boundaries,, Commun. Math. Phys., 294 (2010), 73. doi: 10.1007/s00220-009-0954-6. Google Scholar [12] E. Feireisl, Local decay of acoustic waves in the low Mach number limits on general unbounded domains under slip boundary conditions,, Commun. Partial Differential Equations, 36 (2011), 1778. doi: 10.1080/03605302.2011.602168. Google Scholar [13] E. Feireisl, Low Mach number limits of compressible rotating fluids,, J. Math. Fluid Mechanics, 14 (2012), 61. doi: 10.1007/s00021-010-0043-9. Google Scholar [14] E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system,, Arch. Rational Mech. Anal., 204 (2012), 683. doi: 10.1007/s00205-011-0490-3. Google Scholar [15] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids,, J. Math. Fluid Mech., 3 (2001), 358. doi: 10.1007/PL00000976. Google Scholar [16] E. Feireisl, A. Novotný and Y. Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids,, Indiana Univ. Math. J., 60 (2011), 611. doi: 10.1512/iumj.2011.60.4406. Google Scholar [17] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I,", Springer-Verlag, (1994). Google Scholar [18] I. Gallagher, Résultats récents sur la limite incompressible,, Séminaire Bourbaki. Vol. 2003/2004, (2005), 29. Google Scholar [19] P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system,, J. Math. Fluid Mech., 13 (2011), 137. doi: 10.1007/s00021-009-0006-1. Google Scholar [20] X. P. Hu and D. H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows,, SIAM J. Math. Anal., 41 (2009), 1272. doi: 10.1137/080723983. Google Scholar [21] S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions,, Comm. Math. Phys., 297 (2010), 371. doi: 10.1007/s00220-010-0992-0. Google Scholar [22] S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients,, SIAM J. Math. Anal., 42 (2010), 2539. doi: 10.1137/100785168. Google Scholar [23] T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation,, Arch. Rational Mech. Anal., 25 (1967), 188. Google Scholar [24] T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow,, J. Funct. Anal., 56 (1984), 15. doi: 10.1016/0022-1236(84)90024-7. Google Scholar [25] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. Google Scholar [26] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids,, Comm. Pure Appl. Math., 34 (1981), 481. doi: 10.1002/cpa.3160340405. Google Scholar [27] H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations,, Tohoku Math. J. (2), 41 (1989), 471. doi: 10.2748/tmj/1178227774. Google Scholar [28] P. Kukučka, Singular limits of the equations of magnetohydrodynamics,, J. Math. Fluid Mech., 13 (2011), 173. doi: 10.1007/s00021-009-0007-0. Google Scholar [29] Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows,, J. Differential Equations, 251 (2011), 1990. doi: 10.1016/j.jde.2011.04.016. Google Scholar [30] P.-L. Lions, "Mathematical Topics in Fluid Dynamics. Vol. 2. Compressible Models,", Oxford Lecture Series in Mathematics and its Applications, 10 (1998). Google Scholar [31] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, J. Math. Pures Appl. (9), 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6. Google Scholar [32] N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199. doi: 10.1016/S0294-1449(00)00123-2. Google Scholar [33] N. Masmoudi, Examples of singular limits in hydrodynamics,, in, (2007). doi: 10.1016/S1874-5717(07)80006-5. Google Scholar [34] S. Schochet, The mathematical theory of low Mach number flows,, M2ANMath. Model Numer. Anal., 39 (2005), 441. doi: 10.1051/m2an:2005017. Google Scholar [35] P. Secchi, On the equations of ideal incompressible magnetohydrodynamics,, Rend. Sem. Univ. Padova, 90 (1993), 103. Google Scholar [36] H. F. Smith and C. D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian,, Comm. Partial Differential Equations, 25 (2000), 2171. doi: 10.1080/03605300008821581. Google Scholar [37] R. S. Strichartz, A priori estimates for the wave equation and some applications,, J. Functional Analysis, 5 (1970), 218. Google Scholar
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