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 and Continuous Dynamical Systems, 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), 1807-1835. doi: 10.1137/S0036141003431554.

[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), 1675-1683. doi: 10.1081/PDE-120024528.

[3]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.

[4]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179. doi: 10.1007/BF00250353.

[5]

R. Danchin, Low Mach number limit for viscous compressible flows, M2AN Math. Model Numer. Anal., 39 (2005), 459-475. doi: 10.1051/m2an:2005019.

[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), 87-124. doi: 10.1002/mma.1670100202.

[7]

B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Commun. Partial Differential Equations, 22 (1997), 977-1008. doi: 10.1080/03605309708821291.

[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), 595-629. doi: 10.1007/s00220-006-0052-y.

[9]

J. Edward and D. Pravica, Bounds on resonances for the Laplacian on perturbations of half-space, SIAM J. Math. Anal., 30 (1999), 1175-1184. doi: 10.1137/S003614109733172X.

[10]

D. M. Èĭdus, The principle of limiting amplitude, (in Russian) Usp. Mat. Nauk, 24 (1969), 91-156.

[11]

E. Feireisl, Incompressible limits and propagation of acoustic waves in large domains with boundaries, Commun. Math. Phys., 294 (2010), 73-95. doi: 10.1007/s00220-009-0954-6.

[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), 1778-1796. doi: 10.1080/03605302.2011.602168.

[13]

E. Feireisl, Low Mach number limits of compressible rotating fluids, J. Math. Fluid Mechanics, 14 (2012), 61-78. doi: 10.1007/s00021-010-0043-9.

[14]

E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 204 (2012), 683-706. doi: 10.1007/s00205-011-0490-3.

[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), 358-392. doi: 10.1007/PL00000976.

[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), 611-631. doi: 10.1512/iumj.2011.60.4406.

[17]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I," Springer-Verlag, New York, 1994.

[18]

I. Gallagher, Résultats récents sur la limite incompressible, Séminaire Bourbaki. Vol. 2003/2004, Astérisque, Exp. No. 926, vii, (2005), 29-57.

[19]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146. doi: 10.1007/s00021-009-0006-1.

[20]

X. P. Hu and D. H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294. doi: 10.1137/080723983.

[21]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys., 297 (2010), 371-400. doi: 10.1007/s00220-010-0992-0.

[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), 2539-2553. doi: 10.1137/100785168.

[23]

T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal., 25 (1967), 188-200.

[24]

T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28. doi: 10.1016/0022-1236(84)90024-7.

[25]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.

[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), 481-524. doi: 10.1002/cpa.3160340405.

[27]

H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations, Tohoku Math. J. (2), 41 (1989), 471-488. doi: 10.2748/tmj/1178227774.

[28]

P. Kukučka, Singular limits of the equations of magnetohydrodynamics, J. Math. Fluid Mech., 13 (2011), 173-189. doi: 10.1007/s00021-009-0007-0.

[29]

Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations, 251 (2011), 1990-2023. doi: 10.1016/j.jde.2011.04.016.

[30]

P.-L. Lions, "Mathematical Topics in Fluid Dynamics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publication, The Clarendon Press, Oxford University Press, New York, 1998.

[31]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), 77 (1998), 585-627. doi: 10.1016/S0021-7824(98)80139-6.

[32]

N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199-224. doi: 10.1016/S0294-1449(00)00123-2.

[33]

N. Masmoudi, Examples of singular limits in hydrodynamics, in "Handbook of Differential Equations: Evolutionary Equations. Vol. III" (eds. C. Dafermos and E. Feireisl), Elsevier/North-Holland, Amsterdam, 2007. doi: 10.1016/S1874-5717(07)80006-5.

[34]

S. Schochet, The mathematical theory of low Mach number flows, M2ANMath. Model Numer. Anal., 39 (2005), 441-458. doi: 10.1051/m2an:2005017.

[35]

P. Secchi, On the equations of ideal incompressible magnetohydrodynamics, Rend. Sem. Univ. Padova, 90 (1993), 103-119.

[36]

H. F. Smith and C. D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations, 25 (2000), 2171-2183. doi: 10.1080/03605300008821581.

[37]

R. S. Strichartz, A priori estimates for the wave equation and some applications, J. Functional Analysis, 5 (1970), 218-235.

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), 1807-1835. doi: 10.1137/S0036141003431554.

[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), 1675-1683. doi: 10.1081/PDE-120024528.

[3]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.

[4]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179. doi: 10.1007/BF00250353.

[5]

R. Danchin, Low Mach number limit for viscous compressible flows, M2AN Math. Model Numer. Anal., 39 (2005), 459-475. doi: 10.1051/m2an:2005019.

[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), 87-124. doi: 10.1002/mma.1670100202.

[7]

B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Commun. Partial Differential Equations, 22 (1997), 977-1008. doi: 10.1080/03605309708821291.

[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), 595-629. doi: 10.1007/s00220-006-0052-y.

[9]

J. Edward and D. Pravica, Bounds on resonances for the Laplacian on perturbations of half-space, SIAM J. Math. Anal., 30 (1999), 1175-1184. doi: 10.1137/S003614109733172X.

[10]

D. M. Èĭdus, The principle of limiting amplitude, (in Russian) Usp. Mat. Nauk, 24 (1969), 91-156.

[11]

E. Feireisl, Incompressible limits and propagation of acoustic waves in large domains with boundaries, Commun. Math. Phys., 294 (2010), 73-95. doi: 10.1007/s00220-009-0954-6.

[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), 1778-1796. doi: 10.1080/03605302.2011.602168.

[13]

E. Feireisl, Low Mach number limits of compressible rotating fluids, J. Math. Fluid Mechanics, 14 (2012), 61-78. doi: 10.1007/s00021-010-0043-9.

[14]

E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Rational Mech. Anal., 204 (2012), 683-706. doi: 10.1007/s00205-011-0490-3.

[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), 358-392. doi: 10.1007/PL00000976.

[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), 611-631. doi: 10.1512/iumj.2011.60.4406.

[17]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I," Springer-Verlag, New York, 1994.

[18]

I. Gallagher, Résultats récents sur la limite incompressible, Séminaire Bourbaki. Vol. 2003/2004, Astérisque, Exp. No. 926, vii, (2005), 29-57.

[19]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., 13 (2011), 137-146. doi: 10.1007/s00021-009-0006-1.

[20]

X. P. Hu and D. H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294. doi: 10.1137/080723983.

[21]

S. Jiang, Q. Ju and F. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys., 297 (2010), 371-400. doi: 10.1007/s00220-010-0992-0.

[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), 2539-2553. doi: 10.1137/100785168.

[23]

T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal., 25 (1967), 188-200.

[24]

T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28. doi: 10.1016/0022-1236(84)90024-7.

[25]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.

[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), 481-524. doi: 10.1002/cpa.3160340405.

[27]

H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations, Tohoku Math. J. (2), 41 (1989), 471-488. doi: 10.2748/tmj/1178227774.

[28]

P. Kukučka, Singular limits of the equations of magnetohydrodynamics, J. Math. Fluid Mech., 13 (2011), 173-189. doi: 10.1007/s00021-009-0007-0.

[29]

Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations, 251 (2011), 1990-2023. doi: 10.1016/j.jde.2011.04.016.

[30]

P.-L. Lions, "Mathematical Topics in Fluid Dynamics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publication, The Clarendon Press, Oxford University Press, New York, 1998.

[31]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), 77 (1998), 585-627. doi: 10.1016/S0021-7824(98)80139-6.

[32]

N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199-224. doi: 10.1016/S0294-1449(00)00123-2.

[33]

N. Masmoudi, Examples of singular limits in hydrodynamics, in "Handbook of Differential Equations: Evolutionary Equations. Vol. III" (eds. C. Dafermos and E. Feireisl), Elsevier/North-Holland, Amsterdam, 2007. doi: 10.1016/S1874-5717(07)80006-5.

[34]

S. Schochet, The mathematical theory of low Mach number flows, M2ANMath. Model Numer. Anal., 39 (2005), 441-458. doi: 10.1051/m2an:2005017.

[35]

P. Secchi, On the equations of ideal incompressible magnetohydrodynamics, Rend. Sem. Univ. Padova, 90 (1993), 103-119.

[36]

H. F. Smith and C. D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations, 25 (2000), 2171-2183. doi: 10.1080/03605300008821581.

[37]

R. S. Strichartz, A priori estimates for the wave equation and some applications, J. Functional Analysis, 5 (1970), 218-235.

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