American Institute of Mathematical Sciences

December  2012, 5(4): 743-767. doi: 10.3934/krm.2012.5.743

Global existence in critical spaces for the compressible magnetohydrodynamic equations

 1 Department of Mathematics and Physics, Xiamen University of Technology, Xiamen, Fujian 361024, China 2 School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China

Received  January 2012 Revised  March 2012 Published  November 2012

In this paper, we are concerned with the global existence and uniqueness of the strong solutions to the compressible Magnetohydrodynamic equations in $\mathbb{R}^N(N\ge3)$. Under the condition that the initial data are close to an equilibrium state with constant density, temperature and magnetic field, we prove the global existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations.
Citation: Qing Chen, Zhong Tan. Global existence in critical spaces for the compressible magnetohydrodynamic equations. Kinetic and Related Models, 2012, 5 (4) : 743-767. doi: 10.3934/krm.2012.5.743
References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations," Springer Verlag, 2011. [2] G. Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376. [3] G. Q. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamics equations, Z. Angew. Math. Phys., 54 (2003), 608-632. [4] Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible Magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438-4451. [5] Q. Chen and Z. Tan, Cauchy problem for the compressible Magnetohydrodynamic equations,, Preprint., (). [6] R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233. [7] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. [8] R. Danchin, Global existence in critical spaces for flows of compressible viscous and Heat-Conductive gases, Arch. Ration. Mech. Anal., 160 (2001), 1-39. [9] B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629. [10] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-315. [11] E. Feireisl, A. Novotný and H. Petleltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech., 3 (2001), 358-392. [12] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660. [13] J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. [14] J. F. Gerebeau, C. L. Bris and T. Lelievre, "Mathematical Methods for the Magnetohydrodynamics of Liquid Metals," Oxford University Press, Oxford, 2006. [15] D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 215-254. [16] X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 253-284. [17] X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238. [18] S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A, Math. Sci., 58 (1982), 384-387. [19] P. L. Lions, "Mathematical Topics in Fluids Mechanics," Vol. 2, Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press University Press, New York, 1998. [20] Ta-tsien Li and Tiehu Qin, "Physics and Partial Differential Equations," 2nd ed., vol. I, Higher Education Press, Beijing, PR China, 2005. [21] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heatconductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342. [22] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. [23] R. Moreau, "Magnetohydrodynamics," Kluwer Academic Publishers, Dordredht, 1990. [24] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford University Press, New York, 2004. [25] Z. Tan and Y. J. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.

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References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations," Springer Verlag, 2011. [2] G. Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), 344-376. [3] G. Q. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamics equations, Z. Angew. Math. Phys., 54 (2003), 608-632. [4] Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible Magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438-4451. [5] Q. Chen and Z. Tan, Cauchy problem for the compressible Magnetohydrodynamic equations,, Preprint., (). [6] R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233. [7] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. [8] R. Danchin, Global existence in critical spaces for flows of compressible viscous and Heat-Conductive gases, Arch. Ration. Mech. Anal., 160 (2001), 1-39. [9] B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629. [10] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-315. [11] E. Feireisl, A. Novotný and H. Petleltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech., 3 (2001), 358-392. [12] J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal., 69 (2008), 3637-3660. [13] J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409. [14] J. F. Gerebeau, C. L. Bris and T. Lelievre, "Mathematical Methods for the Magnetohydrodynamics of Liquid Metals," Oxford University Press, Oxford, 2006. [15] D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys., 56 (2005), 215-254. [16] X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 253-284. [17] X. Hu and D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238. [18] S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A, Math. Sci., 58 (1982), 384-387. [19] P. L. Lions, "Mathematical Topics in Fluids Mechanics," Vol. 2, Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press University Press, New York, 1998. [20] Ta-tsien Li and Tiehu Qin, "Physics and Partial Differential Equations," 2nd ed., vol. I, Higher Education Press, Beijing, PR China, 2005. [21] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heatconductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342. [22] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. [23] R. Moreau, "Magnetohydrodynamics," Kluwer Academic Publishers, Dordredht, 1990. [24] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford University Press, New York, 2004. [25] Z. Tan and Y. J. Wang, Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with coulomb force, Nonlinear Anal., 71 (2009), 5866-5884.
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