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Global classical solutions of 3D isentropic compressible MHD with general initial data

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In this paper, we consider the Cauchy problem of three-dimensional isentropic compressible magnetohydrodynamic equations with general initial data which could be either vacuum or non-vacuum under the assumption that the viscosity coefficient μ is large enough.

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References

  1. Adams R.A.: Sobolev Space. Academic Press, New York (1975)

    Google Scholar 

  2. Cabannes H.: Theoretical Magnetofluiddynamics. Academic press, New York (1970)

    Google Scholar 

  3. Deng M., Zhang P., Zhao J.: Global classical solution to the three-dimensional isentropic compressible Navier- Stokes equations with general initial data. Sci. China Math. 55, 2457–2468 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  5. Feireisl E., Novotny A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hoff D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120, 215–254 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hoff D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Ration. Mech. Anal. 132, 1–14 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hu X., Wang D.: Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch. Ration. Mech. Anal. 197, 203–238 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang X.D., Li J., Xin Z.-P.: Global well-posedness of classical solutions with large oscillations and vacuum to the threedimensional isentropic compressible Navier–Stokes equations. Commun. Pure Appl. Math. 65(4), 549–585 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jiang S., Ju Q.C., Li F.C.: Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Commun. Math. Phys. 297, 371–400 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, PhD thesis, Kyoto University (1983)

  12. Ladyzenskaja O.A., Solonnikov V.A., Ural’ceva N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1968)

    Google Scholar 

  13. Li H., Xu X., Zhang J.: Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum. SIAM J. Math. Anal. 45(3), 1356–1387 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lions P.-L.: Mathematical Topics in Fluid Dynamics, vol. 2. Compressible Models. Oxford Science Publication, Oxford (1996)

    Google Scholar 

  15. Liu, S., Yu, H., Zhang, J.: Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum. Journal of Differ. Equ. 254(1), 229–255 (2013)

  16. Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)

    MathSciNet  MATH  Google Scholar 

  17. Novotný A., Straškraba I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  18. Woods L.C.: Principles of Magnetoplasma Dynamics. Oxford University Press, Oxford (1987)

    Google Scholar 

  19. Zlotnik A.A.: Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations. Differ. Equ. 36, 701–716 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Yang Liu.

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Liu, Y. Global classical solutions of 3D isentropic compressible MHD with general initial data. Z. Angew. Math. Phys. 66, 1777–1797 (2015). https://doi.org/10.1007/s00033-015-0515-0

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  • DOI: https://doi.org/10.1007/s00033-015-0515-0

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