Abstract
This paper is devoted to study the strong relaxation limit of multi-dimensional isentropic Euler equations with relaxation. Motivated by the Maxwell iteration, we generalize the analysis of Yong (SIAM J Appl Math 64:1737–1748, 2004) and show that, as the relaxation time tends to zero, the density of a certain scaled isentropic Euler equations with relaxation strongly converges towards the smooth solution to the porous medium equation in the framework of Besov spaces with relatively lower regularity. The main analysis tool used is the Littlewood–Paley decomposition.
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Brenier, Y., Yong, W.-A.: Derivation of particle, string, and membrane motions from the Born–Infeld electromagnetism, J. Math. Phys. 46, 062305(1–17) (2005)
Chemin, J.-Y.: Perfect incompressible fluids. Oxford Lecture Series in Mathematics and its Applications, vol. 14. The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie
Coulombel J.-F., Goudon T.: The strong relaxation limit of the multidimensional isothermal Euler equations. Trans. Am. Math. Soc. 359, 637–648 (2007)
Fang D.Y., Xu J.: Existence and asymptotic behavior of C 1 solutions to the multidimensional compressible Euler equations with damping. Nonlinear Anal. TMA 70, 244–261 (2009)
Hanouzet B., Natalini R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Ration. Mech. Anal. 169, 89–117 (2003)
Junca S., Rascle M.: Strong relaxation of the isothermal Euler system to the heat equation. Z. Angew. Math. Phys. 53, 239–264 (2002)
Marcati P., Milani A.: The one-dimensional Darcy’s law as the limit of a compressible Euler flow. J. Differ. Equ. 84, 129–147 (1990)
Marcati P., Milani A., Secchi P.: Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system. Manuscripta Math. 60, 49–69 (1988)
Marcati P., Rubino B.: Hyperbolic to parabolic relaxation theory for quasilinear first order systems. J. Differ. Equ. 162, 359–399 (2000)
Markowich P.A., Ringhofer C., Schmeiser C.: Semiconductor Equations. Springer, Vienna (1990)
Simon J.: Compact sets in the space L p(0, T; B). Ann. Math. Pura Appl. 146, 65–96 (1987)
Sideris T., Thomases B., Wang D.H.: Long time behavior of solutions to the 3D compressible Euler with damping. Commun. PDE 28, 953–978 (2003)
Xu, J.: Energy-transport limit of the hydrodynamic model for semiconductor. Math. Models Methods Appl. Sci. 20(8) (2010, to appear)
Xu J.: Relaxation-time limit in the isothermal hydrodynamic model for semiconductors. SIAM J. Math. Anal. 40, 1979–1991 (2009)
Xu, J., Fang D.Y.: A note on the relaxation-time limit of the isothermal Euler equations. Boundary Value Problems 2007, Article ID 56945, 10 pages (2007)
Xu J., Yong W.A.: Zero-relaxation limit of non-isentropic hydrodynamic model for semiconductor. Discrete Contin. Dyn. Syst. 25, 1319–1332 (2009)
Yong W.A.: Diffusive relaxation limit of multi-dimensional isentropic hydrodynamic models for semiconductor. SIAM J. Appl. Math. 64, 1737–1748 (2004)
Yong W.A.: Entropy and global existence for hyperbolic balance laws. Arch. Ration. Mech. Anal. 172, 247–266 (2004)
Yong W.A.: Singular pertubations of first-order hyperbolic systems with stiff source terms. J. Differ. Equ. 155, 89–132 (1999)
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The research of Jiang Xu is supported by NUAA’s Scientific Fund for the Introduction of Qualified Personnel.
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Xu, J. Strong relaxation limit of multi-dimensional isentropic Euler equations. Z. Angew. Math. Phys. 61, 389–400 (2010). https://doi.org/10.1007/s00033-009-0034-y
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DOI: https://doi.org/10.1007/s00033-009-0034-y