Abstract
We consider the compressible Euler equations with damping. The singular behavior of the flow near vacuum and the large-time states are of particular interest. A class of solutions is constructed and shown to converge to the self-similar solutions of the porous media equation. The porous media equation is derived from the Euler equations through Darcy’s law. Thus we have justified Darcy’s law for the compressible flow time-asymptotically.
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D.G. Aronson, L.A. Caffarelli and S. Kamin, How an initially stationary interface begins to move in porous media flow. SIAM J. Math. Anal.,14 (1983), 639–658.
G.J. Barenblatt, On one class of solutions of the one-dimensional problem of non-stationary filtration of a gas in a porous medium. Prikl. Mat. i Mekh.,17 (1953), 739–742.
L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm. Math. Phys.,143 (1992), 599–605.
T.-P. Liu, Nonlinear hyperbolic-parabolic p.d.e. Nonlinear Analysis. Academia Sinica, R.O.C.,(eds. Liu, F.-C. and Liu, T.-P.), World Scientific, (1991), 161–170.
T. Makino, On a local existence theorem for the evolution equations of gaseous stars. Pattern and Waves. North-Holland, (1986), 459–479.
T. Makino, S. Ukai and S. Kawashima, Sur la solution à support compact de l’équation d’Euler compressible. Japan J. Appl. Math.,3 (1986), 249–257.
T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics. Publ. Math. D’Orsay, (1978), 46–53.
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Research supported in part by NSF and Army Basic Research Grants.
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Liu, TP. Compressible flow with damping and vacuum. Japan J. Indust. Appl. Math. 13, 25–32 (1996). https://doi.org/10.1007/BF03167296
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DOI: https://doi.org/10.1007/BF03167296