Abstract
We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation
with m >1 and, uo a continuous, nonnegative function.
It is well known that, across a moving interface \(x = eta eft( t ight)\) of the solution u(x,t), the derivatives υt and υχ of the pressure \(v = eft( {m/eft( {m - 1} ight)} ight){u^{m - 1}}\) have jump discontinuities. We prove that each moving part of the interface is a C ∞-curve and that υ is C ∞ on each side of the moving interface (and up to it). We also prove that for solutions with compact support the pressure becomes a concave function of x after a finite time. This fact implies sharp convergence rates for the solution and the interfaces as t → ∞.
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For James Serrin on his sixtieth birthday
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© 1989 Springer-Verlag Berlin Heidelberg
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Aronson, D.G., Vazquez, J.L. (1989). Eventual C∞-Regularity and Concavity for Flows in One-Dimensional Porous Media. In: Analysis and Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83743-2_35
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DOI: https://doi.org/10.1007/978-3-642-83743-2_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50917-2
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