A central limit theorem for solutions of the porous medium equation |
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Authors: | Giuseppe Toscani |
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Institution: | (1) Department of Mathematics, University of Pavia, via Ferrata 1, 27100 Pavia, Italy |
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Abstract: | We study the large–time behavior of the second moment (energy)
for the flow of a gas in a N-dimensional porous medium with initial density v0(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation vt = vm where m > 1 is a physical constant. Assuming that
for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy EB(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/EB(t) converges to 1 at the (optimal) rate t–2/(N(m-1)+2). A simple corollary of this result is a central limit theorem for the scaled solution E(t)N/2v(E(t)1/2x, t). |
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Keywords: | 35K55 35K60 35K65 35B40 |
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