New dissipated energies for the thin fluid film equation

Richard S. Laugesen

doi:10.3934/cpaa.2005.4.613

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2005, Volume 4, Issue 3: 613-634. Doi: 10.3934/cpaa.2005.4.613

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New dissipated energies for the thin fluid film equation

Richard S. Laugesen^1,

1.
Department of Mathematics, University of Illinois, Urbana, IL 61801

Received: August 31, 2004

Revised: December 31, 2004

Published: August 2005

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Abstract

The thin fluid film evolution $h_t = -(h^n h_{x x x})_x$ is known to conserve the fluid volume $\int h dx$ and to dissipate the "energies" $\int h^{1.5-n} dx$ and $\int h_x^2 dx$. We extend this last result by showing the energy $\int h^p h_x^2 dx$ is dissipated for some values of $p < 0$, when $\frac{1}{2} < n < 3$. For example when $n=1$, the Hele-Shaw equation $h_t = -(h h_{x x x})_x$ dissipates $\int h^{-1/2} h_x^2 dx$.

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Mathematics Subject Classification: 35K55,76D08.

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