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September  2005, 4(3): 613-634. doi: 10.3934/cpaa.2005.4.613

New dissipated energies for the thin fluid film equation

1. 

Department of Mathematics, University of Illinois, Urbana, IL 61801, United States

Received  September 2004 Revised  January 2005 Published  June 2005

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 $\int h^{-1/2} h_x^2 dx$.
Citation: Richard S. Laugesen. New dissipated energies for the thin fluid film equation. Communications on Pure & Applied Analysis, 2005, 4 (3) : 613-634. doi: 10.3934/cpaa.2005.4.613
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