# American Institute of Mathematical Sciences

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