New dissipated energy for the unstable thin film equation

Marina Chugunova; Roman M. Taranets

doi:10.3934/cpaa.2011.10.613

Article Contents

2011, Volume 10, Issue 2: 613-624. Doi: 10.3934/cpaa.2011.10.613

This issue Previous Article Breaking of resonance for elliptic problems with strong degeneration at infinity Next Article On the collapsing sandpile problem

New dissipated energy for the unstable thin film equation

Marina Chugunova^1, and
Roman M. Taranets^2,

1.
University of Toronto, Department of Mathematics, 40 St. George Str., Toronto, Ontario M5S 2E4
2.
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, 74 R. Luxemburg Str., Donetsk, 83114

Received: January 31, 2010

Revised: July 31, 2010

Published: February 2011

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Abstract

The fluid thin film equation $h_t = - (h^n h_{x x x})_x - a_1 (h^m h_x)_x$ is known to conserve mass $\int h dx$, and in the case of $a_1 \leq 0$, to dissipate entropy $\int h^{3/2 - n} dx$ (see [8]) and the $L^2$-norm of the gradient $\int h_x^2 dx$ (see [3]). For the special case of $a_1 = 0$ a new dissipated quantity $\int h^{\alpha} h_x^2 dx $ was recently discovered for positive classical solutions by Laugesen (see [15]). We extend it in two ways. First, we prove that Laugesen's functional dissipates strong nonnegative generalized solutions. Second, we prove the full $\alpha$-energy $\int (\frac{1}{2} h^\alpha h_x^2 - $ $ \frac {a_1 h^{\alpha + m - n + 2}}{(\alpha + m - n + 1)(\alpha + m - n + 2)} ) dx $ dissipation for strong nonnegative generalized solutions in the case of the unstable porous media perturbation $a_1> 0$ and the critical exponent $m = n+2$.

Keywords:

Fourth-order degenerate parabolic equations,
thin liquid films,
energy,
entropy.

Mathematics Subject Classification: Primary: 35K55, 35K35, 35Q35; Secondary: 76D08.

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References

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