New dissipated energy for nonnegative weak solution of unstable thin-film equations

TL;DR

This paper extends the understanding of energy dissipation in thin-film equations by proving new dissipative properties of generalized solutions, especially under unstable conditions and specific parameter choices.

Contribution

It proves that Laugesen's dissipative functional applies to generalized solutions and establishes full energy dissipation for certain unstable thin-film equations.

Findings

Laugesen's functional dissipates strong nonnegative generalized solutions.

Full $ ext{α}$-energy dissipates for unstable cases with $a_1 > 0$ and $m = n+2$.

New dissipative quantities are identified for thin-film equations.

Abstract

The fluid thin film equation $h_t = - (h^n h_{xxx})_x - a_1\,(h^m h_x)_x$ 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\,\bigl(\frac{1}{2} \,h^\alpha \, h_x^2\, - \frac{a_1\,h^{\alpha + m - n + 2}}{(\alpha + m - n + 1)(\alpha + m - n + 2)} \bigr)\, dx $ dissipation for strong nonnegative generalized solutions in the case of the unstable porous media perturbation $a_1> 0$ and the critical exponent $m = n+2$.