# New dissipated energy for nonnegative weak solution of unstable thin-film equations - Mathematics > Analysis of PDEs

New dissipated energy for nonnegative weak solution of unstable thin-film equations - Mathematics > Analysis of PDEs - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: The fluid thin film equation $h t = - h^n h {xxx} x - a 1\,h^m h x x$ isknown to conserve mass $\int\,h \, dx$, and in the case of $a 1 \leq 0$, todissipate entropy $\int\,h^{3-2 - n}\,dx$ see 8 and the $L^2$-norm of thegradient $\int\,h x^2\,dx$ see 3. For the special case of $a 1 = 0$ a newdissipated quantity $\int\, h^{\alpha}\,h x^2\,dx$ was recently discovered forpositive classical solutions by Laugesen see 15. We extend it in two ways.First, we prove that Laugesen-s functional dissipates strong nonnegativegeneralized 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 forstrong nonnegative generalized solutions in the case of the unstable porousmedia perturbation $a 1> 0$ and the critical exponent $m = n+2$.

Autor: Marina Chugunova, Roman M. Taranets

Fuente: https://arxiv.org/