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We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the $d$-dimensional cube, for arbitrary $d \geq 1$. The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution., 27 pages, minor changes

A class of Kac-like kinetic equations on the real line is considered, with general smoothing transforms as collisional kernels. These equations have been introduced recently e.g. in the context of econophysics (Cordier et al 2005 J. Stat. Phys. 120 253–77) or as models for granular gases with a background heat bath (Carrillo et al 2009 Discrete Contin. Dyn. Syst. 24 59–81). We show that the stationary solutions to these equations are not smooth in general, and we characterize their (finite) Sobolev regularity in dependence of the properties of the collisional kernel. Moreover, we prove that any initial Sobolev regularity below a well-defined threshold is uniformly propagated in time by the transient weak solutions, implying their strong convergence to the steady state. The applied techniques differ from the classical ones developed for the Kac equation as the models at hand neither dissipate the entropy nor the Fisher information. Instead, the proof relies on direct estimates on the collisional operator.

A new approach to the construction of entropies and entropy productions for a large class of nonlinear evolutionary PDEs of even order in one space dimension is presented. The task of proving entropy dissipation is reformulated as a decision problem for polynomial systems. The method is successfully applied to the porous medium equation, the thin film equation and the quantum drift–diffusion model. In all cases, an infinite number of entropy functionals together with the associated entropy productions is derived. Our technique can be extended to higher-order entropies, containing derivatives of the solution, and to several space dimensions. Furthermore, logarithmic Sobolev inequalities can be obtained.