Derivation of the fourth-order DLSS equation with nonlinear mobility via chemical reactions

Abstract

We provide a derivation of the one-dimensional fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider the rate equation on the discretized circle for a process in which pairs of particles occupying the same site simultaneously jump to the two neighboring sites; the reverse process involves pairs of particles at adjacent sites simultaneously jumping back to the site located between them. Depending on the rates, in the vanishing-mesh-size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. Via EDP convergence, we identify the limiting gradient structure to be driven by entropy with respect to a generalization of diffusive transport with nonlinear mobility. Interestingly, the DLSS equation with power-type mobility shares qualitative similarities with the fast diffusion and porous medium equation, since we find traveling wave solutions with algebraic tails or compactly supported polynomials, respectively.