Derivation of the fourth--order DLSS equation with nonlinear mobility via chemical reactions
Authors
- Mielke, Alexander
ORCID: 0000-0002-4583-3888 - Schlichting, André
ORCID: 0000-0003-4140-491X - Stephan, Artur
ORCID: 0000-0001-9871-3946
2020 Mathematics Subject Classification
- 35A15 35K55 35A35 47J35 65M08
Keywords
- Fourth-order nonlinear evolution equation, gradient-flow equation in continuity-equation format, energy-dissipation principle, chemical reaction network, discrete-to-continuum limit
DOI
Abstract
We provide a derivation of the 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.
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