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

TL;DR

The paper derives a fourth-order DLSS-type equation with nonlinear mobility from a microscopic discrete reaction-network model on a circle, encoded by a rate equation with flux $J_{\alpha,k}$ and an entropy $E_N$. It then develops a rigorous continuous gradient-flow framework driven by entropy, proving energy-dissipation balance and performing EDP convergence to the continuum equation $\partial_t \rho = \Delta j$ with $j = -\rho^\alpha \partial_{xx}\log\rho$, i.e. $\partial_t \rho = \Delta\big(-\rho^\alpha \partial_{xx}\log\rho\big)$. The main contributions are the discrete-to-continuum derivation, the introduction of a relaxed slope $\mathcal S_\alpha$ and a chain-rule-based route to weak solutions, and the demonstration that traveling-wave and similarity profiles exhibit alpha-dependent behavior linking to fast diffusion and porous medium dynamics. The results provide a thermodynamically consistent variational framework (EDB/EDI) for all $\alpha>0$, generalize existing schemes (MRSS) beyond $\alpha=1$, and offer a solid foundation for further analysis of the corresponding nonlinear mobility DLSS equation and its dynamical features.

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.

Figures (2)

• Figure 1: Numerically obtained similarity profiles $\Phi_\alpha$ for $\alpha\in \{-1,0,0.5,1,2,3,4,5\}$ normalized by $\Phi_\alpha(0)=1$. For $\alpha<1$ the solutions have algebraic decay like $|y|^{-3/(1{-}\alpha)}$; for $\alpha=1$ we have $\Phi_1(y)=\mathrm e^{-y^2/4}$, for $\alpha>1$ the solutions have compact support and behave like $(y_\alpha{-}y)^{3/(\alpha-1)}$ to the left of the right boundary of the support $[-y_\alpha,y_\alpha]$.
• Figure 2: Numerically obtained solutions to \ref{['eq:RRE']} for $\alpha\in \{2,4,7\}$ (from left to right). Starting from a discrete bump function $c^0_k = \max\{*\}{ 0 ,1-((N/2-k)/(\ell\,N))^2}$ with $\ell=0.1$ and $N=2^{10}$. First line show the overall evolution and propagation of fronts, whereas the second line are zooms towards the tip of the support. We emphasize that in accordance to our result about positivity of solutions to the discrete system, the obtained numerical are also positive.

Theorems & Definitions (40)

• Definition 2.2: Stolarsky mean
• Remark 2.3: Discussion on the assumptions and examples
• Lemma 2.4: Existence of activity $\sigma_\alpha$
• proof
• Proposition 2.5: Well-posedness of \ref{['eq:RRE']}
• proof
• Proposition 2.6
• proof
• Remark 2.7
• Lemma 3.1: Relaxed slope