Drift-diffusion equations with saturation

TL;DR

This work develops a unified theory for drift-diffusion equations with saturation-type nonlinear mobility $m(\rho)$ on bounded domains, where $0\le \rho \le \alpha$ and $m(0)=m(\alpha)=0$. By constructing approximations, it proves the existence of a $C_0$-semigroup of $L^1$-contractions that dissipates a free-energy $\mathcal{F}$, and it analyzes the long-time behavior via a time-limit operator $S_\infty$ and the $\omega$-limit, including the appearance of free boundaries. It also establishes local minimisers of the free energy, Euler–Lagrange characterisations, and a robust, structure-preserving implicit finite-volume scheme with convergence and long-time accuracy, complemented by numerical experiments. The results extend gradient-flow theory to saturation mobilities in higher dimensions, providing a coherent framework for existence, minimisation, and asymptotics, while revealing rich phenomena such as multiple steady states and kink formation. Open problems include uniqueness of the dissipating solutions, convergence without strong regularity, and extending the framework to nonlocal aggregation terms $W\!*\rho$.

Abstract

We focus on a family of nonlinear continuity equations for the evolution of a non-negative density $ρ$ with a continuous and compactly supported nonlinear mobility $\mathrm{m}(ρ)$ not necessarily concave. The velocity field is the negative gradient of the variation of a free energy including internal and confinement energy terms. Problems with compactly supported mobility are often called saturation problems since the values of the density are constrained below a maximal value. Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $ω$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. This problem has a formal gradient-flow structure, and we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Furthermore, we analyse a structure preserving implicit finite-volume scheme and discuss its convergence and long-time behaviour.

Figures (7)

• Figure 1: $T_{0,\alpha} \circ (U')^{-1} (\zeta)$ for $U(s)= \frac{s^m}{m-1}$ and different choices of the exponent $m$.
• Figure 2: Example of steady states for $U(s)=s^2$ and $V$, the potential above, not radially increasing. The blue steady state is not an $L^1$-local minimiser of the free energy \ref{['eq:Free eenrgy ee=0 Omega']}. On the other hand, the green steady state is an $L^1$-local minimiser, since there is only one constant involved, $C$; see \ref{['thm:Euler-Lagrange']}.
• Figure 3: Double-well potential and the levels of energy. Since the initial data $\rho_0$ (green) is such that $0 \leq \rho_0 \leq \overline{\rho}$ (red), it converges to $S_\infty \rho_0$ (blue).
• Figure 4: Free energy of \ref{['eq:Saddle point']} for different values of $M_1$.
• Figure 5: $\mathrm{m}(\rho) = \rho(1-\rho)$, $U(\rho) = \rho^2$ and $V(x) = 10x^2$. $\Delta t = \Delta x = 2^{-7}$. Left: profiles at different times. Right: distance from $\rho_t$ to $\rho_2$.

Theorems & Definitions (92)

• Definition 2.1: Weak solution
• Definition 2.2
• Lemma 2.3
• Theorem 2.4: Well-posedness of \ref{['eq:the problem regularised']}
• Theorem 2.5: Existence for \ref{['eq:the problem Omega']}
• Theorem 2.6: Euler-Lagrange condition
• Remark
• Definition 2.7
• Theorem 2.8: Long-time behaviour for \ref{['eq:the problem Omega']} and \ref{['eq:the problem regularised']}
• Theorem 2.9: The global attractors of \ref{['eq:the problem regularised']}