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On a repulsion model with Coulomb interaction and nonlinear mobility

Antonin Chodron de Courcel, Charles Elbar

TL;DR

This work analyzes a repulsion model with Coulomb interaction and nonlinear mobility on the torus, given by $\partial_t u - \nabla \cdot (u^m \nabla \mathsf{g} * u)=0$. The authors prove the existence of entropy solutions and a weak-strong uniqueness principle, and they establish fundamental dissipation and monotonicity properties. In the fast-diffusion regime $0<m<1$ they derive a universal lower barrier, while for $m\ge 1$ they exploit decreasing rearrangements to characterize the evolution of the support, including instantaneous growth or waiting-time phenomena, and they prove exponential convergence of solutions to the spatial average in multiple norms. The paper also discusses extensions to multi-population systems and polytropic pressure laws, and develops a Hamilton–Jacobi–type framework for the rearranged density to study front propagation. Overall, the results advance the mathematical understanding of nonlocal fluxes with nonlinear mobility and their long-time behavior in periodic domains.

Abstract

We study a scalar conservation law on the torus in which the flux $\mathbf{j}$ is composed of a Coulomb interaction and a nonlinear mobility: $\mathbf{j} = -u^m\nabla\mathsf{g}\ast u$. We prove existence of entropy solutions and a weak-strong uniqueness principle. We also prove several properties shared among entropy solutions, in particular a lower barrier in the fast diffusion regime $m\lt 1$. In the porous media regime $m\ge 1$, we study the decreasing rearrangement of solutions, which allows to prove an instantaneous growth of the support and a waiting time phenomenon. We also show exponential convergence of the solutions towards the spatial average in several topologies.

On a repulsion model with Coulomb interaction and nonlinear mobility

TL;DR

This work analyzes a repulsion model with Coulomb interaction and nonlinear mobility on the torus, given by . The authors prove the existence of entropy solutions and a weak-strong uniqueness principle, and they establish fundamental dissipation and monotonicity properties. In the fast-diffusion regime they derive a universal lower barrier, while for they exploit decreasing rearrangements to characterize the evolution of the support, including instantaneous growth or waiting-time phenomena, and they prove exponential convergence of solutions to the spatial average in multiple norms. The paper also discusses extensions to multi-population systems and polytropic pressure laws, and develops a Hamilton–Jacobi–type framework for the rearranged density to study front propagation. Overall, the results advance the mathematical understanding of nonlocal fluxes with nonlinear mobility and their long-time behavior in periodic domains.

Abstract

We study a scalar conservation law on the torus in which the flux is composed of a Coulomb interaction and a nonlinear mobility: . We prove existence of entropy solutions and a weak-strong uniqueness principle. We also prove several properties shared among entropy solutions, in particular a lower barrier in the fast diffusion regime . In the porous media regime , we study the decreasing rearrangement of solutions, which allows to prove an instantaneous growth of the support and a waiting time phenomenon. We also show exponential convergence of the solutions towards the spatial average in several topologies.
Paper Structure (23 sections, 19 theorems, 182 equations, 2 figures)

This paper contains 23 sections, 19 theorems, 182 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Related works
  4. Main results
  5. Possible extensions
  6. Systems of interactions
  7. Polytropic pressure law
  8. Existence of entropy solutions and weak-strong uniqueness
  9. Viscous approximation
  10. A compactness result
  11. Weak-strong uniqueness
  12. Local well-posedness
  13. Properties of entropy solutions
  14. Dissipation estimates
  15. Asymptotic behaviour
  16. ...and 8 more sections

Key Result

Theorem 1.2

Suppose $u_{0}\in L^{\infty}({\mathbb{T}}^d)$ is nonnegative, with the additional condition $u_{0}>0$ when $0<m<1$. Then, there exists at least one entropy solution $u$ of eq:PDE in the sense of Definition def:weaksolution. This solution moreover satisfies

Figures (2)

  • Figure 1: Evolution of the system for $m=4$ at different times. Initially (at $t=0.000)$, the support is localized and remains the same up to $t=1.100$. After this point, and once enough pressure has built, the support starts growing.
  • Figure 2: Characteristics of the equation \ref{['eq:primitive']}, where no rarefaction wave appears, and of \ref{['eq:primitiveCarillo']}, where a rarefaction wave does appear. The initial condition is $u_0(x) := 0.4^{-1}\times (\mathbf{1}_{0.2 < x < 0.4} + \mathbf{1}_{0.6<x<0.8})$, and $m=2$. Time is the $y$ axes, and space is the $x$ axes.

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.2: Existence of entropy solutions
  • Proposition 1.3: Weak strong uniqueness
  • Remark 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: Growth of the support
  • Remark 1.10
  • ...and 37 more