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Fractional cross-diffusion in a bounded domain: existence, weak-strong uniqueness, and long-time asymptotics

Nicola De Nitti, Nicola Zamponi

Abstract

We study a fractional cross-diffusion system that describes the evolution of multi-species populations in the regime of large-distance interactions in a bounded domain. We prove existence and weak-strong uniqueness results for the initial-boundary value problem and analyze the convergence of the solutions to equilibrium via relative entropy methods.

Fractional cross-diffusion in a bounded domain: existence, weak-strong uniqueness, and long-time asymptotics

Abstract

We study a fractional cross-diffusion system that describes the evolution of multi-species populations in the regime of large-distance interactions in a bounded domain. We prove existence and weak-strong uniqueness results for the initial-boundary value problem and analyze the convergence of the solutions to equilibrium via relative entropy methods.
Paper Structure (7 sections, 9 theorems, 135 equations)

This paper contains 7 sections, 9 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Main results
  4. Preliminary results
  5. Existence of weak solutions
  6. Weak-strong uniqueness
  7. Long-time asymptotics

Key Result

Theorem 1

Let us assume that the initial data $(u_{0,1}, \dots, u_{0,n}) \eqqcolon u_0 : \Omega\to [0,+\infty)^n$ satisfies $\int_\Omega u_i|\log u_i|\,\mathrm{d} x < \infty$ for $i\in\{1,\ldots,n\}$. Then system 1--1.bc has a weak solution $u : \Omega\times (0,+\infty)\to\mathbb{R}^n$ with non-negative compo for $i\in\{1,\ldots,n\}$, with a suitable $r>1$, and the weak formulation of 1--1.bc, for every $\

Theorems & Definitions (18)

  • Theorem 1: Global existence of weak solutions
  • Theorem 2: Weak-strong uniqueness
  • Theorem 3: Long-time behaviour
  • Lemma 1: Sobolev's embedding
  • proof
  • Lemma 2: Sobolev's embedding
  • proof
  • Corollary 1
  • proof
  • Lemma 3
  • ...and 8 more