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Existence Theory for a Cross-Diffusion System with Independent Drifts: Mixing Dynamics

Alpár R. Mészáros, Guy Parker

Abstract

We consider a cross-diffusion system for which the diffusion of each species is governed solely by the aggregate density through a pressure law of logarithmic or fast diffusion type. The model is set over a one dimensional bounded interval, equipped with no-flux boundary conditions, and accommodates for the presence of potential drifts which are allowed to differ across each species. We establish the global existence of solutions without having to assume the total mixing of solutions. As a consequence, we give a full resolution of the PDE systems recently studied by the authors and by Elbar--Santambrogio, by allowing a general class of initial data with finite bounded variation, with no further structural assumptions on their supports.

Existence Theory for a Cross-Diffusion System with Independent Drifts: Mixing Dynamics

Abstract

We consider a cross-diffusion system for which the diffusion of each species is governed solely by the aggregate density through a pressure law of logarithmic or fast diffusion type. The model is set over a one dimensional bounded interval, equipped with no-flux boundary conditions, and accommodates for the presence of potential drifts which are allowed to differ across each species. We establish the global existence of solutions without having to assume the total mixing of solutions. As a consequence, we give a full resolution of the PDE systems recently studied by the authors and by Elbar--Santambrogio, by allowing a general class of initial data with finite bounded variation, with no further structural assumptions on their supports.
Paper Structure (21 sections, 26 theorems, 364 equations)

This paper contains 21 sections, 26 theorems, 364 equations.

Table of Contents

  1. Introduction
  2. Motivation and Background Literature
  3. Contribution
  4. Method
  5. Main Theorem
  6. The Viscous Approximation
  7. An Inhomogeneous Ratio Variable
  8. Bounding the Non-Linearity
  9. The Transformation of Variables
  10. The Evolution Equation
  11. An Inequality Between Ratios
  12. Aggregate Estimates
  13. Deriving a BV Estimate
  14. Coefficient Bounds
  15. The Energy Dissipation Inequality
  16. ...and 6 more sections

Key Result

Theorem 1.2

Consider System eq:cdid equipped with a logarithmic or fast-diffusive type pressure. Further, suppose that the initial data and drifts of System eq:cdid are suitably regular and satisfy the compatibility condition Then, there exists a weak solution to System eq:cdid in the sense of Definition eq:weakcdid.

Theorems & Definitions (51)

  • Theorem 1.2
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Proposition 3.8
  • proof
  • ...and 41 more