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Fluid relaxation approximation of the Busenberg--Travis cross-diffusion system

J. A. Carrillo, X. Chen, B. Du, A. Jüngel

Abstract

The Busenberg--Travis cross-diffusion system for segregating populations is approximated by the compressible Navier--Stokes--Korteweg equations on the torus, including a density-dependent viscosity and drag forces. The Korteweg term can be associated to the quantum Bohm potential. The singular asymptotic limit is proved rigorously using compactness and relative entropy methods. The novelty is the derivation of energy and entropy inequalities, which reduce in the asymptotic limit to the Boltzmann--Shannon and Rao entropy inequalities, thus revealing the double entropy structure of the limiting Busenberg--Travis system.

Fluid relaxation approximation of the Busenberg--Travis cross-diffusion system

Abstract

The Busenberg--Travis cross-diffusion system for segregating populations is approximated by the compressible Navier--Stokes--Korteweg equations on the torus, including a density-dependent viscosity and drag forces. The Korteweg term can be associated to the quantum Bohm potential. The singular asymptotic limit is proved rigorously using compactness and relative entropy methods. The novelty is the derivation of energy and entropy inequalities, which reduce in the asymptotic limit to the Boltzmann--Shannon and Rao entropy inequalities, thus revealing the double entropy structure of the limiting Busenberg--Travis system.

Paper Structure

This paper contains 18 sections, 11 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Motivation and model setting
  3. Key ideas
  4. State of the art
  5. Main results
  6. Proof of Theorem \ref{['thm.ex']}: existence of solutions
  7. Local existence of solutions
  8. Approximate energy inequality
  9. Approximate entropy inequality
  10. Further uniform estimates
  11. Limit $N\to\infty$
  12. Proof of Theorem \ref{['thm.eps']}
  13. Proof of Theorem \ref{['thm.eps2']}: relative entropy inequality
  14. Energy equality for the limit system
  15. Difference of mass balance equations
  16. ...and 3 more sections

Key Result

Theorem 1

Let Assumptions (A1)--(A2) hold. Then there exists a weak solution $(\rho,u)$ to 1.mass--1.ic with $\rho=(\rho_1,\rho_2)$ and $u=(u_1,u_2)$ such that where $C_1(d)>0$ is a constant only depending on the space dimension $d$, $C_2(\rho^0,u^0)>0$ depends on the initial data (but not on $\varepsilon$), and we recall definitions 1.E and 1.H of $E$ and $H$. Moreover, we have the regularity Therefore,

Theorems & Definitions (19)

  • Definition 1
  • Theorem 1: Existence of solutions
  • Theorem 2: Limit $\varepsilon\to 0$
  • Theorem 3: Limit $\varepsilon\to 0$, $k_1=k_2$
  • Remark 4: Generalizations
  • Lemma 5
  • Lemma 6: Energy inequality
  • proof
  • Corollary 7: Uniform estimates I
  • Lemma 8: Entropy inequality
  • ...and 9 more