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Renormalised solutions to reaction-diffusion systems with interface conditions: Global existence and weak-strong uniqueness

Katharina Hopf, Bao Quoc Tang

Abstract

We introduce an extension of the concept of renormalised solutions for entropy-dissipating reaction-diffusion systems due to J. Fischer (Arch. Ration. Mech. Anal. 218, 2015) to systems coupled by nonlinear interface conditions. For this notion of solution, we establish global existence as well as a weak-strong stability estimate. Our framework allows to handle entropy-dissipating interfacial transmission rates without growth restrictions, including power-law nonlinearities as arising in the thermodynamic modelling of dissipative bulk-interface systems via generalised gradient structures. Our analysis relies on suitable extensions of the species' densities across the interface as well as on a non-local truncated variant of the relative entropy.

Renormalised solutions to reaction-diffusion systems with interface conditions: Global existence and weak-strong uniqueness

Abstract

We introduce an extension of the concept of renormalised solutions for entropy-dissipating reaction-diffusion systems due to J. Fischer (Arch. Ration. Mech. Anal. 218, 2015) to systems coupled by nonlinear interface conditions. For this notion of solution, we establish global existence as well as a weak-strong stability estimate. Our framework allows to handle entropy-dissipating interfacial transmission rates without growth restrictions, including power-law nonlinearities as arising in the thermodynamic modelling of dissipative bulk-interface systems via generalised gradient structures. Our analysis relies on suitable extensions of the species' densities across the interface as well as on a non-local truncated variant of the relative entropy.
Paper Structure (27 sections, 12 theorems, 190 equations, 2 figures)

This paper contains 27 sections, 12 theorems, 190 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Renormalised solutions
  3. Objectives and strategy
  4. State-of-the art
  5. Outline of the paper
  6. A gradient structure for reaction--diffusion systems with interface conditions
  7. Example for interface conditions
  8. Main results
  9. Hypotheses and general model equations
  10. Assumptions on the geometry
  11. Assumptions on coefficients functions and entropic structure
  12. Notations and model equations
  13. Geometric preliminaries
  14. Extension across flat interface
  15. General extension near interface points
  16. ...and 12 more sections

Key Result

Theorem 2.4

Assume hypotheses hp:geo.basic, hp:C1ext and it:HP.diff--it:HP.int. Then for any initial data $\boldsymbol{u}_0$ that is componentwise non-negative with $H(u_0)=\sum_{\sigma\in\{\pm\}}\int_{\Omega_\sigma}h(u^\sigma_0)\,\dd x<\infty$, there exists a global dissipative renormalised solution $\boldsymb

Figures (2)

  • Figure 1: Examples of admissible bulk-interface geometries.
  • Figure 2: Two-step extension procedure for $\beta\in\partial\Gamma$ in the case of a flat interface.

Theorems & Definitions (27)

  • Remark 2.1
  • Definition 2.2: Dissipative renormalised solutions
  • Remark 2.3
  • Theorem 2.4: Existence of dissipative renormalised solutions
  • Remark 2.5: Regularity hypotheses on the geometry
  • Definition 2.6: Strong solution
  • Theorem 2.7: Weak--strong uniqueness
  • Remark 2.8: Existence of a strong solution
  • Lemma 3.1: Partial coercivity of truncated relative entropy density
  • proof : Proof of Lemma \ref{['l:coerc.relen']}
  • ...and 17 more