Multi-species Balance Laws

Publications

Articles in Refereed Journals

• D. Brust, K. Hopf, J. Fuhrmann, A. Cheilytko, M. Wullenkord, Ch. Sattler, Transport of heat and mass for reactive gas mixtures in porous media: Modeling and application, Chemical Engineering Journal, 516 (2025), pp. 162027/1--162027/33, DOI 10.1016/j.cej.2025.162027 .
  Abstract
  We present a modeling framework for multi-component, reactive gas mixtures and heat transport in porous media based on the Maxwell--Stefan and Darcy equations for multi-component diffusion and forced, viscous flow through porous media. Analysis of the model equations reveals thermodynamic con- sistency and uniqueness of steady states, while their mathematical structure facilitates discretization via the Finite-Volume approach resulting in an open- source based implementation of the modeling framework in Julia. The model allows to impose boundary conditions that accurately reflect the conditions prevailing in a photo-thermal chemical reactor that is subsequently intro- duced as a case study for the modeling framework. Comparison of numerical with experimental results reveals good agreement. Improvement options for the physical reactor are derived from simulation results demonstrating the practical utility of the modeling framework. Additionally, the framework is used for the simulation of thermodiffusion in a ternary gas mixture and has been verified with published numerical results with very good agreement.

• K. Hopf, A. Jüngel, Convergence of a finite volume scheme and dissipative measure-valued--strong stability for a hyperbolic-parabolic cross-diffusion system, Numerische Mathematik, 157 (2025), pp. 951--992, DOI 10.1007/s00211-025-01474-7 .
  Abstract
  This article is concerned with the approximation of hyperbolic-parabolic cross-diffusion systems modeling segregation phenomena for populations by a fully discrete finite-volume scheme. It is proved that the numerical scheme converges to a dissipative measure-valued solution of the PDE system and that, whenever the latter possesses a strong solution, the convergence holds in the strong sense. Furthermore, the “parabolic density part” of the limiting measure-valued solution is atomic and converges to its constant state for long times. The results are based on Young measure theory and a weak-strong stability estimate combining Shannon and Rao entropies. The convergence of the numerical scheme is achieved by means of discrete entropy dissipation inequalities and an artificial diffusion, which vanishes in the continuum limit.

Preprints, Reports, Technical Reports

• M.I. Gau, K. Hopf, Well-posedness and relaxation in a simplified model for viscoelastic phase separation via Hilbertian gradient flows, Preprint no. 3212, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3212 .
  Abstract, PDF (481 kByte)
  This article is concerned with a gradient-flow approach to a Cahn--Hilliard model for viscoelastic phase separation introduced by Zhou et al. (Phys. Rev. E, 2006) in its variant with constant mobility. By means of time-incremental minimisation and generalised contractivity estimates, we establish the global well-posedness of the Cauchy problem for moderately regular initial data. For general finite-energy data we obtain the existence of gradient-flow solutions and a stability estimate of weak--strong type. We further study the asymptotic behaviour for relaxation time and bulk modulus depending on a small parameter. Depending on the scaling, we recover the Cahn--Hilliard, the mass-conserving Allen--Cahn or the viscous Cahn--Hilliard equation. A challenge in the well-posedness analysis is the failure of semiconvexity of the appropriate driving functional, which is caused by a phase-dependence of the bulk modulus.

• M.I. Gau, M. Liero, Derivation of a thermo-visco-elastic plate model at small strains, Preprint no. 3209, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3209 .
  Abstract, PDF (339 kByte)
  We investigate a three-dimensional thermo-visco-elastic model with Kelvin--Voigt rheology under small strains confined to a thin domain. The model comprises a quasistatic linear momentum equation, with viscous stresses adhering to a Kelvin--Voigt viscosity law, coupled with a nonlinear heat equation governing temperature. The heat equation incorporates source terms arising from viscous dissipation and adiabatic heat sources due to thermal expansion. The model ensures thermodynamic consistency, maintaining energy conservation, positive temperature, and entropy production. We analyze the asymptotic behavior of solutions as the domain thickness approaches zero, deriving an effective two-dimensional model. This derivation involves rescaling the domain to a fixed thickness and establishing uniform a priori estimates relative to the plate's thickness. In the limit, the temperature becomes vertically constant, and displacement are of Kirchhoff--Love type, enabling meaningful interpretation of the limiting objects within the plate's two-dimensional cross-section. The mechanical equations consist of two parabolic equations, one for the membrane part and one for the bending part. Notably, the viscosity law in the limiting model departs from the Kelvin--Voigt form, reflecting nontrivial kinematic constraints on the rescaled out-of-plane strains. The bending of the plate does not depend on the temperature in the limit.

Talks, Poster

• G. Heinze, Finite-volume-type approximations of coupled multi-species systems from a gradient flow perspective, Oberseminar Analysis, Universität Bielefeld, January 14, 2026.

• K. Hopf, On the Cauchy problem for rank-deficient cross-diffusion: Normal form and measure-valued solutions, Nonlinear Diffusion Models: Analytical & Numerical Challenges, January 19 - 23, 2026, Leiden University, Netherlands, January 19, 2026.

• K. Hopf, A symptotic structures and fractional interface laws in viscoelastic phase separation, Colloquium of the Research Training Group 2339 ``Interfaces, Complex Structures, and Singular Limits'', Universität Regensburg, June 27, 2025.

• K. Hopf, Analysis of entropic cross-diffusion systems of hyperbolic-parabolic type, Mixtures: Modeling, Analysis and Computing, February 5 - 7, 2025, Charles University, Prague, Czech Republic, February 6, 2025.

• K. Hopf, Dissipative systems with cross coupling: Structure, analysis, and variational perspectives, Forschungskolloquium der Fakultät für Mathematik, Universität Regensburg, July 14, 2025.

• K. Hopf, On the Cauchy problem for rank-deficient cross-diffusion: Normal form and measure-valued solutions, Technische Universität Darmstadt, November 13, 2025.

• M.I. Gau, Gradient structures of viscoelastic phase separation, Gradient Flows Face-to-Face 5, September 16 - 19, 2025, Universidad de Granada, Spain, September 18, 2025.

• M.I. Gau, Viscoelastic phase separation: Well-posedness and singular limit to viscous Cahn--Hilliard equation, 24th GAMM Seminar on Microstructures, January 30 - 31, 2025, Humboldt-Universität zu Berlin, January 30, 2025.

• M.I. Gau, Viscoelastic phase separation: Well-posedness and singular limit to viscous Cahn--Hilliard equation, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Section 14.05 ``Applied Analysis'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 10, 2025.