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Publications

Articles in Refereed Journals

  • K. Hopf, Singularities in $L^1$-supercritical Fokker--Planck equations: A qualitative analysis, Annales de l'Institut Henri Poincare. Analyse Non Lineaire, 41 (2024), pp. 357--403, DOI 10.4171/AIHPC/85 .
    Abstract
    A class of nonlinear Fokker--Planck equations with superlinear drift is investigated in the L1-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis -- the main focus being on isotropic solutions, in which case the unique minimiser of the free energy will be shown to be the global attractor. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D Kaniadakis--Quarati model for Bose--Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model.

Preprints, Reports, Technical Reports

  • K. Hopf, M. Kniely, A. Mielke, On the equilibrium solutions of electro-energy-reaction-diffusion systems, Preprint no. 3157, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3157 .
    Abstract, PDF (447 kByte)
    Electro-energy-reaction-diffusion systems are thermodynamically consistent continuum models for reaction-diffusion processes that account for temperature and electrostatic effects in a way that total charge and energy are conserved. The question of the long-time asymptotic behavior in electro-energy-reaction-diffusion systems and the characterization of their equilibrium solutions leads to a maximization problem of the entropy on the manifold of states with fixed values for the linear charge and the nonlinear convex energy functional. As the main result, we establish the existence, uniqueness, and regularity of solutions to this constrained optimization problem. We give two conceptually different proofs, which are related to different perspectives on the constrained maximization problem. The first one is based on the Lagrangian approach, whereas the second one employs the direct method of the calculus of variations.

  • K. Hopf, J. King, A. Münch, B. Wagner, Interface dynamics in a degenerate Cahn--Hilliard model for viscoelastic phase separation, Preprint no. 3149, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3149 .
    Abstract, PDF (474 kByte)
    The formal sharp-interface asymptotics in a degenerate Cahn--Hilliard model for viscoelastic phase separation with cross-diffusive coupling to a bulk stress variable are shown to lead to non-local lower-order counterparts of the classical surface diffusion flow. The diffuse-interface model is a variant of the Zhou--Zhang--E model and has an Onsager gradient-flow structure with a rank-deficient mobility matrix reflecting the ODE character of stress relaxation. In the case of constant coupling, we find that the evolution of the zero level set of the order parameter approximates the so-called intermediate surface diffusion flow. For non-constant coupling functions monotonically connecting the two phases, our asymptotic analysis leads to a family of third order whose propagation operator behaves like the square root of the minus Laplace--Beltrami operator at leading order. In this case, the normal velocity of the moving sharp interface arises as the Lagrange multiplier in a constrained elliptic equation, which is at the core of our derivation. The constrained elliptic problem can be solved rigorously by a variational argument, and is shown to encode the gradient structure of the effective geometric evolution law. The asymptotics are presented for deep quench, an intermediate free boundary problem based on the double-obstacle potential.

  • 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, Preprint no. 3139, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3139 .
    Abstract, PDF (3958 kByte)
    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.

Talks, Poster

  • 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.

  • K. Hopf, On rank-deficient cross-diffusion and the interface dynamics in viscoelastic phase separation, Applied Analysis Complex Systems & Dynamics Seminar, Universität Graz, Institut für Mathematik und Wissenschafltiches Rechnen, Austria, November 13, 2024.

  • K. Hopf, On the equilibrium solutions in a model for electro-energy-reaction-diffusion systems, Modelling, PDE Analysis and Computational Mathematics in Materials Science, September 22 - 27, 2024, Prague, Czech Republic, September 27, 2024.

  • M.I. Gau, Dimension reduction of a thermo-visco-elastic problem at small strains, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs'' 2024, September 16 - 19, 2024, Czech Academy of Sciences, Prague, Czech Republic, September 16, 2024.

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