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search.filters.applied.f.access: openAccess × DDC: 510 × Has files: Yes × Subject: gradient structure ×

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Now showing 1 - 4 of 4
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    EDP-convergence for nonlinear fast–slow reaction systems with detailed balance*
    (Bristol : IOP Publ., 2021) Mielke, Alexander; Peletier, Mark A.; Stephan, Artur
    We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP-convergence, i.e. convergence in the sense of the energy-dissipation principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.
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    Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Disser, Karoline
    We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and e.g. slow, fast and entropic diffusion processes under mass conservation. The main results are global well-posedness and exponential stability of equilibria. As a part of the proof, we show bulk-interface maximum principles and a bulk-interface Poincaré inequality. The method of proof for global existence is a simple but very versatile combination of maximal parabolic regularity of the linearization, a priori L1-bounds and a Schaefer fixed point argument. This allows us to extend the setting e.g. conditions and external forces.
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    Mathematical modeling of semiconductors: From quantum mechanics to devices
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Kantner, Markus; Mielke, Alexander; Mittnenzweig, Markus; Rotundo, Nella
    We discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of so-called GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.
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    EDP-convergence for nonlinear fast-slow reaction systems with detailed balance
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Mielke, Alexander; Peletier, Mark A.; Stephan, Artur
    We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.