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- ItemExploring families of energy-dissipation landscapes via tilting: three types of EDP convergence(Berlin ; Heidelberg : Springer, 2021) Mielke, Alexander; Montefusco, Alberto; Peletier, Mark A.We introduce two new concepts of convergence of gradient systems (Q,Eε,Rε) to a limiting gradient system (Q,E0,R0). These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of (Q,Eε,Rε). It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences (Q,Eε,Rε). In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.
- ItemEDP-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, ArturWe 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.
- ItemEDP-convergence for nonlinear fast–slow reaction systems with detailed balance*(Bristol : IOP Publ., 2021) Mielke, Alexander; Peletier, Mark A.; Stephan, ArturWe 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.