We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba-Jaumann derivative. Moreover, the stress tensor obey
s a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.
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 fa
st 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.
We introduce two new concepts of convergence of gradient systems $(mathbf Q, mathcal E_varepsilon,mathcal R_varepsilon)$ to a limiting gradient system $(mathbf Q, mathcal E_0,mathcal R_0)$. 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 $(mathbf Q, mathcal E_varepsilon,mathcal R_varepsilon)$. 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 $(mathbf Q, mathcal E_varepsilon,mathcal R_varepsilon)$. 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.
We show that all Lindblad operators (i.e. generators of quantum semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect
to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems. In particular we discuss the coupling of a quantum dot coupled to macroscopic charge carriers.
We discuss a new notion of distance on the space of finite and nonnegative measures which can be seen as a generalization of the well-known Kantorovich-Wasserstein distance. The new distance is based on a dynamical formulation given by an Onsager ope
rator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and its properties. In fact the distance can be equivalently described by an optimal transport problem on the cone space over the underlying metric space. We give a construction of geodesic curves and discuss their properties.
We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transpo
rt problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins f
or these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary $Gamma$-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.
