Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

Abstract

Abstract 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 convergencewith 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 newconcepts 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. Continuum Mech. Thermodyn. (2021) 33:611–637 https://doi.org/10.1007/s00161-020-00932-x