TY - JOUR
T1 - Exploring families of energy-dissipation landscapes via tilting
T2 - three types of EDP convergence
AU - Mielke, Alexander
AU - Montefusco, Alberto
AU - Peletier, Mark A.
PY - 2021/5
Y1 - 2021/5
N2 - We introduce two new concepts of convergence of gradient systems (Q, Eε, Rε) to a limiting gradient system (Q, E, R). 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.
AB - We introduce two new concepts of convergence of gradient systems (Q, Eε, Rε) to a limiting gradient system (Q, E, R). 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.
KW - Energy-dissipation principle
KW - Evolutionary Gamma-convergence
KW - Generalized gradient systems
KW - Large deviations
KW - Variational evolution
UR - http://www.scopus.com/inward/record.url?scp=85099966054&partnerID=8YFLogxK
U2 - 10.1007/s00161-020-00932-x
DO - 10.1007/s00161-020-00932-x
M3 - Article
AN - SCOPUS:85099966054
SN - 0935-1175
VL - 33
SP - 611
EP - 637
JO - Continuum Mechanics and Thermodynamics
JF - Continuum Mechanics and Thermodynamics
IS - 3
ER -