Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence
Alexander Mielke, Alberto Montefusco, Mark A. Peletier
Abstract
Abstract We introduce two new concepts of convergence of gradient systems $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Q</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>ε</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>ε</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math> to a limiting gradient system $$({\mathbf{Q}},{{\mathcal {E}}}_0,{{\mathcal {R}}}_0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Q</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math> . 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 $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Q</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>ε</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>ε</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math> . 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 $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Q</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>ε</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>ε</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math> . 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.