Speaker
Description
Many mathematical models for active interfaces, biological membranes, and other free boundary problems combine geometric evolution with conserved quantities living on evolving surfaces. This motivates the development of general variational frameworks for deriving energy-dissipative evolution equations.
In this talk, we present a general variational framework for constructing coupled $(L^2,H^{-1})$-gradient flows on evolving surfaces. A key ingredient is the scalar Truesdell time derivative, which provides a natural evolution operator for conserved scalar quantities on moving manifolds. Combined with a compatible notion of surface variations, this yields evolution equations that simultaneously guarantee density conservation and energy dissipation.
Rather than focusing on a particular application, the framework serves as a general modeling principle for free boundary problems coupling geometric evolution with conserved surface fields. It systematically generates coupled evolution equations from a prescribed surface energy and naturally incorporates both normal and tangential surface motion. Several examples illustrate how classical geometric flows arise as special cases and demonstrate the versatility of the proposed framework for a broad class of coupled free boundary problems.