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Daily Overview |
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Gvp3: Geometric variational problems
Session Topics: Geometric variational problems
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Existence Theory for Geometric Evolution Equations: Short-Time, Long-Time, and Beyond University Salzburg, Austria IIn this talk we revisit short-time existence theory for geometric evolution equations, presenting new results for the elastic flow and geometric curvature energies of curves. These allow the flows to start from initial curves with merely Hölder continuous unit tangent, beyond the regularity required in the classical theory, but can also be used to prove long-time existence in subcritical settings or construct a blow-up limit at a singularity. The Verigin problem with phase transition as a gradient flow 1: TU Wien, Austria; 2: Heidelberg University, Germany; 3: Bonn University, Germany In this talk, we study the modeling of a compressible two-phase flow in a porous medium, known as the Verigin problem with phase transition. This evolution can be regarded as a gradient flow with respect to the 2-Wasserstein distance of a suit- able energy. Our aim is to construct weak solutions of the underlying PDE using the minimizing movement scheme, with the Wasserstein distance as dissipation. We show that, in the limit as the time step tends to zero, we obtain distributional solutions that satisfy an optimal energy-dissipation inequality. This talk is based on joint work with Tim Laux and Alice Marveggio. Existence and convergence of the area-constrained elastic flow University of Bonn, Germany
The Bernoulli model describes the bending behaviour of an elastic rod -- represented by a smooth immersed curve $\gamma: \mathbb{S}^1 \to \mathbb{R}^2$ -- in terms of its stored elastic energy. To predict this behaviour, it is classical to study the evolution of curves with fixed length moving by the negative $L^2$-gradient of the elastic energy. This talk focuses on the less-known flow subject to fixed enclosed area rather than fixed length.
While local and global existence hold for smooth initial data, establishing convergence requires a uniform bound on the length of the evolving curves. In contrast to the length-constrained flow, this bound is non-trivial. Indeed, there exist initial configurations for which the length diverges to infinity. This talk presents strategies to obtain length bounds through initial energy assumptions. Pseudolocality 1: Universität Ulm, Germany; 2: Universität Konstanz, Germany We investigate a pseudolocality property for parabolic differential equations. It characterises, whether we can find for any given positive constant a time interval such that the absolute value of any solution, that vanishes initially on the unit ball, stays bounded at the origin by the given constant throughout the whole time interval. We investigate several parabolic differential equations and find those that fulfil and those that violate this property. | ||
