Abstract : Gradient flows, a type of dynamics where systems follow steepest descent paths of various functionals, are ubiquitous in many areas of science and technology. Their mathematical understanding is still developing. Ideas like evolutionary variational inequalities, notions of slope, or very weak definitions originating from dynamical systems allow for far-reaching generalizations. Nonetheless, obstacles such as lack of convexity, non-trivial weights, or complicated geometric settings still cause difficulties. We would like to gather experts within the broad limits we stated, dealing with well-posedness and properties of gradient flows in non-classical cases, as well as singular limits or asymptotics.
Organizer(s) : Yoshikazu Giga, Michal Lasica, Piotr Rybka
[05191] Existence for a class of fourth-order quasilinear parabolic systems
Format : Talk at Waseda University
Author(s) :
Michal Lasica (Polish Academy of Sciences)
Yoshikazu Giga (the University of Tokyo)
Abstract : We consider a class of nonlinear fourth-order parabolic systems of PDEs formally arising as gradient flows of $p$-Dirichlet type energies with respect to $H^{-1}$ metrics weighted by spatial derivatives of the solution up to second order. PDEs with such structure appear for example in modeling of thermal fluctuations in crystal surfaces. We prove global existence of weak solutions. Our tools include a variant of Galerkin scheme, monotonicity methods, and interpolation.
[03572] Evolutionary limit of gradient flows in heterogeneous Wasserstein space
Format : Online Talk on Zoom
Author(s) :
Yuan Gao (Purdue University)
Abstract : The Fokker-Planck equation with fast oscillated coefficients can be regarded as a gradient flow in Wasserstein space with heterogeneous medium. We will use an evolutionary variational approach to obtain the homogenized dynamics, which preserves the gradient flow structure in a limiting homogenized Wasserstein space. Equivalent formulations for heterogeneous Wasserstein distance and their limits will also be discussed.
[04719] The fourth-order total variation flow in R^n
Format : Talk at Waseda University
Author(s) :
Hirotoshi Kuroda (Hokkaido University)
Yoshikazu Giga (the University of Tokyo)
Michał Łasica (Polish Academy of Sciences)
Abstract : We characterize the solution in terms of what is called the Cahn-Hoffman vector field, and introduce a notion of calibrability of a set in our fourth-order setting.
This notion is related to whether a characteristic function preserves its form throughout the evolution.
If $n \neq 2$, all annuli are calibrable.
In the case $n=2$, if an annulus is too thick, it is not calibrable.
[05426] Global existence for the p-Sobolev flow
Format : Online Talk on Zoom
Author(s) :
Masashi Misawa (Kumamoto University)
Abstract : We shall talk about the global existence of the p-Sobolev flow.
The p-Sobolev flow is regarded as the heat flow associated with
the Sobolev inequality and the nonlinear eigenvalue problem
corresponding to it. We shall study the asymptotic behavior
of the p-Sobolev flow and present the so-called volume concentration
phenomenon at time-infinity.
[02587] Weak solutions to gradient flows in metric measure spaces
Format : Talk at Waseda University
Author(s) :
Jose M Mazon (Universitat de Valencia)
Abstract : We show how to introduce the notion of weak solutions in metric measure spaces in the model case of the
$p$-Laplacian evolution equation, including the borderline case $p = 1$, i.e., the total variation flow. For $p > 1$, it has been previously studied as the gradient flow in $L^2 $of the $p$-Cheeger energy. Using the first-order differential structure on a metric measure space introduced by Gigli, we characterise the subdifferential in $L^2$ of the $p$-Cheeger energy. This leads to a new definition of solutions to the $p$-Laplacian evolution equation in metric measure spaces, in which the gradient is replaced by a vector field, defined via Gigli’s differential structure, satisfying some compatibility conditions.
[04233] Duality methods for gradient flows of linear growth functionals
Format : Online Talk on Zoom
Author(s) :
Wojciech Górny (University of Vienna; University of Warsaw)
Jose M Mazón (University of València)
Abstract : We study gradient flows in $L^2$ of general convex and lower semicontinuous functionals with linear growth. Typical examples of such evolution equations are the time-dependent minimal surface equation and the total variation flow. Classical results concerning characterisation of solutions require a special form or differentiability of the Lagrangian; we apply a duality-based method to formulate a general definition of solutions, prove their existence and uniqueness, and reduce the regularity and structure assumptions on the Lagrangian.
[04556] Convergence of Sobolev gradient trajectories to elastica
Format : Talk at Waseda University
Author(s) :
Shinya Okabe (Tohoku University)
Abstract : In this talk we consider the $H^2(ds)$-gradient flow for the modified elastic energy defined on closed immersed curves in $\mathbb{R}^n$. We prove (i) the existence of a unique global-in-time solution to the flow; (ii) the full limit convergence of solutions to elastica without any additional reparametrization and translation. The main ingredients of the proof of (ii) are a Lojasiewicz--Simon's gradient inequality and the completeness of a $H^2(ds)$-Riemannian metric space.
[05251] Recent advances in Sobolev gradient flows of plane curves
Format : Talk at Waseda University
Author(s) :
Glen Wheeler (University of Wollongong)
Abstract : In this talk, we discuss recent progress made alongside Shinya Okabe, Philip Schrader, and Valentina-Mira Wheeler in studying the relationship between the classical curve shortening flow and the triviality of the L^2(ds) metric.
Our method is to study gradient flows with different metrics (that give rise to non-trivial metric spaces).
Our investigation with global analysis reveals intriguing behavior.
