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In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in \cite{AWW2016}. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is the only obstruction for long-time existence of the spinor flow. This generalizes the blow up criteria obtained in \cite{Sc2018} for surfaces to general dimensions. As another application of the estimates, we also obtain a lower bound for the existence time in terms of the initial data. Our estimates are based on an observation that, up to pulling back by a one-parameter family of diffeomorphisms, the metric part of the spinor flow is equivalent to a modified Ricci flow., Comment: 32 pages

In this work, we investigate a variational formulation for a time-fractional Fokker-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [26]. We propose a JKO type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.

We apply the penalty method to the curve straightening flow of inextensible planar open curves generated by the Kirchhoff bending energy. Thus we consider the curve straightening flow of extensible planar open curves generated by a combination of the Kirchhoff bending energy and a functional penalizing deviations from unit arc-length. We start with the governing equations of the explicit parametrization of the curve and derive an equivalent system for the two quantities indicatrix and arc-length. We prove existence and regularity of solutions and use the indicatrix/arc-length representation to compute the energy dissipation. We prove its coercivity and conclude exponential decay of the energy. Finally, by an application of the Lions-Aubin Lemma, we prove convergence of solutions to a limit curve which is the solution of an analogous gradient flow on the manifold of inextensible open curves. This procedure also allows us to characterize the Lagrange multiplier in the limit model as a weak limit of force terms present in the relaxed model.

We derive Kramers' formula as singular limit of the Fokker-Planck equation with double-well potential. The convergence proof is based on the Rayleigh principle of the underlying Wasserstein gradient structure and complements a recent result by Peletier, Savar\'e and Veneroni., Comment: revised proofs, 12 pages, 1 figure

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in C(infinity) up to gauge to a critical point of the Seiberg-Witten functional.

Constructing the visible and invisible regions of an observer due to the presence of obstacles in the environment has played a central role in many applications. It can also be a first step. In this paper, we adopt a visibility algorithm that can produce a variety of general information to handle the optimization of visibility information. Through the use of level set tools, gradient flow, finite differencing, and solvers for ordinary differential equations, we introduce a set of distinct algorithms for several model problems involving the optimization of visibility information.

In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient flow lines between monopoles are identified with holomorphic data on an associated ruled surface, and a dimension formula for such flows is calculated.

In their classical 1937 paper, Kolmogorov, Petrovsky and Piskunov proved that for a particular class of reaction-diffusion equations on the real line the solution of the initial value problem with the initial data in the form of a unit step propagates at long times with constant velocity equal to that of a certain special traveling wave solution. This type of a propagation result has since been established for a number of general classes of reaction-diffusion-advection problems in cylinders. Here we show that in problems without advection or in the presence of transverse advection by a potential flow these results do not rely on the specifics of the problem. Instead, they are a consequence of the fact that the equation considered is a gradient flow in an exponentially weighted L2-space generated by a certain functional, when the dynamics is considered in the reference frame moving with constant velocity along the cylinder axis. We show that independently of the details of the problem only three propagation scenarios are possible in the above context: no propagation, a “pulled” front, or a “pushed” front. The choice of the scenario is completely characterized via a minimization problem.

This paper is the sequel to my recent paper [10]. It will provide technical details of our gradient flow construction and related problems, which are essential for our construction of Lagrangian torus fibrations in [10] and subsequent papers [11, 13, 14].

We consider the L2 gradient flow for the Willmore functional. In [5] it was proved that the curvature concentrates if a singularity develops. Here we show that a suitable blowup converges to a nonumbilic (compact or noncompact) Willmore surface. Furthermore, an L∞ estimate is derived for the tracefree part of the curvature of a Willmore surface, assuming that its L2 norm (the Willmore energy) is locally small. One consequence is that a properly immersed Willmore surface with restricted growth of the curvature at infinity and small total energy must be a plane or a sphere. Combining the results we obtain long time existence and convergence to a round sphere if the total energy is initially small.

We investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space. The results are applied to an infinite dimensional problem involving the symplectίc action function.