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We study the gradient flow of the potential energy on the infinite-dimensional Riemannian manifold of spatial curves parametrized by the arc length, which models overdamped motion of a falling inextensible string. We prove existence of generalized solutions to the corresponding nonlinear evolutionary PDE and their exponential decay to the equilibrium. We also observe that the system admits solutions backwards in time, which leads to non-uniqueness of trajectories.

Optimal transport theory has been a powerful tool for the analysis of parabolic equations viewed as gradient flows of volume forms according to suitable transportation metrics. In this paper, we present an example of gradient flows for closed (d-1)-forms in the Euclidean space R^d. In spite of its apparent complexity, the resulting very degenerate parabolic system is fully integrable and can be viewed as the Eulerian version of the heat equation for curves in the Euclidean space. We analyze this system in terms of ``relative entropy" and ``dissipative solutions" and provide global existence and weak-strong uniqueness results.

For large classes of non-convex subsets $Y$ in ${\mathbb R}^n$ or in Riemannian manifolds $(M,g)$ or in RCD-spaces $(X,d,m)$ we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $(Y,d_Y,m_Y)$ exists - despite the fact that the entropy is not semiconvex - and coincides with the heat flow on $Y$ with Neumann boundary conditions., to appear in Calc.Var.PDE

We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time dependent energy functionals in both settings. In particular we are interested in the case when the underlying spaces are metric measure spaces and the energy functional is given by the time-dependent Boltzmann entropy or the time-dependent Cheeger's energy. Under the assumption that each static space satisfies a lower Ricci curvature bound, we prove existence and uniqueness of the entropy gradient flow and we identify it with the forward adjoint heat flow introduced in "Super-Ricci Flows for Metric Measure Spaces" by Kopfer/Sturm. We identify the gradient flow for the time-dependent Cheeger's energy with the heat flow introduced by Kopfer/Sturm., Revised version with corrected typos and section about EVI

This article considers nonlocal heat flows into a singular target space. The problem is the parabolic analogue of a stationary problem that arises as the limit of a singularly perturbed elliptic system. It also provides a gradient flow approach to an optimal eigenvalue partition problem that, in light of the work of Caffarelli and Lin, is equivalent to a constrained harmonic mapping problem. In particular, we show that weak solutions of our heat equation converge as time approaches infinity to stationary solutions of this mapping problem. For these weak solutions, we also prove Lipschitz continuity and regularity of free interfaces, making use of a parabolic Almgren-type monotonicity formula.

A local existence and uniquness of the gradient flow of one dimensional Dirichlet energy on the Wasserstein space is proved. The proofs are based on a relaxation of displacement convexity in the Wasserstein space and can be applied to a family of higher order energy functionals which are not displacement convex in the standard sense. As the result a local well-posedness of the corresponding nonlinear evolution equations including the thin-film equation and the quantum drift diffusion equation are proved.

In this paper we develop a general critical point theory to deal with existence and locations of multiple critical points produced by minimax methods in relation to multiple invariant sets of the associated gradient flow. The motivation is to study non-trivial nodal solutions with each component sign-changing for a class of nonlinear Schrodinger systems which arise from Bose–Einstein condensates theory. Our general method allows us to obtain infinitely many mixed states of nodal solutions for the repulsive case.

In this article, we study a gradient flow associated with a gauged harmonic map energy in dimension two. Particularly, we focus on the bubbling analysis at the first singular time.

A fully coupled system of two second-order parabolic degenerate equations arising as a thin film approximation to the Muskat problem is interpreted as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of weak solutions. The availability of two Liapunov functionals turns out to be a central tool to obtain the needed regularity to identify the Euler-Lagrange equation in the variational scheme.

In this paper we present a method to untangle smooth knots by a gradient flow for a suitable energy. We show that the flow of smooth initial knots remains smooth for all time and approaches asymptotically an “optimal embedding” in its isotopy type. The method is to set up a gradient flow for the total energy of knots, which consists of bending energy and the Mobius energy of knots.

We prove existence and uniqueness of the gradient flow of the Entropy functional under the only assumption that the functional is λ-geodesically convex for some $${\lambda\in\mathbb {R}}$$ . Also, we prove a general stability result for gradient flows of geodesically convex functionals which Γ−converge to some limit functional. The stability result applies directly to the case of the Entropy functionals on compact spaces.

Divergence-Lq and divergence-measure tensor fields are introduced and their properties are analyzed. These measure-theoretic tools are then used to study the gradient flow of linear growth functionals of maps into the unit sphere. A BV-solution concept is introduced and the existence of such a solution is established for the flow using the energy method together with a regularization and a penalization technique. Based on the analytical results, practical fully discrete finite element methods are then proposed for computing weak solutions of the gradient flow, and the convergence of the proposed numerical method is also proved.

We consider the dynamics of an inextensible elastic closed wire in the plane under uniform high pressure. In 1967, Tadjbakhsh and Odeh (J. Math. Anal. Appl. 18:59–74, 1967) posed a variational problem to determine the shape of a buckled elastic ring under uniform pressure. In order to comprehend a dynamics of the wire, we consider the following two mathematical questions: (i) can we construct a gradient flow for the Tadjbakhsh–Odeh functional under the inextensibility condition?; (ii) what is a behavior of the wire governed by the gradient flow near every critical point of the Tadjbakhsh–Odeh variational problem? For (i), first we derive a system of equations which governs the gradient flow, and then, give an affirmative answer to (i) by solving the system involving fourth order parabolic equations. For (ii), we first prove a stability and instability of each critical point by considering the second variation formula of the Tadjbakhsh–Odeh functional. Moreover, we give a lower bound of its Morse index. Finally we prove a dynamical aspects of the wire near each equilibrium state.

We formulate an appropriate gradient flow in order to study the evolution of the Q-curvature to a prescribed function on a 4-manifold. For a class of prescribed functions, we show convergence and describe the asymptotic behaviour at infinity.

We are interested in the coarsening of a spatial distribution of two phases, driven by the reduction of interfacial energy and limited by diffusion, as described by the Mullins–Sekerka model. We address the regime where one phase covers only a small fraction of the total volume and consists of many disconnected components (“particles”). In this situation, the energetically more advantageous large particles grow at the expense of the small ones, a phenomenon called Ostwald ripening. Lifshitz, Slyozov and Wagner formally derived an evolution for the distribution of particle radii. We extend their derivation by taking into account that only particles within a certain distance, the screening length, communicate. Our arguments are rigorous and are based on a homogenization within a gradient flow structure.

We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction such that micro phase separation results in an ensemble of small balls of one component. Mean-field models for the evolution of a large ensemble of such spheres have been formally derived in Glasner and Choksi (Physica D, 238:1241-1255, 2009), Helmers et al. (Netw Heterog Media, 3(3):615-632, 2008). It turns out that on a time scale of the order of the average volume of the spheres, the evolution is dominated by coarsening and subsequent stabilization of the radii of the spheres, whereas migration becomes only relevant on a larger time scale. Starting from the free boundary problem restricted to balls we rigorously derive the mean-field equations in the early time regime. Our analysis is based on passing to the homogenization limit in the variational framework of a gradient flow. © 2010 Springer-Verlag.

We introduce the transportation-annihilation distance $W_p^\sharp$ between subprobabilities and derive contraction estimates with respect to this distance for the heat flow with homogeneous Dirichlet boundary conditions on an open set in a metric measure space. We also deduce the Bochner inequality for the Dirichlet Laplacian as well as gradient estimates for the associated Dirichlet heat flow. For the Dirichlet heat flow, moreover, we establish a gradient flow interpretation within a suitable space of charged probabilities. In order to prove this, we will work with the \emph{doubling} of the open set, the space obtained by gluing together two copies of it along the boundary., Comment: To appear in Calc. Var. PDE