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We explore equivariant dynamics under the symmetric group $S_N$ of all permutations of $N$ elements. Specifically we study one-parameter vector fields, up to cubic order, which commute with the standard real $(N-1)$-dimensional irreducible representation of $S_N$. The parameter is the linearization at the trivial 1-cluster equilibrium of total synchrony. All equilibria are cluster solutions involving up to three clusters. The resulting global dynamics is of gradient type: all bounded solutions are cluster equilibria and heteroclinic orbits between them. In the limit of large $N$, we present a detailed analysis of the web of heteroclinic orbits among the plethora of 2-cluster equilibria. Our focus is on the global dynamics of 3-cluster solutions with one rebel cluster of small size. These solutions describe slow relative growth and decay of 2-cluster states. For $N\rightarrow\infty$, the limiting heteroclinic web defines an integrable \emph{rebel flow} in the space of 2-cluster equilibrium configurations. We identify and study the seven qualitatively distinct global rebel flows which arise in this setting. Applications include oscillators with all-to-all coupling, and electrochemistry. For illustration we consider synchronization clusters among $N$ complex Stuart-Landau oscillators with complex linear global coupling., 46 pages, 21 figures

The normalized gradient flow, i.e., the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697] or the imaginary time evol...

The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constrai...

We study the emergent behavior of the matrix ensemble on the unitary group in which a state of each oscillator is influenced by the relative distances. Thus, the coupling strength becomes a time-de...

In this paper, we propose and analyze a gradient flow based model for electronic structure calculations. First, based on an extended gradient flow proposed in this paper, we propose a Kohn--Sham gr...

We consider a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths. We construct a weak solution of the flow by means of an implicit variational scheme. We show long-time existence of the evolution and convergence to a critical point of the energy., 27 pages

We present several essential improvements to the powerful scalar auxiliary variable (SAV) approach. Firstly, by using the introduced scalar variable to control both the nonlinear and the explicit l...

We propose numerical schemes for a class of Keller--Segel equations. The discretization is based on the gradient flow structure. The resulting first-order scheme is mass conservative, bound preserv...

In this paper we derive optimal algebraic-in-time relaxation rates to the kink for the Cahn-Hilliard equation on the line. We assume that the initial data have a finite distance---in terms of either a first moment or the excess mass---to a kink profile and capture the decay rate of the energy and the perturbation. Our tools include Nash-type inequalities, duality arguments, and Schauder estimates.

A novel algorithmic approach is presented for the problem of partitioning a connected undirected weighted graph under constraints such as cardinality or membership requirements or must-link and can...

We present an exponential synchronization for the finite-dimensional Kuramoto model for identical oscillators under the effect of frustration. In the presence of frustration, the total phase balance law and gradient-flow structure are destroyed. Hence the gradient flow approach cannot be applied for the emergence of phase-locking, and the Lyapunov functional approach based on $\ell^2$-energy and phase diameter only show some results in a restricted set of initial configurations smaller than a half circle. In this paper, we show that an initial phase configuration whose order parameter is bounded below by some positive value, which is only depending on the size of frustration, leads to the complete phase synchronization or the bipolar state exponentially fast. This extends earlier results on the asymptotic phase-locking for the identical oscillators in which initial phase configurations are confined in a half circle.

We consider in this paper gradient flows with disparate terms in the free energy that cannot be efficiently handled with the scalar auxiliary variable (SAV) approach, and we develop the multiple sc...

We consider the problem of stabilizing a matrix by a correction of minimal norm: Given a square matrix that has some eigenvalues with positive real part, find the nearest matrix having no eigenvalu...

We introduce a novel spatio-temporal discretization for nonlinear Fokker--Planck equations on the multidimensional unit cube. This discretization is based on two structural properties of these equations: the first is the representation as a gradient flow of an entropy functional in the $L^2$-Wasserstein metric; the second is the Lagrangian nature, meaning that solutions can be written as the push-forward transformation of the initial density under suitable flow maps. The resulting numerical scheme is entropy diminishing and mass conserving. Further, we are able to prove consistency in the sense that if the discrete solutions are regular independently of the mesh size, then they converge to a classical solution. Finally, we present results from numerical experiments in space dimension d=2.

We derive a fractional Cahn--Hilliard equation (FCHE) by considering a gradient flow in the negative order Sobolev space $H^{-\alpha}$, $\alpha\in [0,1]$, where the choice $\alpha=1$ corresponds to the classical Cahn--Hilliard equation while the choice $\alpha=0$ recovers the Allen--Cahn equation. The existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fractional order $\alpha$ and that it indeed reduces the free energy. We then turn to the delicate question of the $L_\infty$ boundedness of the solution and establish an $L_\infty$ bound for the FCHE in the case where the nonlinearity is a quartic polynomial. As a consequence of the estimates, we are able to show that the Fourier--Galerkin method delivers a spectral rate of convergence for the FCHE in the case of a semidiscrete approximation scheme. Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order $\alpha$. It is...

In this work we study the stability of a stationary solution to the thin-film equation with linear mobility and partial wetting boundary conditions. The method used is strongly based on the gradient-flow structure of the problem. We obtain natural relaxation rates of perturbations to the stationary solution by showing that the energy is in fact convex in a neighborhood around the stationary solution.

This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e. Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically non-linear diffusion equations that model for instance porous media or crowd evolutions. These gradient flows define a suitable notion of weak solutions for these evolutions and they can be approximated in a stable way using discrete flows. These discrete flows are implicit Euler time stepping according to the Wasserstein metric. A bottleneck of these approaches is the high computational load induced by the resolution of each step. Indeed, this corresponds to the resolution of a convex optimization problem involving a Wasserstein distance to the previous iterate. Following several recent works on the approximation of Wasserstein distances, we consider a discrete flow induced by an entropic regularization of the transportation coupling. This entropic regularization allows one to trade the initial Wasserstein fidelity term for a Kulback-Leibler divergence, which is easier to deal with numerically. We show how KL proximal schemes, and in particular Dykstra's algorithm, can be used to compute each step of the regularized flow. The resulting algorithm is both fast, parallelizable and versatile, because it only requires multiplications by a Gibbs kernel. On Euclidean domains discretized on an uniform grid, this corresponds to a linear filtering (for instance a Gaussian filtering when $c$ is the squared Euclidean distance) which can be computed in nearly linear time. On more general domains, such as (possibly non-convex) shapes or on manifolds discretized by a triangular mesh, following a recently proposed numerical scheme for optimal transport, this Gibbs kernel multiplication is approximated by a short-time heat diffusion.

Controlling droplet shape via surface tension has numerous technological applications, such as droplet lenses and lab-on-a-chip. This motivates a partial differential equation-constrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate. We use shape differential calculus to derive an $L^2$ gradient flow approach to compute equilibrium shapes for sessile droplets on substrates. We then develop a gradient-based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint (i.e., the liquid-solid interface has a desired shape). Moreover, we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint. Numerical results are also presented to showcase the method.

We derive the curvature driven flow of closed-loop pore structures which arise in quasi-stationary states in the $H^{-1}$ gradient flow of the weakly functionalized Cahn--Hilliard free energy which models amphiphilic mixtures. We extend this result to a sharp-interface reduction for the competitive evolution of disjoint collections of bilayer interfaces and closed-loop pores. In particular, for a mixture of spherical bilayers and circular closed pores we explicitly identify two regimes: one in which spherical bilayers extinguish and the circular pores arrive at a common radius, and a complimentary regime in which spherical bilayers of differing radii stably coexist with common-radius, closed-loop, circular pores.