Related papers: A mean field type flow
We derive continuity equation and exact expression for flow probability density in a space with arbitrary deformed algebra leading to minimal length. In coordinate representation the flow probability density is presented as infinite series…
We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by…
We flow a hypersurface in Euclidean space by mean curvature flow with a Neumann boundary condition, where the boundary manifold is any torus of revolution. If we impose the conditions that the initial manifold is compatible and does not…
We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the…
We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general…
We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion…
The well-posedness of a phase-field approximation to the Willmore flow with volume constraint is established. The existence proof relies on the underlying gradient flow structure of the problem: the time discrete approximation is solved by…
Gradient flows play a substantial role in addressing many machine learning problems. We examine the convergence in continuous-time of a \textit{Fisher-Rao} (Mean-Field Birth-Death) gradient flow in the context of solving convex-concave…
We derive the system of differential equations for the gradient flow characterizing the training process of linear in-context learning in full generality. Next, we explore the geometric structure of the gradient flows in two instances,…
We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as…
We apply the parabolic flow method to solving complex quotient equations on closed K\"ahler manifolds. We study the parabolic equation and prove the convergence. As a result, we solve the complex quotient equations.
Predicting the amount of time required for a transient groundwater response to take place is a practical question that is of interest in many situations. This time scale is often called the response time. In the groundwater hydrology…
Flow instability and turbulent transition can be well explained using a new proposed theory--Energy gradient theory [1]. In this theory, the stability of a flow depends on the relative magnitude of energy gradient in streamwise direction…
We are interested in the gradient flow of a general first order convex functional with respect to the $L^1$-topology. By means of an implicit minimization scheme, we show existence of a global limit solution, which satisfies an…
We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then…
We present a causal trajectory interpretation for the massive vector field, based on the flows of rest energy and a conserved density defined using the time-like eigenvectors and eigenvalues of the stress-energy-momentum tensor. This work…
In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates,…
In this paper, we provide a proof of Hamilton's extrinsic pinching theorem using the mean curvature flow approach.
Our study of a basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics in the case of constant but non-equal densities of the phases, begun by the first two authors is continued. We extend our…
We consider a variational model for heterogeneous phase separation, based on a diffuse interface energy with moving wells. Our main result identifies the asymptotic behavior of the first variation of the phase field energies as the width of…