Related papers: The parabolic flows for complex quotient equations
We prove the convergence of a particle method for the approximation of diffusive gradient flows in one dimension. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles and preserves the…
In this paper, we consider Hessian equations with its structure as a combination of elementary symmetric functions on closed K\"ahler manifolds. We provide a sufficient and necessary condition for the solvability of these equations, which…
The paper addresses the numerical approximation of two variants of hyperbolic mean curvature flow of surfaces in $\mathbb R^3$. For each evolution law we propose both a finite element method, as well as a finite difference scheme in the…
In this paper the Orlicz-Minkowski problem for torsional rigidity, a generalization of the classical Minkowski problem, is studied. Using the flow method, we obtain a new existence result of solutions to this problem for general measures.
In this article, we show how to construct a numerical method for solving hyperbolic problems, whether linear or nonlinear, using a continuous representation of the variables and their mean value in each triangular element. This type of…
We study the convergence behavior of the general inverse $\sigma_k$-flow on K\"{a}hler manifolds with initial metrics satisfying the Calabi Ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic…
We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of…
We present the method for computation of fluid flows that are characterized by the large degree of expansion/contraction and in which the fluid velocity is dominated by the bulk component associated with the expansion/contraction and/or…
In this paper we review recent results by the author on the problem of quantization of measures. More precisely, we propose a dynamical approach, and we investigate it in dimensions 1 and 2. Moreover, we discuss a recent general result on…
We study a parabolic complex Monge-Amp\`{e}re type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain…
By using a formulation of a class of compressible viscous flows with a heat source via vorticity and expansion-rate, we study the Oberbeck-Boussinesq flows. To this end we establish a new integral representation for solutions of parabolic…
Optimal transport theory has been a powerful tool for the analysis of parabolic equationsviewed as gradient flows of volume forms according to suitable transportation metrics.In this paper, we present an example of gradient flows for closed…
This article studies special solutions to symplectic curvature flow in dimension four. Firstly, we derive a local normal form for static solutions in terms of holomorphic data and use this normal form to show that every complete static…
We develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. These are applied to homogenize convection dominated parabolic equations with rapidly…
The classical approach to visualizing a flow, in terms of its streamlines, motivates a topological/soft-analytic argument for constrained variational equations. In its full generality, that argument provides an explicit formula for…
In this article we derive both Hamilton type and Souplet-Zhang type gradient estimations for a system of semilinear equations along a geometric flow on a weighted Riemannian manifold.
It is shown that the complex Bernoulli differential equations admitting the supplementary structure of a Lie-Hamilton system related to the book algebra $\mathfrak{b}_2$ can always be solved by quadratures, providing an explicit solution of…
Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge-B\"acklund transformation, underlying symmetries among superficially…
We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and…
A modified version of the conditional symmetry method, together with the classical method, is used to obtain new classes of elliptic solutions of the isentropic ideal compressible fluid flow in (3+1) dimensions. We focus on those types of…