相关论文: Variational evolution problems and nonlocal geomet…
We investigate a singularly perturbed, non-convex variational problem arising in materials science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan M\"uller, where it is proven that the…
We study numerical methods for the nonlinear partial differential equation that governs the motion of level sets by affine curvature. We show that standard finite difference schemes are nonlinearly unstable. We build convergent finite…
We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough…
In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that is, backward stochastic evolution equations, stochastic Volterra type evolution…
The concept of continuous topological evolution, based upon Cartan's methods of exterior differential systems, is used to develop a topological theory of non-equilibrium thermodynamics, within which there exist processes that exhibit…
We consider the modified Monge-Kantorovich problem with additional restriction: admissible transport plans must vanish on some fixed functional subspace. Different choice of the subspace leads to different additional properties optimal…
This short survey article stems from recent progress on critical cases of stochastic evolution equations in variational formulation with additive, multiplicative or gradient noises. Typical examples appear as the limit cases of the…
Many applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of non-moving porous media,…
Short time existence for a surface diffusion evolution equation with curvature regularization is proved in the context of epitaxially strained three-dimensional films. This is achieved by implementing a minimizing movement scheme, which is…
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown…
This paper mainly investigates the approximation of a global maximizer of the 1-D Monge-Kantorovich mass transfer problem through the approach of nonlinear differential equations with Dirichlet boundary. Using an approximation mechanism,…
We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients,…
We show that the Hellinger-Kantorovich distance can be expressed as the metric infimal convolution of the Hellinger and the Wasserstein distances, as conjectured by Liero, Mielke, and Savar\'e. To prove it, we study with the tools of…
For a parabolic surface partial differential equation coupled to surface evolution, convergence of the spatial semidiscretization is studied in this paper. The velocity of the evolving surface is not given explicitly, but depends on the…
Moving boundary problems allow to model systems with phase transition at an inner boundary. Driven by problems in economics and finance, in particular modeling of limit order books, we consider a stochastic and non-linear extension of the…
We consider a nonlocal differential equation of Kirchhoff type with a convolution coefficient involving variable growth. The novelty of our work lies in allowing a variable exponent in the nonlocal term. By relating the variable growth…
Similar evolutionary variational inequalities appear as convenient formulations for continuous models for sandpile growth, magnetization of type-II superconductors, and evolution of some other dissipative systems characterized by the…
It is shown that, by imposing reparametrization invariance, one may derive a variety of stochastic equations describing the dynamics of surface growth and identify the physical processes responsible for the various terms. This approach…
Space time fractional nonlinear evolution equations have been widely applied for describing various types of physical mechanism of natural phenomena in mathematical physics and engineering. The proposed generalized exp expansion method…
A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot - or, in some cases, only -- periodic solutions. Several examples (ODEs and PDEs) are exhibited.