相关论文: Maximal Solutions and Stochastic Free Boundary For…

We introduce and analyze a nonlocal version of the one-phase Stefan problem in which, as in the classical model, the rate of growth of the volume of the liquid phase is proportional to the rate at which energy is lost through the…

The dissolution of solids has created spectacular geomorphologies ranging from centimeter-scale cave scallops to the kilometer-scale "stone forests" of China and Madagascar. Mathematically, dissolution processes are modeled by a Stefan…

Motivated by applications in economics and finance, in particular to the modeling of limit order books, we study a class of stochastic second-order PDEs with non-linear Stefan-type boundary interaction. To solve the equation we transform…

We consider closed immersed hypersurfaces in $\R^{3}$ and $\R^4$ evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for…

For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry,…

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The…

We consider fully discrete numerical approximations for axisymmetric Willmore flow that are unconditionally stable and work reliably without remeshing. We restrict our attention to surfaces without boundary, but allow for spontaneous…

We consider a curve with boundary points free to move on a line in $\mathbb R^2$, which evolves by the $L^2$--gradient flow of the elastic energy, that is a linear combination of the Willmore and the length functional. For such planar…

The classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing…

Our study of abstract quasi-linear parabolic problems in time-weighted L_p-spaces, begun in [17], is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to…

Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical…

We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve…

In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat…

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the…

We consider the one-phase Stefan problem describing the evolution of melting ice. On the one hand, we focus on understanding the evolution of the free boundary near isolated singular points, and we establish for the first time upper and…

We derive two weak formulations for the supercooled Stefan problem with transport noise on a half-line: one captures a continuously evolving system, while the other resolves blow-ups by allowing for jump discontinuities in the evolution of…

We consider the quasistationary Stokes flow that describes the motion of a two-dimensional fluid body under the influence of surface tension effects in an unbounded, infinite-bottom geometry. We reformulate the problem as a fully nonlinear…

We consider two implicit approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently…

We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of 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…