Related papers: The Stefan problem and concavity

We develop a framework for a unified treatment of well-posedness for the Stefan problem with or without surface tension. In the absence of surface tension, we establish well-posedness in Sobolev spaces for the classical Stefan problem. We…

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a…

A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain…

In this paper, we study the deformation of the 2 dimensional convex surfaces in $\R^{3}$ whose speed at a point on the surface is proportional to $\alpha$-power of positive part of Gauss Curvature. First, for 1/2<\alpha\leq 1$, we show that…

The Cauchy problem for non-isentropic compressible Navier-Stokes/Allen-Cahn system with degenerate heat-conductivity $\kappa(\theta)=\tilde{\kappa}\theta^\beta$ in 1-d is discussed in this paper. This system is widely used to describe the…

We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in $N$-dimensional convex domains, where $N\ge 2$ (indeed, we prove that starting with a negative power concave initial datum may result in…

We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo's definition. We survey…

This paper delves into the Inverse Stefan problem, specifically focusing on determining the time-dependent source coefficient in the parabolic heat equation governing heat transfer in a semi-infinite rod. The problem entails the intricate…

We consider the Stefan problem, firstly with regular data and secondly with irregular data. In both cases is given a proof for the convergence of an approximation obtained by regularising the problem. These proofs are based on weak…

We develop a high-order energy method to prove asymptotic stability of flat steady surfaces for the Stefan problem with surface tension - also known as the Stefan problem with Gibbs-Thomson correction.

We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: \[ \partial_t \beta(u)-\Delta_p u\ni 0\quad\text{ for…

We consider the three-dimensional radial Stefan problem which describes the evolution of a radial symmetric ice ball with free boundary \begin{equation*} \left\{\begin{aligned} &\partial_{t}u-\partial_{rr}u-\frac{2}{r}\partial_{r}u=0 \quad…

We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space-time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such…

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…

We derive the quantitative modulus of continuity $$ \omega(r)=\left[ p+\ln \left( \frac{r_0}{r} \right) \right]^{-\alpha (n,p)}, $$ which we conjecture to be optimal, for solutions of the $p$-degenerate two-phase Stefan problem. Even in the…

We consider the problem of heat conduction with phase change, that is essential for permafrost modeling in Land Surface Models and Dynamic Global Vegetation Models. These models require minimal computational effort and an extremely robust…

In this paper we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative…

Unlike the heat equation or the Laplace equation, solutions of the wave equation on general domains have no known stochastic representation. This short note gives a simple solution to this well known problem in arbitrary dimensions. The…

We study the supercooled Stefan problem in arbitrary dimensions. First, we study general solutions and their irregularities, showing generic fractal freezing and nucleation, based on a novel Markovian gluing principle. In contrast, we then…

Stefan problems relevant to burning oil-water systems are formulated. Two moving boundary sub-problems are defined: burning liquid surface and formation of a distillation ("hot zone") layer beneath it. The basic model considers a heat…