M. K. Korten, C. N. Moore
We consider local solutions of the two-phase Stefan problem with a "mushy" region. We show that if a (distributional) solution u is locally square integrable then the temperature is continuous.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, $L_p$-maximal regularity theory, and the implicit function theorem.
Free boundary regularity in nonlinear one-phase Stefan problem
本文针对非线性算子的一相Stefan问题，借助Hodograph变换与线性化技术证明平坦自由边界具有时空$C^{1,\alpha}$正则性，并在算子光滑且凹或凸时得到自由边界光滑性。
Free boundary regularity for the inhomogeneous one-phase Stefan problem
本文证明不均匀单相Stefan问题解的平坦自由边界具有C^{1,α}正则性，采用 hodograph 变换和线性化技术，沿De Silva等人方法推导正则性。
Generic regularity in time for solutions of the Stefan problem in 4+1 dimensions
该研究证明4+1维Stefan问题解的自由边界在几乎每个时间点是C^∞类3维流形，通过分解奇异集为Σ^∞（豪斯多夫维数0投影）和抛物可数(n-2)可求长的Σ^*实现。
We provide perturbative estimates for the one-phase Stefan free boundary problem and obtain the regularity of flat free boundaries via a linearization technique in the spirit of the elliptic counterpart established by the first author.
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 freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Holder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root…
We study a Hele-Shaw problem with a mushy region obtained as a Mesa type limit of one phase Stefan problems in exterior domains. We study the convergence, determine some of the qualitative properties and regularity of the unique limiting solution, and prove regularity of the free boundary of this limit under very general conditions on the initial data. Indeed, our results handle changes in topology and multiple injection slots.
On the Geometry of Solutions of the Fully Nonlinear Inhomogeneous One-Phase Stefan Problem
本文在新非退化假设下刻画了带平坦自由边界的单相非齐次完全非线性Stefan问题解的几何性质，延续了线性情形正则性研究并验证了相关平坦性定义的合理性。
Regularity of the free boundary for the supercooled Stefan problem in arbitrary dimensions
本文研究了超冷却Stefan问题中自由边界的正则性，证明了其结构的稳健性，并揭示了奇异点总是以无限速度发生。
The Fractional Stefan Problem: Global Regularity of the Bounded Selfsimilar Solution"
本文研究了分数阶Stefan问题中自相似解的正则性，发现临界指数$ s=1/2 $决定解的正则性，亚临界情况下具有更高正则性，而临界及超临界情况下解在自由边界处不满足Lipschitz连续性。
We prove strong convergence to singular limits for a linearized fully inhomogeneous Stefan problem subject to surface tension and kinetic undercooling effects. Different combinations of $σ\to σ_0$ and $δ\toδ_0$, where $σ,σ_0 \ge 0$ and $δ,δ_0 \ge 0$ denote surface tension and kinetic undercooling coefficients respectively, altogether lead to five different types of singular limits. Their strong convergence is based on uniform maximal regularity estimates.
A transmission problem arising from the two-phase Stefan problem
本文研究源于两相非齐次Stefan问题线性化极限的抛物型传输问题，证明了经典解的存在唯一性、广义传输问题的Harnack不等式及界面附近的$C^{1,α}$估计，可用于推导带分布源的两相Stefan问题平坦自由边界的正则性。
We show uniqueness of solutions to the two-phase Stefan problem which have signed measures as initial data.
The non-local in space two-phase Stefan problem (a prototype in phase change problems) can be formulated via a singular nonlinear parabolic integro-differential equation which admits a unique weak solution. This formulation makes Stefan problem to be part of the General Filtration Problems; a class which includes the Porous Medium Equation. In this work, we prove that the weak solutions to both Stefan and Porous Media problems are continuous.
A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered. An integral relationship between the temperature and the free boundary is obtained which is equivalent to the fractional Stefan condition. Moreover, an exact solution of similarity type expressed in terms of Wright functions is given.
Loss of embeddedness for the one-phase quasistationary Stefan problem in 2D
本文构造了一个光滑且嵌入的初始状态例子，证明其在2D准稳态Stefan问题（含Gibbs-Thomson修正和动力学过冷）下会在有限时间内丧失嵌入性。
We investigate the regularizing behavior of two-phase Stefan problem near initial data. The main step in the analysis is to establish that in any given scale, the scaled solution is very close to a Lipschitz profile in space-time. We introduce a new decomposition argument to generalize the preceding ones by Choi, Jerion and Kim([CJK1], [CJK2]) and by Choi and Kim([CK]) on one-phase problems.
In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space one-phase Stefan problem in terms of the three parametric Mittag-Leffer function $E_{α,m;l}(z)$. We consider Dirichlet and Newmann conditions at the fixed face, involving Caputo fractional space derivatives of order $0 < α< 1$. We recover the solution for the classical one-phase Stefan problem when the order of the Caputo derivatives approaches one.
Free boundary regularity and well-posedness of physical solutions to the supercooled Stefan problem
研究超冷却Stefan问题物理解的正则性和适定性，证明自由边界在空间变量下为C^1，在除外闭集外为C^∞，并证明跳跃点集无积累点，通过加权障碍问题建立正则性和非退化估计，控制未来跳跃的发生。
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 introduce a new velocity variable which extends the velocity of the moving free-boundary into the interior domain. The equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature. Solutions to the classical Stefan problem are then constructed as the limit of solutions to a carefully chosen sequence o…
A Convergence result for a Stefan problem with phase relaxation
针对带相位松弛的Stefan模型，论文在初始相位受约束且边界条件更一般的条件下，证明松弛参数趋于零时其解强收敛到经典弱Stefan问题的解。
We obtain a generalized Neumann solution for the two-phase fractional Lamé-Clapeyron-Stefan problem for a semi-infinite material with constant boundary and initial conditions. In this problem, the two governing equations and a governing condition for the free boundary include a fractional time derivative in the Caputo sense of order $0<\al\leq 1$. When $ \al \nearrow $ 1 we recover the classical Neumann solution for the two-phase Lamé-Clapeyron-Stefan problem given through the error function.
Asymptotic analysis for the Stefan problem with a doubly nonlinear phase relaxation
研究具有双非线性相松弛的斯蒂芬问题，通过递增连续函数和极大反单调图确定相函数收敛速率，证明松弛问题解的存在唯一性并对斯蒂芬问题作渐近分析。
An elliptic regularization approach to the Stefan problem
本文通过椭圆正则化方案证明了具有有限能量、可能非空糊状区域和空间依赖熔化温度的两相Stefan问题弱解的存在性，并建立了比较原理。
We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.
We study the vanishing viscosity limit of a nonlinear diffusion equation describing chemical reaction interface or the spatial segregation interface of competing species, where the diffusion rate for the negative part of the solution converges to zero. As in the standard one phase Stefan problem, we prove that the positive part of the solution converges uniformly to the solution of a generalized one phase Stefan problem. This information is then employed to determine the limiting equation for the negative part, which is an ordinary differential equation.
Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile
针对带Neumann边界条件的一维单相Stefan问题，构造了唯一自相似解，并证明任意初值下的解轨道均收敛于该自相似解。
We study the Cauchy-Dirichlet problem associated to a phase transition modeled upon the degenerate two-phase Stefan problem. We prove that weak solutions are continuous up to the parabolic boundary and quantify the continuity by deriving a modulus. As a byproduct, these a priori regularity results are used to prove the existence of a so-called physical solution.
We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.
We study a nonlocal version of the one-phase Stefan problem which develops mushy regions, even if they were not present initially, a model which can be of interest at the mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The created mushy regions have the size of the support of this kernel. If the kernel is suitably rescaled, such regions disappear and the solution converges to the solution of the usual local version of the one-phase Stefan problem. We prove that the model is well posed, and give several qualitative properties. In particular, the long-tim…
We study a nonlocal version of the two-phase Stefan problem, which models a phase transition problem between two distinct phases evolving to distinct heat equations. Mathematically speaking, this consists in deriving a theory for sign-changing solutions of the equation, ut = J * v - v, v = Γ(u), where the monotone graph is given by Γ(s) = sign(s)(|s|-1)+ . We give general results of existence, uniqueness and comparison, in the spirit of [2]. Then we focus on the study of the asymptotic behaviour for sign-changing solutions, which present challenging difficulties due to the non-monotone evoluti…
We analyze the state constrained inverse Stefan type parabolic free boundary problem as an optimal control problem in the Sobolev-Besov spaces framework. Boundary heat flux, density of heat sources, and free boundary are components of the control vector. Cost functional is the sum of the $L_2$-norm declinations of the temperature measurement at the final moment, the phase transition temperature, the final position of the free boundary, and the penalty term, taking into account the state constraint on the temperature. We prove the existence of optimal control, Frechet differentiability, and opt…
On solutions of an ill-posed Stefan problem
本文研究具有递增Riemann初值和负潜热的多相Stefan问题，通过自相似解的变分公式给出了解存在唯一性的精确条件。
Logarithmic continuity for the Nonlocal degenerate two-phase Stefan problem
本文针对非局部退化两相Stefan问题，建立了弱解的振荡估计，并在内蕴尺度圆柱体中获得了定量连续性模，从而构造了初边值问题的连续弱解。
We derive the quantitative modulus of continuity $$ ω(r)=\left[ p+\ln \left( \frac{r_0}{r} \right) \right]^{-α(n,p)}, $$ which we conjecture to be optimal, for solutions of the $p$-degenerate two-phase Stefan problem. Even in the classical case $p=2$, this represents a twofold improvement with respect to the 1984 state-of-the-art result by DiBenedetto and Friedman [J. reine angew. Math., 1984], in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent $α(n,p)$.
The two-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain, composed of two regions separated by an a priori unknown moving boundary which is transported by the difference (or jump) of the normal derivatives of the temperature in each phase. We establish local-in-time well-posedness and a global-in-time stability result for arbitrary sufficiently smooth domains and small initial temperatures. To this end, we develop a higher-o…
We consider a one-dimensional one-phase inverse Stefan problem for the heat equation. It consists in recovering a boundary influx condition from the knowledge of the position of the moving front, and the initial state. We derived a logarithmic stability estimate that shows that the inversion is ill-posed. The proof is based on integral equations and unique continuation for holomorphic functions. We also proposed a direct algorithm with a regularization term to solve the nonlinear inverse problem. Several numerical tests using noisy data are provided with relative errors analysis.
From the one-dimensional consolidation of fine-grained soils with threshold gradient, it can be derived a special type of Stefan problems where the seepage front, due to the presence of this threshold gradient, exhibits the features of a moving boundary. In this kind of problems, in contrast with the classical Stefan problem, the latent heat is considered to depend inversely with the rate of change of the seepage front. In this paper a one-phase Stefan problem with a latent heat that not only depends on the rate of change of the free boundary but also on its position is studied. The aim of thi…
We consider a family of multi-phase Stefan problems for a certain 1-d model of cell-to-cell adhesion and diffusion, which takes the form of a nonlinear forward-backward parabolic equation. In each material phase the cell density stays either high or low, and phases are connected by jumps across an `unstable' interval. We develop an existence theory for such problems which allows for the annihilation of phases and the subsequent continuation of solutions. Stability results for the long-time behaviour of solutions are also obtained, and, where necessary, the analysis is complemented by numerical…
We study the long-time behavior of solutions of the one-phase Stefan problem in inhomogeneous media in dimensions $n \geq 2$. Using the technique of rescaling which is consistent with the evolution of the free boundary, we are able to show the homogenization of the free boundary velocity as well as the locally uniform convergence of the rescaled solution to a self-similar solution of the homogeneous Hele-Shaw problem with a point source. Moreover, by viscosity solution methods, we also deduce that the rescaled free boundary uniformly approaches a sphere with respect to Hausdorff distance.
We obtain for the two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material an equivalence between the temperature and convective boundary conditions at the fixed face in the case that an inequality for the convective transfer coefficient is satisfied. Moreover, an inequality for the coefficient which characterizes the solid-liquid interface of the classical Neumann solution is also obtained. This inequality must be satisfied for data of any phase-change material, and as a consequence the result given in Tarzia, Quart. Appl. Math., 39 (1981), 491-497 is also recovered when a heat fl…
We study the large-time behavior of solutions of a one-phase Stefan-type problem with anisotropic diffusion in periodic media on an exterior domain in a dimension $n \geq 3$. By a rescaling transformation that matches the expansion of the free boundary, we deduce the homogenization of the free boundary velocity together with the homogenization of the anisotropic operator. Moreover, we obtain the convergence of the rescaled solution to the solution of the homogenized Hele-Shaw-type problem with a point source and the convergence of the rescaled free boundary to a self-similar profile with respe…
Taking into account the recent works \cite{RoTaVe:2020} and \cite{Rys:2020}, we consider a phase-change problem for a one dimensional material with a non-local flux, expressed in terms of the Caputo derivative, which derives in a space-fractional Stefan problem. We prove existence of a unique solution to a phase-change problem with the fractional Neumann boundary condition at the fixed face $x=0$, where the domain, at the initial time, consists of liquid and solid. Then we use this result to prove the existence of a limit solution to an analogous problem with solid initial domain, when it is n…
Optimal control of the singular nonlinear parabolic PDE which is a distributional formulation of multidimensional and multiphase Stefan-type free boundary problem is analyzed. Approximating sequence of finite-dimensional optimal control problems is introduced via finite differences. Existence of the optimal control and the convergence of the sequence of discrete optimal control problems both with respect to functional and control is proved. In particular, convergence of the method of finite differences, and existence, uniqueness and stability estimations are established for the singular PDE pr…
In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $Ω\subset\mathbb{R}^d$ with time-dependent Dirichlet boundary condition for the temperature $\vartheta=\vartheta(x,t)$, $\vartheta=g$ on $Ω^c\times]0,T[$, and initial condition $η_0$ for the enthalpy $η=η(x,t)$, given in $Ω\times]0,T[$ by \[\frac{\partial η}{\partial t} +\mathcal{L}_A^s \vartheta= f\quad\text{ with }η\in β(\vartheta),\] where $\mathcal{L}_A^s$ is an anisotropic fractional operator defined in the distributional sense by \[\langle\mathcal{L}_A^su,v\rangle=\int_{\mathbb{R}^d}AD^su\cdot D^sv…
The supercooled Stefan problem: fractal freezing and the fine structure of maximal solutions
本文研究了过冷Stefan问题，揭示了最大解的分形冻结特性及非通用性，并通过障碍问题理论分析了其正则性及稳定性。
We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also pres…
The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems. After a general overview, we present some recent results on the structure of singular free boundary points. Then, we show some selected applications to the generic smoothness of the free boundary in the stationary obstacle problem (Schaeffer's conjecture), and to the smoothness of the free boundary in the one-phase Stefan problem for almost every time.
Extinction rates for nonradial solutions to the Stefan problem
研究非径向解Stefan问题中自由边界在奇点处的正则性及灭绝速率的估计。
The Stefan problem in a thermomechanical context with fracture and fluid flow
本研究将经典仅考虑传热的Stefan问题拓展至热力学范畴，结合欧拉描述、粘弹性流变学、相场模型与L¹理论，严格证明了含断裂、流体流动及相变动力学效应的不完全熔化问题弱解的存在性。
We prove existence and uniqueness of strong solutions to the two-phase Stefan problem with Gibbs-Thomson law where the free interface forms a ninety degree contact angle with the fixed boundary. We also discuss existence of global solutions and convergence to equilibria.
Well-posedness for a two-phase Stefan problem with radiation
本文研究冰融化的一维两相Stefan类自由边界问题，其固相温度方程含非局部积分算子，结合Banach不动点定理与抛物理论证明了局部适定性，并通过构造稳态上下解建立了一大类初始数据的全局适定性理论。
The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), a modified heat balance integral method, the refined integral method (RIM) . Taking advantage of the exact analytical solutions we compare and test the accuracy of the approximate solutions. The…
Traveling waves for a two-phase Stefan problem with radiation
本文研究具有辐射传热的一维两相Stefan自由边界问题的行波解，证明其存在固相膨胀的行波解，并利用最大值原理、爆破极限和刘维尔型定理分析解的性质。
We derive the fractional version of one-phase one-dimensional Stefan model. We assume that the diffusive flux is given by the time-fractional Riemann-Liouville derivative, i.e. we impose the memory effect in the examined model. Furthermore, we find a special solution to this problem.
On behavior of free boundaries to generalized two-phase Stefan problems for parabolic partial differential equation systems
本文研究了广义双相Stefan问题中自由边界的行为，通过提升解的正则性，克服了自由边界增长速率依赖于水分含量以及水分含量边界条件包含温度的困难，建立了局部时间内的解的存在性和唯一性，并在某些初始数据符号条件下推导了解存在最大区间的结果。
We construct examples for the one-phase Stefan problem which show that $α$-concavity of the solution is in general not preserved in time, for $0 \le α<1/2$. In particular, this shows that, in contrast to the case of the heat equation for a fixed convex domain, log concavity is not preserved for solutions of the Stefan problem.
On self-similar solutions of a multi-phase Stefan problem
本文研究多相Stefan问题的自相似解，证明单变量与径向多维情形下确定自由边界的非线性代数系统为梯度型且对应显式强制势函数，一维情形下势严格凸保证解唯一，多维情形下解的唯一性可由一般理论得出。
Existence of weak solutions to two-phase Stefan problems for parabolic partial differential equation system
本文针对面包烘焙过程的两相Stefan问题，在放宽初始函数条件后，证明了弱解的存在性，弥补了强解仅适用于严格条件下的局限。
In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。