Uniqueness for solutions of the two-phase Stefan problem with signed measures as data

一维超冷却Stefan问题最大解的唯一性

Uniqueness of the maximal solution of the supercooled Stefan problem in 1D

本文研究了一维超冷却Stefan问题最大解的唯一性，证明其在弱测度收敛下具有稳定性，同时指出该问题缺乏单调性和L^1-Lipschitz稳定性。

关于一个不适定Stefan问题的解

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。

Solvability of some Stefan type problems

In this paper, we interest on some class of Stefan type problems. We prove the existence and uniqueness of renormalized solution in anisotropic Sobolev spaces with data belongs to $L^1- data,$ based on the properties of the renormalized trunctions and the generalized monotonicity method in the functional spaces.

Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness

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…

Existence and uniqueness of solution for two one-phase Stefan problems with variable thermal coefficients

One dimensional Stefan problems for a semi-infinite material with temperature dependent thermal coefficients are considered. Existence and uniqueness of solution are obtained imposing a Dirichlet or a Robin type condition at fixed face $x=0$. Moreover, it is proved that the solution of the problem with the Robin type condition converges to the solution of the problem with the Dirichlet condition at the fixed face. Computational examples are provided.

关于多相Stefan问题的自相似解

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。

The Hele-Shaw problem as a "Mesa" limit of Stefan problems: Existence, uniqueness, and regularity of the free boundary

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.

具有双非线性相松弛的斯蒂芬问题的渐近分析

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。

A nonlocal two phase Stefan problem

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…

A free boundary problem of Stefan type with nonlocal diffusion

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 interphase. We prove existence and uniqueness for the problem posed on the line, and on the half-line with constant Dirichlet data, and in the radial case in several dimensions. We also describe the asymptotic behaviour of both the solution and its free boundary. The model may be of interest to describe the spreading of populations in hostile environments.

源于两相Stefan问题的一个传输问题

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。

Recovering the initial condition in the One-Phase Stefan problem

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…

Signed Radon measure-valued solutions of flux saturated scalar conservation laws

We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.

随机Stefan问题潜热消失极限：一个误差估计

The vanishing latent heat limit of a stochastic Stefan problem : An error estimate

本文研究随机Stefan问题在潜热系数趋于零时的极限，通过引入时间白噪声和空间有色噪声，证明了弱解的存在唯一性，并基于正潜热与零潜热解之间的误差估计（即使在确定性情形下也新颖）建立了极限行为。

具有表面张力的 Stefan 问题：径向对称下物理解的唯一性

Stefan problem with surface tension: uniqueness of physical solutions under radial symmetry

本文研究带表面张力的径向对称 Stefan 问题，利用三维 Brown 运动击中概率分析与自由边界凸性，证明了 [NS23] 所引入物理解的唯一性。

Explicit solution for Stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face using Kummer functions

An explicit solution of a similarity type is obtained for a one-phase Stefan problem in a semi-infinite material using Kummer functions. Motivated by [D.A. Tarzia, Relationship between Neumann solutions for two phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions, Thermal Sci.(2016) DOI 10.2298/TSCI 140607003T, In press], and [Y. Zhou, L.J. Xia, Exact solution for Stefan problem with general power-type latent heat using Kummer function, Int. J. Heat Mass Transfer, 84 (2015) 114-118], we consider a phase-change problem with a latent heat defined as a power fu…

自由边界正则性与超冷却Stefan问题物理解的适定性

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。

Uniqueness of signed measures solving the continuity equation for Osgood vector fields

Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this result to signed measure-valued solutions, under a quantitative two-sided Osgood condition on the velocity field. Our results extend those obtained for log-Lipschitz vector fields by Bahouri and Chemin.

Optimal Control of Singular Parabolic PDEs Modeling Multiphase Stefan-type Free Boundary Problems

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…

Uniqueness for the inverse fixed angle scattering problem

We present a uniqueness result in dimensions $2$ and $3$ for the inverse fixed angle scattering problem associated to the Schrödinger operator $-Δ+q$, where $q$ is a small real valued potential with compact support in the Sobolev space $W^{β,2}$ with $β>0.$ This result improves the known result, due to Stefanov, in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension $4$, but for more regular potentials in $W^{β,2}$ with $β>2/3$.

在存在唯一流的情况下连续性方程的有符号测度值解的非唯一性

Non-uniqueness of signed measure-valued solutions to the continuity equation in presence of a unique flow

本文通过两个反例证明：即使向量场具有唯一的同胚流，带符号测度值连续性方程仍可存在零初始数据的非平凡解，从而否定叠加原理的相应推广。

Auxiliary functions in the study of Stefan-like problems with variable thermal properties

We address the existence and uniqueness of the so-called modified error function that arises in the study of phase-change problems with specific heat and thermal conductivity given by linear functions of the material temperature. This function is defined from a differential problem that depends on two parameters which are closely related with the slopes of the specific heat and the thermal conductivity. We identify conditions on these parameters which allow us to prove the existence of the modified error function. In addition, we show its uniqueness in the space of non-negative bounded analyti…

关于广义双相Stefan问题中自由边界行为的研究

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。

Explicit solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms

A one-phase Stefan problem for a semi-infinite material is investigated for special functional forms of the thermal conductivity and specific heat depending on the temperature of the phase-change material. Using the similarity transformation technique, an explicit solution for these situations are showed. The mathematical analysis is made for two different kinds of heat source terms, and the existence and uniqueness of the solutions are proved.

过冷Stefan问题的一种变换方法

A transform approach to the supercooled Stefan problem

本文通过伊藤公式建立初始数据拉普拉斯变换与解变换的关系，推广过冷Stefan问题至非可积初始数据，获得全局解存在性、唯一性扩展及渐近行为结果。

A Wasserstein norm for signed measures, with application to nonlocal transport equation with source term

We introduce the optimal transportation interpretation of the Kantorovich norm on thespace of signed Radon measures with finite mass, based on a generalized Wasserstein distancefor measures with different masses.With the formulation and the new topological properties we obtain for this norm, we proveexistence and uniqueness for solutions to non-local transport equations with source terms, whenthe initial condition is a signed measure.

Homogenization of the Stefan problem, with application to maple sap exudation

The technique of periodic homogenization with two-scale convergence is applied to the analysis of a two-phase Stefan-type problem that arises in the study of a periodic array of melting ice bars. For this "reduced model" we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, which requires solving a macroscale problem for the global temperature field and a reference cell problem at each point in space which captures the underlying phase change process occurring on the microscale. We state a corresponding strong formulation of the limit probl…

Nonlinear Stefan Problem for one-phase generalized heat equation with heat flux and convective boundary condition

In this article we consider a mathematical model of an initial stage of closure electrical contact that involves a metallic vaporization after instantaneous exploding of contact due to arc ignition with power $P_0$ on fixed face $z=0$ and heat transfer in material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component with heat flux and convective boundary conditions prescribed at the known free boundary $z = α(t)$. The temperature field in the liquid region of such kind of material can be modelled by Stefan …

半直线上带固定边界不同边界条件的多相Stefan问题

On a multi-phase Stefan problem in the half-line with different boundary conditions at the fixed boundary

该研究针对半直线上带不同边界条件的多相Stefan问题，通过变分方法建立解的存在唯一性，确定了Dirichlet-to-Neumann映射的性质，并推广到Robin边界情形，还研究了负连接系数的不适定问题。

Zero kinetic undercooling limit in the supercooled Stefan problem

We study the solutions of the one-phase supercooled Stefan problem with kinetic undercooling, which describes the freezing of a supercooled liquid, in one spatial dimension. Assuming that the initial temperature lies between the equilibrium freezing point and the characteristic invariant temperature throughout the liquid our main theorem shows that, as the kinetic undercooling parameter tends to zero, the free boundary converges to the (possibly irregular) free boundary in the supercooled Stefan problem without kinetic undercooling, whose uniqueness has been recently established in [DNS19], [L…

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component is considered. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modelled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature …

有符号测度的最优运输：存在性、唯一性与分形结构

Optimal transport of signed measures: existence, uniqueness and fractal structure part I: the separated-support case research announcement, with a numerical validation report

本文研究有符号实测度的最优运输理论，通过位置惩罚参数λ区分同号与异号运输，证明存在性和唯一性，并展示最优运输保持分形集的Hausdorff维度和Ahlfors正则性。

具有振荡初始条件的过冷 Stefan 问题的适定性

Well-posedness of the supercooled Stefan problem with oscillatory initial conditions

本文通过引入平均条件替代局部单调性，证明了具有振荡初始条件的过冷 Stefan 问题物理解的唯一性，并验证了该条件适用于多种振荡密度。

确定时间相关源系数和热流函数的逆Stefan问题

Inverse Stefan problems of determining the time-dependent source coefficient and heat flux function

本文针对半无限长杆热传导的逆Stefan问题，提出确定时间相关源系数和热流函数的方法，并建立了弱解的存在性、唯一性及连续依赖性。

Nonlinear diffusion equations as asymptotic limits of Cahn--Hilliard systems on unbounded domains via Cauchy's criterion

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $Ω\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-Δ+1)β(u) = g \quad \mbox{in}\ Ω\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $β$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $β$ even tho…

带有符号测度数据的半线性 Schrödinger 方程的边值问题

Boundary value problems with signed measure data for semilinear Schrödinger equations

本文研究了有界 Lipschitz 域上半线性 Schrödinger 方程的边值问题，推导了具有边界迹的符号上下解表示公式，并据此解决了带符号测度数据的关联边值问题的存在性与唯一性。

Stefan问题与最优布朗鞅输运的自由目标

The Stefan problem and free targets of optimal Brownian martingale transport

本文提出并解决自由目标最优布朗停时问题，其最优目标具有普适性，统一生成Stefan问题的不稳定（冻结）与稳定（融化）解，并首次获得不适定冻结问题的全局存在性与弱强唯一性。

两相源与反应系数的斯蒂芬型问题

Two-phase source and reaction coefficient Stefan type problems

研究抛物型热方程逆两相斯蒂芬型问题，通过变换和傅里叶谱展开推导未知系数重构公式，分两种表述识别源和反应系数，利用移动边界的斯蒂芬条件确定系数，建立解的性质，示例验证方法适用性及系数在噪声下的稳定性。

在本构约束下波动方程初始源和声速的同时恢复

Simultaneous Recovery of the Initial Source and Sound Speed for the Wave Equation under a Constitutive Constraint

研究从单个边界测量对标量波动方程初始源和声速同时恢复，通过本构关系耦合未知量，利用微局部和卡尔曼框架，在特定几何条件下建立全局唯一性和利普希茨稳定性，还获部分数据结果。

A two-scale Stefan problem arising in a model for tree sap exudation

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multi-scale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the g…

Ice formation in the Arctic during summer: false-bottoms

The only source of ice formation in the Arctic during summer is a layer of ice called false-bottoms between an under-ice melt pond and the underlying ocean. Of interest is to give a mathematical model in order to determine the simultaneous growth and ablation of false-bottoms, which is governed by both of heat fluxes and salt fluxes. In one dimension, this problem may be considered mathematically as a two-phase Stefan problem with two free boundaries. Our main result is to prove the existence and uniqueness of the solution from the initial condition.

Convergence in $C(\lbrack0,T\rbrack;L^2(Ω))$ of weak solutions to perturbed doubly degenerate parabolic equations

We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic $p$-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original problem as the perturbed data approach the original data. We do not assume uniqueness or additional regularity of the solution. However, when uniqueness is known, our result demonstrates that the weak solution is uniformly temporally stable to pert…

p-Laplacian type equations involving measures

This is a survey on problems involving equations $-\operatorname{div}{\Cal A}(x,\nabla u)=μ$, where $μ$ is a Radon measure and ${\Cal A}:\bold {R}^n\times\bold R^n\to \bold R^n$ verifies Leray-Lions type conditions. We shall discuss a potential theoretic approach when the measure is nonnegative. Existence and uniqueness, and different concepts of solutions are discussed for general signed measures.

非局部方程的渗流类型方程分布解的唯一性与性质

Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type

本文研究非局部渗流类型方程初始值问题中分布解的唯一性、存在性及性质，探讨了非局部渗流方程、快速扩散方程和Stefan问题等方程类的解的唯一性和存在性，同时得到L^1收缩性和先验估计。

带维度扭曲的有符号分形测度的最优输运：变分刻画

Optimal transport of signed fractal measures with dimensional distortion: a variational characterization

本文扩展了有符号测度的最优输运理论，通过引入惩罚项控制源和目标之间分形支撑的豪斯多夫维度扭曲，证明了最优输运映射的存在唯一性、推广的Monge-Ampère方程以及双Legendre-Fenchel变分刻画。

含焦耳热源与非线性热系数的闭合电接触热现象数学模型

Mathematical model of thermal phenomena of closure electrical contact with Joule heat source and nonlinear thermal coefficients

针对含金属汽化的闭合电接触加热问题，提出基于Stefan型问题的数学模型，采用相似变量变换法简化问题，并证明了解的存在性与唯一性。

小数据$3D$可压缩欧拉解的两个全局稳定性定理：具有梯度爆破的唯一极大全局双曲发展及全时空光滑全局存在性

Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime

本文证明两个互补的小数据稳定性定理，分别给出球对称3D可压缩欧拉方程激波形成解的MGHD唯一存在性及带$1/r$密度尾部数据的全时空光滑全局存在性。