论文标题
在各向异性的分数stefan型问题上
On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions
论文作者
论文摘要
在这项工作中，我们考虑Lipschitz有限域中的分数Stefan型问题$ω\ subset \ subset \ mathbb {r}^d $，与时间相关的dirichlet边界条件$ \ vartheta = \ vartheta = \ vartheta（x，x，x，t）$，$ \ vartheta = g $ ody $ o $在$ω\ times] 0，t [$ by \ [\ frac {\ frac {\partialη} {\ partial t} +\ mathcal {l} _a^s \ s \ s \ vartheta = f \ quad quad \ quad pext { $ \ MATHCAL {l} _a^s $是由\ [\ langle \ Mathcal {langle \ Mathcal {L} _a^su，v \ rangle = \ rangle = \ int _ {\ int _ {最大单调图，$ a（x）$是一种对称，严格的椭圆形和均匀界限的矩阵，而$ d^s $是分销riesz分数梯度，价格为$ 0 <s <1 $。我们显示了具有相应弱的规则性的独特弱解的存在。我们还将收敛性视为$ s \ nearrow 1 $对经典的本地问题，渐近行为为$ t \ to \ infty $，以及两阶段Stefan型问题的收敛性通过改变最大单调图形$β$来改变单相Stefan-type问题。
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 \[\langle\mathcal{L}_A^su,v\rangle=\int_{\mathbb{R}^d}AD^su\cdot D^sv\,dx,\] $β$ is a maximal monotone graph, $A(x)$ is a symmetric, strictly elliptic and uniformly bounded matrix, and $D^s$ is the distributional Riesz fractional gradient for $0<s<1$. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $s\nearrow 1$ towards the classical local problem, the asymptotic behaviour as $t\to\infty$, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $β$.
