Well-posedness for the Classical Stefan Problem and the Zero Surface Tension Limit
Well-posedness for the Classical Stefan Problem and the Zero Surface Tension Limit
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经典 Stefan 问题和零表面张力极限的适定性
DOI：
10.1007/s00205-016-1041-8
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发表时间：
2011
影响因子：
2.5
通讯作者：
S. Shkoller
中科院分区：
文献类型：
--
作者：
Mahir Hadžić;S. Shkoller
我们开发了一个框架来统一处理有或没有表面张力的Stefan问题的适定性。在没有表面张力的情况下，我们建立了经典Stefan问题在Sobolev空间中的适定性。我们引入了一个新的速度变量，它将移动自由边界的速度扩展到内部区域。用该速度所满足的方程代替温度所满足的热方程进行分析。然后，经典Stefan问题的解被构造为对速度方程精心选择的近似序列的解的极限，其中移动的自由边界被正则化，并且边界条件被修改以保持原始问题的基本非线性结构。利用我们的方法，我们同时找到了适定所需的稳定性条件，并得到了移动自由边界正则性的新估计。最后，我们证明了具有正表面张力的Stefan问题的解σ收敛到经典Stefan问题的解，即σ→0。
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 of approximations to the velocity equation, in which the moving free-boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem. With our methodology, we simultaneously find the required stability condition for well-posedness and obtain new estimates for the regularity of the moving free-boundary. Finally, we prove that solutions of the Stefan problem with positive surface tension $${\sigma}$$σ converge to solutions of the classical Stefan problem as $${\sigma \to 0}$$σ→0.
具有表面张力和零表面张力极限的自由边界可压缩3-D欧拉方程的适定性
影响因子：
2
作者：
Coutand D
通讯作者：
Coutand D
注入 Hele-Shaw 流解的全局存在与衰减
影响因子：
1
作者：
Cheng C
通讯作者：
Cheng C
具有表面张力的热力学一致 Stefan 问题解的定性行为
影响因子：
2.5
作者：
Jan Prüss;Gieri Simonett;Rico Zacher
通讯作者：
Rico Zacher