Moving Surfaces and Abstract Parabolic Evolution Equations
Moving Surfaces and Abstract Parabolic Evolution Equations
复制标题
移动表面和抽象抛物线演化方程
DOI：
10.1007/978-3-0348-8765-6_10
复制
发表时间：
1999
期刊：
影响因子：
--
通讯作者：
G. Simonett
中科院分区：
文献类型：
--
作者：
J. Escher;G. Simonett
本文的目的是对涉及运动曲面的椭圆和抛物问题的经典解理论的一些最新发展作一综述。这种类型的问题不满足叠加原理的解决方案，因此，进行了固有的非线性结构。事实上，大多数描述曲面演化的方程都是拟线性的，甚至是完全非线性的。此外，这些方程往往具有非局部性质，因此从数学的角度来看，作为分析的第一步，寻找这些非线性方程的弱解似乎是诱人的。然而，在许多应用中，相应的数学模型是在假设存在尖锐光滑表面或移动边界的情况下获得的。当然，我们可以尝试遵循两步过程，首先构造弱解，然后在第二步中，分析弱解的实际正则性。这两个步骤通常与比较原则密切相关。Rence，对于没有比较或最大值原理的奢侈品的问题，上述程序根本不明显。我们提到的拟定常Stefan问题的表面张力，Mullins-Sekerka模型，或表面扩散流不满足比较原则。对于这些问题，直到最近才建立（甚至弱）解的存在性或（甚至经典）解的唯一性，见[34，35，36，40，20，21，22]。另一方面，有移动边界问题，其中一个可以保证存在的弱解（如释放肖流没有表面张力，斯蒂芬问题与吉布斯-汤姆森修正），但实际的规则性弱解决方案仍然远远没有被理解。有一种不同的方法来解决运动曲面的问题，在这种方法中，人们从一开始就寻求一个唯一的经典解。在这种方法中，
It is the purpose of this paper to give a survey over some recent developments in the theory of classical solutions to elliptic and parabolic problems involving moving surfaces. Problems of this type do not satisfy a superposition principle for solutions and, hence, carry an inherent nonlinear structure. In fact, it turns out that most of the equations describing the evolution of surfaces are of quasilinear or even of fully nonlinear type. Additionally, these equations are often of a nonlocal nature.From a mathematical point of view it therefore seems tempting-as a first step in the analysis-to look for weak solutions to these nonlinear equations. In many applications, however, the corresponding mathematical model is obtained under the assumption of the existence of a sharp and smooth surface or moving boundary. Of course, one can try to follow a two-step procedure of first constructing weak solutions and then, in a second step, analyzing the actual regularity of a weak solution. Both steps are often closely tied to a comparison principle. Rence, for problems without the luxury of a comparison or maximum principle the above program is not at all obvious to realize. We mention the quasi-stationary Stefan problem with surface tension, the Mullins-Sekerka model, or the surface diffusion flow which do not satisfy a comparison principle. For these problems neither existence of (even weak) solutions nor uniqueness of (even classical) solutions was established until quite recently, see [34, 35, 36, 40, 20, 21, 22]. On the other hand, there are moving boundary problems for which one can guarantee existence of weak solutions (eg the Rele-Shaw flow without surface tension, the Stefan problem with Gibbs-Thomson corrections) but for which the actual regularity of weak solutions is still far from being understood. There is a different approach to problems with moving surfaces in which one seeks a unique classical solution from the very beginning. In this approach one
DOI：
10.1007/978-94-009-1926-6
发表时间：
1990-03
期刊：
--
影响因子：
--
作者：
Jacob Bear;Y. Bachmat
通讯作者：
Jacob Bear;Y. Bachmat