论文标题
通过da prato-Grisvard理论的自由边界问题
Free Boundary Problems via Da Prato-Grisvard Theory
论文作者
论文摘要
$ \ mathrm {l} _1 $ - 最大规律性理论是针对受da Prato和Grisvard开创性工作启发的抛物线进化方程的。除了自身的兴趣外，该方法还产生了一个框架，允许全球时间控制与流体力学相关的各种问题中欧拉（Eulerian）向拉格朗日坐标的变化。对于自由边界问题，此属性当然是决定性的。通过分析自由边界价值问题，描述了粘性，不可压缩的牛顿液的运动，而没有表面张力，其次是可压缩的无压气体运动的运动。 为此，开发了一种最大$ \ mathrm {l} _1 $ - 型号的stokes和lamé系统。然后将其应用于在初始域与半空间重合的情况下，建立上述自由边界问题的全局，强大的结果，并且相对于合适的缩放不变标准而言，初始速度很小。
An $\mathrm{L}_1$-maximal regularity theory for parabolic evolution equations inspired by the pioneering work of Da Prato and Grisvard is developed. Besides of its own interest, the approach yields a framework allowing global-in-time control of the change of Eulerian to Lagrangian coordinates in various problems related to fluid mechanics. This property is of course decisive for free boundary problems. This concept is illustrated by the analysis of the free boundary value problem describing the motion of viscous, incompressible Newtonian fluids without surface tension and, secondly, the motion of compressible pressureless gases. For this purpose, an endpoint maximal $\mathrm{L}_1$-regularity approach to the Stokes and Lamé systems is developed. It is applied then to establish global, strong well-posedness results for the free boundary problems described above in the case where the initial domain coincides with the half-space, and the initial velocity is small with respect to a suitable scaling invariant norm.
