On phase transitions and fluid flows

摘要（Simonett, 1265579） PI将继续研究存在表面张力的相变和流体流动中的自由边界问题。特别是，PI计划研究(1)具有可变表面张力和表面粘度的两相粘性流体流动的相变，(2)具有动力学过冷的两相粘性流体流动的相变，以及(3)化学势驱动的相变。要研究的其他主题是(4)表面张力存在下的蒸发，(5)相变中固化过程的稳定性和不稳定性。几个世纪以来，两相流的动力学及其分离界面的运动一直是科学界和工业界感兴趣的问题。在没有相变的情况下，这意味着界面与流动平流，这个问题是相当容易理解的。但是，如果存在相变，则所得到的模型从根本上来说更加复杂，因为它们涉及流体动力学方程和相变方程。预期的结果将一方面澄清解的存在唯一性，另一方面建立这些解的定性和渐近性质。PI提出了一种通用而灵活的数学方法，这将为解决更多问题开辟道路。在过去的几十年里，自由边界问题引起了越来越多的关注，这既是因为它的理论兴趣，也是因为它在自然科学和工程科学中的大量应用。通常，自由边界问题由一个或多个偏微分方程组成，这些偏微分方程必须在先验未知的域中求解，并且必须作为问题的一部分确定。一般来说，自由边界问题无论在解析上还是在数值上都比在规定的区域内求解基本微分方程要困难。它们具有固有的非线性结构，因为两个独立的解不能叠加。因此，解决自由边界问题的各种应用和固有困难引发了各种理论方法，并在许多情况下决定了这些方法的发展方向，这并不奇怪。自由边界问题在许多领域都很重要，如材料科学、流体力学、流体力学、热力学、磁动力学、固态物理、等离子体物理、地质学、化学、生物和医学科学。工业中的许多制造过程都会导致自由边界问题，例如电化学加工、粘性烧结、晶体生长、注塑工艺、蚀刻工艺、凝固工艺、铸造工艺和化学气相沉积工艺等。对工程师和数学家来说，适当的数值和解析处理都是一个重大挑战。所提出的工作将对那些不能忽视表面张力的现实问题和应用产生影响。

Abstract (Simonett, 1265579)The PI will continue studying free boundary problems in phase transitions and fluid flows in the presence of surface tension. In particular, the PI plans to study (1) phase transitions in two-phase viscous fluid flows with variable surface tension and surface viscosity, (2) phase transitions in two-phase viscous fluid flows with kinetic undercooling, and (3) phase transitions driven by chemical potentials. Additional topics to be investigated are (4) evaporation in the presence of surface tension, and (5) stability and instability of solidifying processes in phase transitions. The dynamics of two-phase flows and the motion of their separating interfaces has been a problem of scientific and industrial interest for centuries. In the absence of phase transitions, which means that the interface is advected with the flow, this problem is fairly well understood. However, if phase transitions are present, the resulting models are fundamentally more complex, as they involve the equations of fluid dynamics in conjunction with those of phase transitions. The results anticipated will on the one hand clarify the existence and uniqueness of solutions, and on the other establish qualitative and asymptotic properties of these solutions. The PI proposes a mathematical approach that is general and flexible, and that will open up the treatment of many more problems.Over the last decades the subject of free boundary problems has attracted increasing attention, both because of its theoretical interest and because of its numerous applications in the natural and engineering sciences. Typically, a free boundary problem consists of one or more partial differential equations that have to be solved in a domain that is a priori unknown and that has to be determined as part of the problem. Free boundary problems are in general harder to solve, both analytically and numerically, than the underlying differential equations would be in a prescribed domain. They have an inherent nonlinear structure, as two separate solutions cannot be superposed. It is not surprising, then, that the variety of applications and the inherent difficulties of solving free boundary problems have initiated a variety of theoretical methods, and in many cases have determined the direction of the development of these methods. Free boundary problems are important in many fields, such as material sciences, fluid mechanics, hydrodynamics, thermo-mechanics, magneto-dynamics, solid-state physics, plasma physics, geology, chemistry, and the biological and medical sciences. Many manufacturing processes in industry lead to free boundary problems, such as electrochemical machining, viscous sintering, the growth of crystals, injection molding processes, etching processes, solidification processes, casting processes, and chemical vapor deposition processes, to mention a few. The appropriate numerical and analytical treatment is a major challenge, both to the engineer and to the mathematician. The proposed work will have an impact for those real-world problems and applications where surface tension cannot be neglected.