Free Boundary Problems

我们计划研究各种自由边界问题。第一个是关于生长的聚合物晶体。未知数是基质和生长表面的温度。自由边界的速度是温度的给定函数，自由边界点沿这样的方向移动，以使旅行时间最小。我们希望确定解的存在性并确定自由边界的形状。第二个问题涉及非牛顿喷流。我们想通过线性化情况的非线性扰动来逼近它，这是我们最近研究的一种情况。下一个问题是发展自由边界问题的一般分支理论。在这里，我们希望以我们最近所做的处理数学生物学中出现的特殊模型的工作为指导。为了确定分支的稳定性，我们首先研究自由边界问题解的渐近行为，包括Hele-Shaw问题和粘性液滴的演化。随后，我们将考虑数学生物学中出现的更复杂的问题，例如肿瘤和原细胞的进化。最后，我们将研究弹性介质中裂纹的演化。我们希望利用我们推导出的公式来描述应力强度因子的演化。自由边界问题处理的是在区域中求解偏微分方程组，其部分边界事先是未知的；这部分边界被称为自由边界。除了通常规定的初始条件和边界条件外，还在自由边界上附加了一个条件，并试图确定自由边界和微分方程组的解。施加在自由边界上的条件看起来很小的变化，从技术上讲往往会导致一个完全不同的问题。在这一领域，一些特殊的例子历来指导着研究；一些最常见的例子是与空气接触的液体流动、飞机后面的气流、钢的凝固和固体的熔化。由物理模型激发的特殊例子仍然是该领域发展的推动力。目前的建议集中在几个不同的问题上，如聚合物的结晶，非牛顿流体的射流(如喷墨)，生物学中出现的边界问题的分叉，非线性稳定性，以及弹性介质中裂纹的扩展。在所有这些问题中，目的是证明存在一个科学问题的数学解决方案，并确定自由边界的性质。

We plan to study a variety of free boundary problems. The first one dealswith growing polymeric crystallines. The unknowns are temperature in themelt and the growing surface. The speed of the free boundary is a givenfunction of the temperature, and free boundary points move in such adirection so as to minimize the travel time. We wish to establish theexistence of the solution and to determine the shape of the free boundary.The second problem deals with non-Newtonian jets. We want to approach itvia nonlinear perturbation of the linearized case, a case we have recentlystudied. The next problem is to develop a general bifurcation theory forfree boundary problems. Here we expect to be guided by recent work thatwe have done dealing with special models that arise in mathematicalbiology. In order to determine the stability of the bifurcation, we shallfirst study the asymptotic behavior of solutions of free boundaryproblems, including the Hele-Shaw problem and the evolution of viscousdrops. Subsequently we shall consider the more complicated problems thatarise in mathematical biology, such as the evolution of tumors and ofprotocells. Finally, we shall study the evolution of cracks in elasticmedia. We expect to utilize formulas that we have derived for theevolution of the stress intensity factors.Free boundary problems deal with solving partial differential equations ina domain, a part of whose boundary is unknown in advance; that portion ofthe boundary is called a free boundary. In addition to the usuallyprescribed initial and boundary conditions, an additional condition isimposed at the free boundary, and one seeks to determine both the freeboundary and the solution to the differential equations. A seeminglysmall change in the conditions imposed at the free boundary often resultin, technically, an entirely different problem. Special examples havehistorically guided research in this field; some of the most commonlyknown examples are flow of liquid in contact with air, air streams behindan aircraft, solidification of steel, and melting of solid. Specialexamples motivated by physical models continue to be a driving force inthe development of the field. The present proposal focuses on severaldifferent problems dealing with questions such as crystallization ofpolymers, jet flows for non-Newtonian fluid (e.g. inkjets), bifurcation offree boundary problems arising in biology, nonlinear stability, andpropagation of cracks in elastic media. In all these problems, the goalis to prove that there is a mathematical solution to the scientificproblem and to determine properties of the free boundary.