自由边界中若干数学问题的研究

This program is to study the fluid dynamic equations with free boundary. We plan in this program to study two problems: the first is the well-posedness of beach problem, and the second is zero-viscosity limit of the two-fluid system..In the first part, beach problem studies the model whoes free surface intersects with the oblique flat beach. The domain is non-smooth. By now, the well-posedness of the water wave in smooth domain have a lot of results, such as irrotational and rotational, finite depth and infinite depth. But, there are few results on beach problem. Meantime, the non-smooth case is of great physical meaning and has a lot of application. It is the important part of the study of water-wave equations. Here, we will focus on the 2D corner case which has some special characteristic. These characteristic will help us to get the well-posedness of the Darcy's flow and the water-wave equations..Second, we will study the inviscid limit of the two-fluid problem. When the viscosity coefficient equals to zero, the model becomes vortex sheet. Here, we find out that there will appear the boundary layer when the viscosity coefficient goes to zero. As we known, we only can deal with the strong boundary layer under the analytic or Gevrey setting. On one hand, we have used the energy method to get the limit in the half plan under analytic setting. We plan to apply the energy method to the two-fluid system to get the zero-viscosity limit of the two-fluid system.

本项目研究方向是具有自由边界的流体力学方程组。研究课题是浅滩问题的适定性以及两相流的粘性极限。.浅滩问题是指自由面与固定边界相接触的水波方程。此时区域为非光滑的。目前，自由边界问题在光滑区域上的适定性有着丰富的结果，如有旋、无旋，无限深度、有限深度等情况。但是浅滩问题的结果却很稀少。而非光滑区域又是最贴近物理现象，具有很强物理背景和应用价值。我们将关注二维的角型区域，利用角型区域的特殊结构去得到Darcy流和水波方程的适定性。.本项目另一个研究课题是两相流的粘性极限。当粘性系数等于零时，此时方程就变为两相流的水波方程，即涡流层。由于边界条件的不同，此极限过程中会产生边界层。对强边界层情况，目前只能在解析类或者Gevrey类的框架下去证明极限。申请人已通过能量方法得到了在解析框架下的粘性极限的结果。本项目将结合自由边界的特性证明解析框架下的两相流粘性极限。

本项目研究主要内容是几类具有不同边界值的流体力学方程组。.本项目一研究课题具有自由边界的流体力学方程组的适定性。我们给出了浅滩问题的先验估计。浅滩问题是指自由面与固定边界相接触的水波方程。此时区域为非光滑的。目前，自由边界问题在光滑区域上的适定性有着丰富的结果，如有旋、无旋，无限深度、有限深度等情况。但是浅滩问题的结果却很稀少。而非光滑区域又是最贴近物理现象，具有很强物理背景和应用价值。除此之外，我们还得到了具有真空的可压缩Navier-Stokes方程的适定性。正是由于真空的存在，该问题的适定性需要处理由真空所带来的奇性问题。.本项目另一个研究课题是无粘极限。目前，在强边界层下，不可压缩流的无粘极限的结果都是上半平面上。申请人已通过能量方法得到了在解析框架下在光滑有界区域上的无粘极限。.除此之外，本项目还证明了全空间上，可压缩流的长时间行为。