Study on symmetric positive systems

对称正系统研究

基本信息

批准号：
09440059
负责人：
NISHITANI Tatsuo
金额：
$ 2.56万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440059/
关键词：
symmetric system boundary matrix weight function a priori estimate blow up

项目摘要

(i) We have studied symmetric positive systems on a bounded open set assuming the following conditions on the boundary matrix. That is, outside a embedded submanifold of codimension 1 in the boundary, the rank of the boundary matrix is constant. In this case, we found a suitable weight function which is positive outside the above mentioned submanifold so that an a priori estimate for solutions to the boundary value problem is obtained. Using this a priori estimate, we have proved the existence of solution to the boundary value problem which is regular with respect to the normal direction. This is very important to applications to non-linear perturbations. With the aid of this a priori estimate and the existence of smooth solutions, we succeeded to get the behavior of weak solutions near the reference embedded submanifold, which is very sharp as several examples show. These results, as far as concerning two dimentional domains, are fairly satisfactory. If we apply this result to so called Triconi's equation, we get another proof of the uniqueness of solution. As for 3 dimentional domains, there remains one fundamental case which we cound not treat.(ii) In the case that the boundary matrix is zero on the submanifold where the rank of the boundary matrix changes, we clarified the structure of the boundary value problem. On the blown up manifold along the submanifold, we can get an a priori estimate with a simple weight function. Using this a priori estimate in the blown up space, we proved the existence of smooth solution even with respect to the normal direction. Applying the same method that we employed in (i), we are able to examine the behavior of weak solutions near the submanifold. Applying this result we obtained a priori estimate for the linealized MHD equation under some boundary condition which has not been treated before.

（i）我们在边界矩阵上的以下条件上研究了对称的阳性系统。也就是说，在边界中的编码1的嵌入式子手术中，边界矩阵的等级是恒定的。在这种情况下，我们发现了一个合适的权重函数，该功能在上述子手术中之外为正，因此可以获得对边界价值问题解决方案的先验估计。使用此先验估计，我们已经证明了边界值问题的解决方案，这相对于正常方向是规律的。这对于非线性扰动的应用非常重要。借助此，先验估计值和平滑解决方案的存在，我们成功地获得了参考嵌入式子曼物的弱解决方案的行为，正如几个示例所示，这非常敏锐。就二维领域而言，这些结果相当令人满意。如果将此结果应用于所谓的Triconi方程，我们将获得解决方案独特性的另一个证明。至于3个尺寸域，我们仍然存在一个基本情况，我们不对待。（ii）在边界矩阵上边界矩阵的情况下，边界矩阵的等级变化为零，我们阐明了边界值问题的结构。在沿着子手法的吹动的歧管上，我们可以通过简单的重量函数获得先验估计。在吹动空间中，使用此事项是先验估计，即使相对于正常方向，我们也证明了平滑解决方案的存在。应用与我们在（i）中使用的相同方法，我们能够检查子序列附近弱解决方案的行为。应用此结果，我们在某种边界条件下获得了直率的MHD方程的先验估计值，而以前尚未处理过。