Study on symmetric positive systems

对称正系统研究

• 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

(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)我们研究了有界开集上的对称正系统，假设边界矩阵满足下列条件。也就是说，在边界中余维为1的嵌入子流形之外，边界矩阵的秩是常数。在这种情况下，我们找到了一个合适的权函数，这是积极的上述子流形外，使一个先验估计的边值问题的解决方案。利用这个先验估计，我们证明了边值问题解的存在性，该边值问题关于法向是正则的。这对于非线性摄动的应用是非常重要的。借助于这一先验估计和光滑解的存在性，我们成功地得到了弱解在参考嵌入子流形附近的性质，并通过几个例子证明了这一性质是非常尖锐的.这些结果对于二维区域是比较令人满意的。如果把这个结果应用于所谓的Triconi方程，我们得到了解的唯一性的另一个证明。至于三维域，还有一个基本的情况，我们无法处理。(ii)在边界矩阵秩变化的子流形上，当边界矩阵为零时，我们阐明了边值问题的结构。在沿着子流形的爆破流形上，我们可以用一个简单的权函数得到一个先验估计。利用这一先验估计，在爆破空间中，我们证明了光滑解的存在性，甚至关于正常方向。应用我们在（i）中使用的相同方法，我们能够检查子流形附近弱解的行为。应用这一结果，我们得到了线性化MHD方程在某些边界条件下的先验估计。

项目成果

T.Nishitani,M.Takayama: "Characteristic initial boundary value problems for symmetric hyperbolic systems" Osaka J.Math.35・3. 629-657 (1998)

T.Nishitani，M.Takayama：“对称双曲系统的特征初始边值问题”Osaka J.Math.35・3（1998）。

M.Sugimoto: "Estimates for hyperbolic equations of space dimension 3" J.Func.Anal.160・2. 382-407 (1998)

M.Sugimoto：“空间维度 3 的双曲方程的估计”J.Func.Anal.160・2（1998）。

T.Mandai: "A construction of asymptotic solutions and the existence of smooth null-solutions for a class of non-Fuchsian partial differeutial operators" Nagoya Math.J.145・1. 125-142 (1997)

T.Mandai：“一类非 Fuchsian 偏微分算子的渐近解的构造和平滑零解的存在”Nagoya Math.J.145・1（1997）。

K.Kajitani: "The Cauchy problem for Schrodinger type equations with variable coefficients" J.Math.Soc.Japan. 50・1. 179-202 (1997)

K.Kajitani：“具有变系数的薛定谔型方程的柯西问题”J.Math.Soc.Japan 50・1（1997）。

K.Kajitani,M.Mikami: "The Cauchy problem for degenerate parabolic equations in Gevrey class" Ann.Scuola Norm.Sup.Pisa. 26・2. 383-406 (1998)

K.Kajitani，M.Mikami：“Gevrey 类简并抛物线方程的柯西问题”Ann.Scuola Norm.Sup.Pisa 26・2 (1998)。