耗散性双曲平衡律方程组弱解整体存在性应用数学专业论文.docxVIP

南京航空航天大学硕士毕业论文 耗散性双曲平衡律方程组弱解整体存在性 i 摘 要 本文研究带耗散项的双曲守恒律方程组弱解的整体存在性. 对于此类问题，Dafermos 和肖玲 [1]已经有过相关的研究. 在他们的文章中，解整体存在的条件是非齐次项线性部分对应的矩阵具 有严格对角占优性质. 本文的结果与他们的相比，减弱了弱解存在的条件，要求非齐次项线性部 分的特征值实部全大于零. 事实上可以证明，对角占优矩阵的特征值实部必全大于零. 我们运用 改进的 Glimm 格式构造方程组的近似解，以 Riemann 问题的解为框架，在 BV 空间得到近似解 序列的紧性. 证明的关键在于两个方面，一方面是我们对波的相互作用做了更精确的估计；另一 方面就是我们对于波的强度给出了重新定义，从而去掉了严格对角占优条件，克服了局部相互作 用估计过程中遇到的困难. 由于波强度的重新定义，利用波的局部相互作用估计证明 Glimm 泛函单调性时，要对 Glimm 泛函做相应的调整. 对于所构造的近似解的收敛性，和传统的证明方法一样，我们首先证明近似 解序列全变差及其本身的有界，然后利用 Helly 定理证明近似解序列在 BV 空间存在一个收敛的 子序列，最后证明子序列的极限即为方程组的弱解. 关键词：双曲型守恒律方程组，耗散项，Glimm 格式，基本波强度，局部相互作用估计，Glimm 泛函单调性.  Abstract In this paper, we study the global existence of weak solutions to hyperbolic system of balance laws with dissipation. It has been studied by Dafermos and Hsiao[1]. In their paper, by virture of the strength of element wave, the main condition to guarantee the existence of the solution is that the matrix of the linear part of the dissipative terms is a diagonally dominant matrix. Compared with the result in Dafermos and Hsiao[1], we weakened the conditions and only required that the real parts of eigenvalues of the linear part of the dissipative terms (which is an n × n constant matrix) are all positive. In fact, for any matrix which is diagonally dominant, the real parts of it’s eigenvalues are all positive. We used a modi?ed Glimm scheme and give the global existence result in BV space. The key points of the proof this lies in two aspects. First, we get a more precise local interaction estimate on the interaction of the elementary waves. Second, to get the monotonicity of the Glimm functional, we use a new way to de?ne the strength of elementary waves, which can be used to modify the riquriement of on the dissipative terms. Because of the new definition of the strength of the elementary waves, we have to adjust the Glimm functional in the proof of its monotonicity. For the convergence of the sequence of the approximate solution, we first prove that the total variation o