二维Glimm型格式与高维守恒律方程解的爆破及奇性结构的研究

发布时间：2018-04-17 12:49

  本文选题：高维单守恒律方程 + 二维格式　； 参考：《中国科学院研究生院(武汉物理与数学研究所)》2016年博士论文

【摘要】：本文我们主要研究了二维非线性双曲守恒律方程的Cauchy解的相关问题。第二章首先介绍了二维单守恒律方程的概念和相关结论,然后给出了二维T-C变差和二维有界变差空间的概念和相关结论。第三章我们考虑了具有紧支集初值的二维单守恒律方程的Cauchy问题,用Riemann解的结构构造了一个二维格式,并最终证明了该格式的极限为熵弱解,其过程分为以下五个步骤：在第二节中,我们时间以步长At进行分层,在每层开始时,对初值重新定义,使得其变为四片常值的Riemann问题,然后用Riemann问题解来代表一个时间步长At内的解,从而构造出二维格式。在第三节中,我们估计了该二维格式关于空间变量x,y的二维T-C变差,利用熵条件以及T-C变差的性质证明了该二维格式的T-C变差是一致有界的。在第四节中,我们考虑该二维格式关于时间t的一致连续性,我们分别讨论了在一个时间步长内和跨越多个时间步长这两种情况,并得到了一致的估计式。在第五节中,利用第三、四节的结论,我们证明了该格式在R2×R+中几乎处处收敛的意义下趋近于某个极限函数u(x,y,t)。在第六节中,我们证明了该极限函数u(x,y,t)是满足方程的熵条件的。由于我们在每个时间步长都做了一次小扰动,我们需要证明这些小扰动的控制函数趋近于O。第四章我们把该二维格式运用到求解一类无界初值uo(x,y)∈Lloc∞(R2)的Cauchy问题,这里uo(x, y)局部变差有界且满足其中r是极坐标的半径。无界初值和有界初值有着本质的区别,不过我们还是证明了某种条件下的二维Cauchy问题熵弱解的存在唯一性。第五章我们研究了n维非齐次守恒律方程的Cauchy问题及奇性解的结构。第一节介绍了相关概念及前人的结果。第二节研究光滑Cauchy初值的光滑解产生爆破的充分必要条件和爆破时间,并给出光滑解全局存在的充分必要条件。第三节我们计算了两个二维非齐次Riemann解的全局结构及其演化。
[Abstract]:In this paper, we mainly study the Cauchy solution of two-dimensional nonlinear hyperbolic conservation law equation.In the second chapter, the concepts and relevant conclusions of two-dimensional simple conservation law equations are introduced, and then the concepts and conclusions of two-dimensional T-C variation and two-dimensional bounded variation spaces are given.In chapter 3, we consider the Cauchy problem of two-dimensional conservation law equation with the initial value of compact set, construct a two-dimensional scheme by using the structure of Riemann solution, and finally prove that the limit of the scheme is entropy weak solution.The process is divided into the following five steps: in the second section, we delaminate with step size at the beginning of each layer, redefine the initial value so that it becomes a four-piece constant Riemann problem.Then the solution of the Riemann problem is used to represent the solution in a time step at, and a two-dimensional scheme is constructed.In the third section, we estimate the two-dimensional T-C variation of the two-dimensional scheme with respect to the spatial variable xy. By using the entropy condition and the properties of the T-C variation, we prove that the T-C variation of the two-dimensional scheme is uniformly bounded.In the fourth section, we consider the uniform continuity of the two-dimensional scheme with respect to time t. We discuss the two cases of a time step size and a span of multiple time steps respectively, and obtain a consistent estimation formula.In the fifth section, by using the conclusions of the third and fourth sections, we prove that the scheme converges to a certain limit function UX ~ XY ~ (t) in the sense of almost everywhere convergence in R ~ 2 脳 R.In the sixth section, we prove that the limit function u _ XY _ t) satisfies the entropy condition of the equation.Since we make a small perturbation at each time step, we need to prove that the control function of these small perturbations approaches O.In chapter 4, we apply this two-dimensional scheme to solve the Cauchy problem for a class of unbounded initial value UOX (y) 鈭，

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