Hyperbolic Conservation Laws with Application to Blood Flow Problems

该提案的重点是对双曲守恒定律方程的数学研究。它们描述了基本的物理原理，如质量、动量和能量守恒，并在许多应用科学中出现，如生物医学（例如，血液流动的研究）和航空航天工程（例如，研究航天飞机机翼周围超音速流动的稳定性）。本提案的主要目标是了解(1)血液流经柔顺动脉的守恒定律的解的结构，以及(2)在多个空间维度上的超音速和跨音速流动问题。(1)拟人对柔顺动脉血流建模的一维双曲守恒定律的研究，源于使用“支架”置入的假体对腹动脉瘤进行血管内修复后经腹主动脉血流的研究。提议者已经开始研究腹主动脉和支架在脉动血流引起的压力下的动力学。计划是研究两个问题，这两个问题对于理解血流和血管壁之间的流体结构相互作用至关重要：(a)从数学上严格推导出新的改进的一维模型，该模型将考虑血管壁和支架的纵向和径向位移，以及(b)对具有不连续系数的简化模型进行数学分析，以考虑主动脉和假体具有不同的弹性特性。(2)超声速和跨声速多空间流动问题的研究是由于跨声速多空间流动问题的理论尚不发达，尽管其应用具有重要意义。这主要是由于在解中出现了新的奇点，而这些奇点不能用一维技术来分析。本提案的主要目标是通过使用和发展描述跨音速冲击的自由边界问题的技术，以及通过使用和发展亚音速部分产生的混合（双曲-椭圆）系统的分析，研究二维双曲守恒律中自相似解的奇异性和全局结构。该提案的大部分内容是由提案人通过与心脏病专家Z. Krajcer博士（德克萨斯心脏研究所），分子生物学家Doreen Rosenstrauch博士（德克萨斯心脏研究所）和材料科学家K. Ravi-Chandar博士（德克萨斯大学奥斯汀分校）的多学科合作发起的血管假体（支架）工作所推动的。主要目标是推进这一基础数学领域的知识，同时指导与生物医学研究和跨音速流相关的应用研究。这项研究的结果不仅在发展新的数学理论方面具有重要意义，而且在设计可靠的数值方案用于人体动脉血流的高性能计算和某些心血管干预的血管假体的优化设计方面也具有重要意义。

The focus of the proposal is on mathematical study of the equations known as hyperbolic conservation laws. They describe the basic physical principles such as conservation of mass, momentum and energy, and as such arise in many applied sciences such as biomedicine (e.g., study of blood flow) and aerospace engineering (e.g., study of stability of supersonic flow around a wing of a space shuttle). The main goals of this proposal are to understand the structure of solutions of conservation laws that model (1) blood flow through compliant arteries, and (2) supersonic and transonic flow problems in more than one space dimension.(1) The proposer's research on one-dimensional hyperbolic conservation laws modeling blood flow in compliant arteries stems from the study of blood flow through the abdominal aorta after endovascular repair of abdominal aneurysm using inserted prostheses called ``stents''. The proposer has begun an investigation of the dynamics of the abdominal aorta and of the stents, subject to the pressure induced by the pulsatile blood flow. The plan is to work on two problems which are essential in understanding fluid-structure interaction between blood flow and vessel wall: (a) mathematically rigorous derivation of the new, improved one-dimensional models that would account for both the longitudinal and the radial displacements of the vessel wall and and of the stent, and (b) mathematical analysis of the reduced modelswhich have discontinuous coefficients to account for the fact that the aorta and the prostheses have different elastic properties.(2) The research on supersonic and transonic flow problems in more than one space dimension has been motivated by the fact that in spite of the importance of the applications, the theory for transonic flow problems in more than one space dimension is still underdeveloped. This is primarily due to the fact that new singularities appear in the solutions which cannot be analyzed by one- dimensional techniques. The main objective in this proposal is to study singularities and global structure of self-similar solutions in two-dimensional hyperbolic conservation laws by using and by developing the techniques for free-boundary problems that describe transonic shocks, and by using and developing the analysis of mixed (hyperbolic-elliptic) systems arising in the subsonic part of the flow.Much of this proposal has been motivated by the work on vascular prostheses (stents) initiated by the proposer through a multi-disciplinary collaboration with cardiologist Dr. Z. Krajcer (Texas Heart Institute), molecular biologist Dr. Doreen Rosenstrauch (Texas Heart Institute) and material scientist Dr. K. Ravi-Chandar (UT-Austin). The main goal is to advance knowledge of thisfundamental area of mathematics while at the same time guide research in applications related to biomedical research and to transonic flow. The results of this research will be important not only in the development of new mathematical theories, but also in the design of reliable numerical schemes for high-performance computing of blood flow in human arteries and in the optimal design of vascular prostheses for certain cardiovascular interventions.