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 models which 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 this fundamental 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.
