Hyperbolic Conservation Laws in Continuum Physics

提案#0202888PI：Constantine M. Dafermos机构：布朗大学名称：连续介质物理中的双曲守恒律摘要本研究计划是在连续介质物理和双曲守恒律系统理论之间的接口上进行的。 由于Bianchini和Bressan最近在消失粘性方法上的突破，它包括以下几个项目：（a）用消失粘性方法研究含耗散源项的平衡律：（B）尝试将分层方法从标量守恒律推广到标量守恒律系统;（c）借助Bianchini和Bressan的新估计研究经典Riemann问题。双曲守恒律属于一类非线性偏微分方程，它支配具有非线性弹性响应的材料的动力学，包括橡胶等固体和空气等气体。 这些方程的特点是非线性引起了不连续性的发展，类似于人们熟悉的海滩上波浪破碎的现象。 然后，不连续性作为冲击波传播。 建议的研究计划是调查一些方法来构造的冲击的解决方案。 其目的是解决理论性质的问题，例如解的存在性问题，通过一种方法，揭示数学和物理之间的密切联系，同时提出实际计算这些解的算法。

Proposal #0202888PI: Constantine M. DafermosInstitution: Brown UniversityTitle: Hyperbolic Conservation Laws in Continuum PhysicsABSTRACTThe proposed research program lies on the interface between continuum physics and the theory of hyperbolic systems of conservation laws. It involves the following projects, which are motivated by the recent breakthrough of Bianchini and Bressan on the vanishing-viscosity method: (a) The study of balance laws with dissipative source terms through the vanishing-viscosity method; (b) An attempt to extend the layering method from scalar conservation laws to systems thereof; and (c) The study of the classical Riemann problem with the help of the new estimates of Bianchini and Bressan.Hyperbolic conservation laws belong to a class of nonlinear partial differential equations that govern the dynamics of materials with nonlinear elastic response, including solids like rubber and gases like air. The special feature of these equations is that the nonlinearity induces the development of discontinuities, akin to the familiar phenomenon of the breaking of waves on the beach. The discontinuities then propagate as shock waves. The proposed research program is to investigate a number of methods for constructing solutions with shocks. The objective is to settle questions of a theoretical nature, such as the issue of existence of solutions, by an approach that brings out the intimate connection between mathematics and physics, while suggesting, at the same time, algorithms for the actual computation of such solutions.