Hyperbolic Conservation Laws in Continuum Physics

提案#0202888PI：康斯坦丁·M·达费尔莫斯研究所：布朗大学题目：连续介质物理中的双曲守恒定律摘要拟议的研究计划位于连续介质物理和双曲守恒律理论之间的界面上。它涉及以下项目，这是由于Bianchini和Bressan最近在消失粘性方法上的突破：(A)通过消失粘性方法研究具有耗散源项的平衡律；(B)尝试将分层方法从标量守恒律扩展到它的系统；以及(C)借助Bianchini和Bressan的新估计来研究经典的Riemann问题。双曲型守恒律属于一类非线性偏微分方程组，它控制具有非线性弹性响应的材料的动力学，包括橡胶固体和空气等气体。这些方程的特点是，非线性导致了不连续性的发展，类似于常见的海滩上波浪破碎的现象。然后，不连续性以冲击波的形式传播。建议的研究方案是研究一些构造具有冲击的解的方法。其目的是通过一种方法来解决理论性质的问题，例如解的存在问题，通过一种方法来揭示数学和物理之间的密切联系，同时建议用于实际计算这种解的算法。

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.