Conservation Laws with Partial Dissipation in Continuum Mechanics

提案#0207154PI: Yanni ZengInstitution: University of Alabama birmingham标题：在连续介质力学中具有部分耗散的守恒定律本项目涉及守恒定律（平衡定律）的系统，这是连续介质力学中产生的一类广义的非线性偏微分方程（PDEs）。特别感兴趣的是具有部分耗散的方程，包括双曲-抛物型系统，具有消退记忆的双曲型系统，以及具有松弛的双曲型系统。本研究的目的是通过研究偏微分方程解的定性行为来更好地理解其潜在的物理现象。第一个目标是获得具有松弛的双曲型系统的激波剖面的非线性稳定性，包括完全耗散系统和在线性退化场中包含一个非衰减波的系统（复合型系统）。第二个目标是获得双曲抛物型系统初始边值问题的类似稳定性结果，特别是了解边界的影响。第三个目标是将研究扩展到接触不连续面和稀疏波。在该奖项下进行的研究与可压缩粘性流动、粘弹性材料运动、热非平衡气体动力学和磁流体动力学有关。通过包括粘度、热传导、电阻率、分子的激发内部结构、化学反应等等，我们专注于比传统的理想模型更接近现实世界的数学模型。该项目有三个具体目标。第一个目标是了解激波剖面的稳定性，例如，当分子的内部结构和化学反应不再是可以忽略不计的高温气体流动时。例如，在航天飞机重返大气层时就会出现这个问题，这与传统的飞机超音速飞行有很大不同。第二个目标是更好地理解固体表面附近的气体流动。第三个目标是在几种情况下研究更完整的波型。

Proposal #0207154PI: Yanni ZengInstitution: University of Alabama BirminghamTitle: Conservation laws with partial dissipation in continuum mechanicsThis project is concerned with systems of conservation laws (balance laws), a broad class of nonlinear partial differential equations (PDEs) arising from continuum mechanics. Of special interest are equations with partial dissipation, including hyperbolic-parabolic systems, hyperbolic systems with fading memory, and hyperbolic systems with relaxation. The purpose of the research is to obtain a better understanding of the underlying physical phenomena through the study of the qualitative behavior of the solutions of the PDEs. The first objective is to obtain the nonlinear stability of shock profiles for hyperbolic systems with relaxation, including fully dissipative systems and those containing one non-decaying wave in a linearly degenerate field (systems of composite type). The second objective is to obtain a similar stability result for initial boundary-value problems for hyperbolic-parabolic systems, especially to understand the effect of the boundary. The third objective is to extend the study to contact discontinuities and rarefaction waves. The research conducted under this award is relevant for the study of compressible viscous flows, the motion of viscoelastic materials, gas dynamics in thermal nonequilibrium, and magnetohydrodynamics. By including viscosity, heat conduction, electrical resistivity, excited internal structures of molecules, chemical reactions, and so on we focus on mathematical models that are closer to the real world than the traditional ideal ones. The project has three specific objectives. The first objective is to understand the stability properties of shock profiles in, for example, high-temperature gas flows when the internal structure of molecules and chemical reactions are no longer negligible. This problem arises, for example, at the reentry of a space shuttle into the atmosphere and is quite different from the traditional supersonic flight of an airplane. The second objective is to better understand gas flows near a solid surface. The third objective is to study more complete wave patterns in several cases.