Conservation Laws with Partial Dissipation in Continuum Mechanics

本项目研究连续介质力学中描述质量、动量、能量等物理量守恒或平衡的规律，这些规律的数学描述是复杂的非线性偏微分方程组。该系统包含一定的条款representingthe耗散机制，如粘度，热传导，摩擦阻尼，松弛，物种扩散等一方面，系统的非线性倾向于产生奇异性（冲击波）的流动。另一方面，耗散机制缓解了这种趋势，但同时产生了更丰富的波型。流的整体行为则取决于两者之间的竞争。问题的难点在于耗散只存在于系统的某些方程（定律）中。例如，物理学规定，质量守恒不应该有任何耗散。这样一个事实，然后进一步复杂的竞争之间的非线性和耗散。在不同的波向中，一个或另一个占主导地位。这样就很容易理解，一般来说，没有一个明确的解决方案的公式。事实上，甚至解决方案的存在也可能是一个悬而未决的问题。在这个项目中，获奖者将研究什么时候一个解决方案可以一直存在。此外，如果一个解一直存在，它的定性行为是什么？为此，获奖者将首先减少系统到一组简化的，解耦的方程。它们中的每一个都代表一个沿特定方向的波，并且可以显式地求解。这些波一起给出了原系统的时间渐近波型。接下来，获奖者将研究渐近解与实际解的近似程度。通过这样的研究，可以获得对潜在物理现象的理解。本研究中所解决的问题可以通过一个例子来说明。当航天飞机返回地球时，它周围的空气温度变得如此之高，以至于空气中分子的内部结构被激发，发生化学反应。然后，空气失去其局部热力学平衡状态。偏离均衡反过来又为内部变化提供了"驱动力"。空气通过分子碰撞向其局部平衡状态弛豫。这种弛豫过程对声波方向有显著影响，但对粒子路径方向无明显影响。获奖者将研究在这种情况下放松过程如何改变气流的整体行为。

This project is to study laws for continuum mechanics, describing the conservationor balance of physical quantities such as mass, momentum, energy, etc. Themathematical description of those laws are complicated nonlinear systems ofpartial differential equations. The systems contain certain terms representingthe dissipative mechanism such as viscosity, heat conduction, frictional damping,relaxation, species diffusion, etc. On one hand, the nonlinearity of the systemstends to generating singularities (shock waves) in the flows. On the other hand,the dissipative mechanism eases such a tendency, but gives rise to richer wavepatterns at the same time. The overall behavior of the flows then depends onthe competition between the two. The difficulty of the problem lies in the factthat the dissipation exists only in certain equations (laws) of a system. Forinstance, physics dictates that the conservation of mass should not have anydissipation. Such a fact then further complicates the competition betweennonlinearity and dissipation. In different wave directions, one or the otherdominates. It is then easy to understand that in general there is no explicitformula for a solution. In fact, even the existence of a solution can be an openquestion. In this project, the awardee will study when a solution canexist all the time. Furthermore, if a solution exists all the time, what isits qualitative behavior? To this aim, the awardee will first reduce the systeminto a set of simplified, decoupled equations. Each of them represents a wavealong a particular direction, and can be solved explicitly. These waves togethergive the time asymptotic wave pattern of the original system. Next, the awardeewill study how well the asymptotic solution approximates the actual solution.Through such a study, the understanding of the underlying physical phenomena canbe obtained.The problems addressed in this research can be illustrated by an example. When a space shuttle returns to the earth, the temperature of the air around it becomes so high that the internal structure of the molecules in the air gets excited, and chemical reactions occur. The air then loses its local thermodynamic equilibrium state. The departure from equilibrium in turn provides the ``driving force" for internal changes. The air relaxes towards its local equilibrium state throughmolecular collisions. Such relaxation processes have significant influence inthe acoustic directions, but not the particle path direction. The awardeewill study how the relaxation processes change the overall behavior ofthe air flow in this situation.