Mathematical Sciences: On Various Equations of Continuum Mechanics

本工作涉及五个数学理论问题 连续体运动的性质：1.不可压缩流体的运动 自由边界和表面张力。 PI计划表明， 初值问题是适定的。 2.永远存在 的解决方案的运动方程的不可压缩的新， 二维空间中的胡克物质。 3.所有人的存在 不可压缩流体运动方程解的时间 如果初始数据很小，则在三个空间中的材料。 4. 气体方程解的常时存在性 在三个空间中具有小初始数据的动力学。 5.存在 对于所有时间的解决方案的运动方程的完美 三维空间中的流体 这项研究将拓宽我们对这些机制的理解 液体、气体和可变形固体的运动， 可能发生的并发症，如形成冲击， 奇点 连续统形式的动力学研究 许多机械工程领域的理论基础， 空气动力学

This work involves five problems on the mathematical theory of motion of continua: 1. Motion of an incompressible fluid with free boundary and surface tension. The PI plans to show that the initial value problem is well-posed. 2. Existence for all time of solutions to the equations of motion of an incompressible Neo- Hookean material in two space dimensions. 3. Existence for all time of solutions of the equations of motion of an incompressible material in three space if the initial data are small. 4. Existence for all time of solutions to the equations of gas dynamics with small initial data in three space. 5. Existence for all time of solutions to the equations of motion of a perfect fluid in three-space. This study will broaden our understanding of the mechanisms of motion of liquids, gases, and deformable solids and complication that may occur such as formation of shocks and singularities. The dynamical study of the continua forms to theoretical basis for many areas of mechanical engineering and aerodynamics.