Mathematical Sciences: On Various Equations of Continuum Mechanics

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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.