Singular Solutions to Certain Equations in the Physical Sciences

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0071858ZhengIt is proposed to study some nonlinear partial differential equations from fluid dynamics and liquid crystal physics. These equations are the laws of motion of their respective physics. The turbulent nature and/or defects in the materials make the solutions of the equations singular, unstable, and hard to calculate. It is planned to use advanced analytical tools to study the structures of the singular solutions. In the case of the liquid crystal wave equation, for example, the plan is to study it as one of the simplest examples of nonlinear generalizations of the basic linear wave equation. It can be seen from its elegant form that there will be many applications of the equation in the future. Singularities of its solutions have been shown to exist recently by the principal investigator and his collaborators. It is these singularities that block the establishment of a general theory of existence, uniqueness, and stability of solutions. It is planned to do detailed estimates and analysis on the singularities and improve current compactness arguments to form an existence theory of solutions. The result of the investigation will be a clear understanding of the worst possible solutions, and thereby quantify our knowledge of the physics and offer guidance in numerical computations of general solutions.The research will involve the study of some applied mathematical problems in the fields of fluid dynamics (which includes motion of the air and water) and liquid crystal physics in material science. Scientists and engineers have used mathematical equations, called partial differential equations, to model the motions. The turbulent nature and/or defects in the materials show up in the form of singularities and instabilities in the solutions of the equations. It is these singularities and instabilities that often spoil accurate numerical computations of the solutions. It is planned to use the state of the art analytical tools to study the structures of the singular solutions. In the case of a compressible gas such as air, for example, the principal investigator plans to isolate typical singularities (hurricanes, tornadoes, shocks, etc.) and investigate their individual structures. The result of the investigation will be a clear understanding of the worst possible solutions, and thereby quantify our knowledge of the physics and offer guidance in high-performance numerical computations of general solutions.