Mathematical Sciences: Geometric Properties of Solutions of Partial Differential Equations

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Abstract This project concerns certain geometric properties of solutions of partial differential equations. Of particular interest are a study of the size of level sets, the maximal order of vanishing, and growth properties of solutions of various types of partial differential equations. The proposed method is a use of elliptic regularity theory to find simplifications and extensions of the results of H. Donnelly, C. Fefferman, and F.-H. Lin concerning geometric properties of solutions of second order partial differential equations. The PI's method, based on the theorem on elliptic iterates of Lions and Magenes, leads to extensions to a wide class of linear and nonlinear equations of elliptic and parabolic type with analytic coefficients. The PI proposes to address some questions which still remain unresolved, including, in particular, sharp bounds on the size of level sets and the maximal order of vanishing for nonlinear evolution equations and equations with non-analytic coefficients. Nonlinear evolution equations, such as the Navier-Stokes system, arise in in connection with numerous problems of mathematical physics. Numerical and experimental models support the theory that such equations produce chaotic behavior of solutions, i.e., solutions which exhibit complex temporal and spatial oscillations. This project addresses this issues and aims to obtain such intimate information of solutions as their oscillatory behavior and their growth properties. Information on these issues, which is the main purpose of this project, would lead to better understanding of oscillatory phenomena arising in many problems of fluid mechanics and related fields.