Nonlinear PDE in Spaces of Analytic Functions

Information about the domain of analyticity of a solution of a partial differential equation (PDE) can be used to gain insights into underlying physical processes. In this project, we study the equations of magneto-hydrodynamics (MHD) and several other nonlinear models originating in the study of wave motion in fluid mechanics. The MHD equations are of fundamental importance in the study of nuclear fusion. In particular, one of the most important issues in the quest for understanding plasma dynamics is understanding MHD turbulence-driven anomalous transport. A better understanding of the geometry of the solutions is a key issue in constructing more effective fusion devices, including ITER and post-ITER generations.

The proposed research is aimed at extending the use of function-theoretic methods in the study of nonlinear partial differential equations with a view towards applications to important problems in mathematical physics. Information about the domain of anal yticity of a solution of a PDE can be used to gain a quantitative understanding of the structure of the equation, and to obtain insight into underlying physical processes. The study of real-analytic solutions to nonlinear PDE has developed over the last t wo decades, and analytic function spaces have become popular tools for the study of a variety of questions connected with nonlinear evolutionary PDE. In particular, the use of Gevrey-type spaces has given rise to a number of important results in the study of long time dynamics of dissipative equation, such as estimating the asymptotic degrees of freedom, and approximating the global attractors. Here, we want to expand the range of these techniques and applications in the context of dissipative and dispers ive equations. The equations under study include the three-dimensional magneto-hydrodynamics system, and the nonlinear Maxwell-Dirac system. In view will also be nonlocal model equations for fluid flow, such as the Benjamin-Ono equation. In the context of magnetohydrodynamics, the proposed research will focus on the study of possible singularity formation, and scenarios to prevent such a collapse. The Maxwell-Dirac system appears in the study of quantum electrodynamics, and has recently been studied in the context of low-regularity solutions. However, results about the domain of analyticity of spatially-analytic solutions appear to be lacking. Regarding nonlocal evolution equations, our goal will be to obtain a theory of solutions in classes of analytic functions which parallels the existing theory for third-and higher order differential equations. We will also exploit analyticity of solutions in the study of the convergence of numerical approximation schemes.