Partial Differential Equations

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DMS-9970857ABSTRACTOur mathematical research centers around the followingproblems in partial differential equations of boththeoretical and practical interest.The first part is concerned with the regularity of solutionsfor the sum of squares operator, or sublaplacian.We will investigate necessary andsufficient conditions for the local and global analytic regularity of thesublaplacian when the bracket condition is satisfied.Moreover, in the absence of the bracket condition we will studythe weaker properties of global analytic and smooth regularityof the sublaplacian on the torus.Also we will consider regularity problems for CR structures,and principal type operators.The second part is concerned with the studyof the Cauchy problem for nonlinear partial differential equationsof shallow water type under low regularity initial datausing harmonic analysis techniques. In particular weplan to investigate the local and global well-posedness of theinitial value problem for the completely integrableCamassa-Holm equation.And finally, in the third part we plan to show that the exponential map ofan appropriate metric on the group of volume--preserving diffeomorphisms of acompact riemannian manifold is a nonlinear Fredholm map ofindex zero. This would be significant because, as is well known,geodesics of this metric correspond to solutions of the Eulerequations of hydrodynamics.Partial differential equations is a many-faceted subject.Our understanding of the fundamental processes of the natural worldis based to a large extent on partial differential equations.For example, the equations considered in the first part of ourproposal arise in the diffusion of chemicals, in the spread of heat,and in many other physical processes influenced by diffusion.Moreover diffusion processes are closely connected torandom (Brownian) motions. The synthesis of the theory of thesediffusion partial differential equations and probability provides the theoretical framework for studying problemswhere diffusion and randomness are present. An example outsidemathematics and physics is the field of Finance, where powerful techniques fromstochastic analysis have been brought to bear on almost all aspects ofmathematical finance: pricing of financial derivative products suchas options and bonds, hedging, interest rates and so on.The equations in the second and third parts arise from problems inhydrodynamics. They are mathematical models describingfluid flow. The study of these equations will contributein our understanding of wave formation, the structure of theirsingularities, andtheir long time behavior. It may also contribute in the verybig problem of understanding turbulence.At the same time the theory for studying these partial differentialequations problems forms a vast subject that interacts with many otherbranchesof mathematics, such as complex analysis, differential geometry, harmonicanalysis, probability, and mathematical physics.