Mathematical Sciences: Non-Linear Evolution Equations

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8902704 Ponce Three primary themes will be emphasized in this mathematical project, combining elements of partial differential equations, harmonic analysis and mathematical physics. The first involves nonlinear evolution equations focusing on commutator estimates involving fractional derivatives. These estimates are used to obtain local-to-global information on bilinear pseudodifferential operators illustrating, for example, how Euler and Navier-Stokes equations can be continued indefinitely if the deformation tensor and vorticity remains finite. Work will also be done in developing information about the local regularity and asymptotic behavior of solutions to nonlinear dispersive equations. Specific examples include the generalized Korteweg-de Vries equations for which there are no known global existence-uniqueness results. Moreover, numerical calculations suggest a blow-up of solutions when the power of the function in the equation exceeds five. The third area concerns the Cauchy problem for the three-dimensional thermoelasticity system (for a homogeneous medium). The equations form a coupled hyperbolic-parabolic system. An immediate goal of the research is to determine whether the behavior of small solutions is controlled by the hyperbolic part or by the parabolic part. For the pure elastic case, this problem has been studied earlier, however the techniques used are not applicable in the present context. Later work will consider the question of characterizing the nonlinear terms of the systems which can lead to singularities or solutions which have small initial data.