Use of Harmonic Analysis Methods for the Equations of Fluid Motion

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This project applies techniques of harmonic analysis to investigate problems in fluid dynamics that involve important mathematical questions and at the same time reflect intrinsic properties of fluid behavior. The first problem is related to proving partial regularity results for the Navier-Stokes equations with hyper-dissipation. The work will provide an upper bound on the Hausdorff dimension of the set of singular points. Mathematical techniques involve Littlewood-Paley operators and pseudodifferential operators. The second problem considers refined dyadic models for the equations of fluid motion that describe a possible cascade of energy along the dyadic tree. The project employs wavelets to explore blow-up scenarios for these models. The third area of work establishes partial regularity results for a modification of the Navier-Stokes equations involving a nonlinear relation between the viscous stress tensor and the rate of strain, useful in modeling behavior of non-Newtonian fluids. The mathematical techniques involve generalized energy inequalities and microlocal analysis.The proposed activity contains an interdisciplinary approach to questions arising from fluid dynamics. Methods used include sophisticated techniques of harmonic analysis that recently yielded progress on important similar questions. The principal investigator will disseminate results of the project among both pure and applied mathematicians, and she will incorporate results of the research activity into a graduate course.