On well-posedness and regularity properties for fluid equations and nonlinear dispersive equations

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Pavlovi'c plans to gain better understanding of fluid and nonlinear wave and dispersive equationsvia analyzing dispersive equations using harmonic analysis techniques that turned out to be usefulin the context of fluid equations and vice verse. The first group of proposed problems focuses onthe Navier-Stokes equations that describe fundamental properties of viscous fluids. One approachin studying existence of solutions to the Navier-Stokes is to obtain solutions to the correspondingintegral equation. Existence of these solutions in 3D has been proved only locally in time andglobally for small initial data. Hence it is important to understand behavior of these solutions in"critical spaces" that preserve scaling invariance. With her collaborators the PI will investigate thebehavior of solutions to the Navier-stokes equations in critical spaces. Questions of interest includestability of self-similar solutions in critical spaces (which is motivated by the analogy with solitonsin the context of dispersive equations) and a long standing open problem related to well-posednessof the Navier-Stokes equations in the largest critical space. The second group of proposed problemsconcentrates on nonlinear dispersive equations. Many of the important structural properties(e.g. conserved or monotone quantities) of the nonlinear Schrodinger equations (NLS) are at lowregularities, and to exploit these features one needs to establish existence theory at low regularities.Pavlovi'c proposes to continue her work on establishing global well-posedness for certain class ofNLS equations corresponding to low regularity data. The PI will employ and further investigatetools that were useful in recent advances in the field, such as interaction Morawetz estimates.The third group of problems is related to super-critical nonlinear wave and NLS equations. Heresuper-critical refers to equations with conserved quantities at lower regularities than the scalinginvariant norm (the 3D Navier-Stokes is an example). Motivated by her earlier work with Katzon partial regularity of the Navier-Stokes equations, Pavlovi'c proposes to use microlocalizationtechniques in order to obtain a partial regularity result for super-critical NLS and wave equations.Suggested problems involve important mathematical questions such as existence and regularityof solutions to nonlinear PDEs that describe motion of fluid or various wave phenomena. Forinstance, the theory of the Navier-Stokes equations in three dimensions is far from being complete.The outstanding open problems, whose better understanding would have impact in the fields fromoceanography to cosmology, are global existence, uniqueness and regularity of smooth solutions tothe Navier-Stokes in 3D. On the other hand, the NLS and their combinations with the Kortewegde-Vries and wave equations have been proposed as models for many basic wave phenomena.Such a physical relevance of the equations motivates mathematical explorations. The proposedactivity seeks to find an interdisciplinary approach to questions arising from fluid and dispersivePDEs. In particular, the PI plans to analyze dispersive equations using sophisticated techniquesof harmonic analysis that turned out to be useful in the context of fluid equations and vice verse.