Lagrangian-Eulerian Methods for Uniqueness in Hydrodynamic Systems

We present a Lagrangian-Eulerian strategy for proving uniqueness and local existence of solutions in path spaces of limited smoothness for a class of incompressible hydrodynamic models including Oldroyd-B type complex fluid models and zero magnetic resistivity magneto-hydrodynamics equations

Transport in Rotating Fluids

We consider uniformly rotating incompressible Euler and Navier-Stokes equations. We study the suppression of vertical gradients of Lagrangian displacement ("vertical" refers to the direction of the rotation axis). We employ a formalism that relates the total vorticity to the gradient of the back-to-labels map (the inverse Lagrangian map, for inviscid flows, a diffusive analogue for viscous flows). The results include a nonlinear version of the Taylor-Proudman theorem: in a steady solution of the rotating Euler equations, two fluid material points which were initially on a vertical vortex line, will perpetually maintain their vertical separation unchanged. For more general situations, including unsteady flows, we obtain bounds for the vertical gradients of the Lagrangian displacement that vanish linearly with the maximal local Rossby number

Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications

We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak solutions of critical surface quasi-geostrophic equations