We consider spherical averages of the Fourier transform of fractal measures and improve both the upper and lower bounds on the rate of decay. Maximal estimates with respect to fractal measures are deduced for the Schrödinger and wave equations. This refines the almost everywhere convergence of the solution to its initial datum as time tends to zero. A consequence is that the solution to the wave equation cannot diverge on a (d − 1)-dimensional manifold if the data belongs to the energy spaceḢ 1 (R d ) × L 2 (R d ).
We prove smoothing estimates in Morrey-Campanato spaces for a Helmholtz equationwith fully variable coefficients, of limited regularity, defined on the exterior of a starshaped compact obstacle in R n , n ≥ 3, with Dirichlet boundary conditions. The principal part of the operator is a long range perturbation of a constant coefficient operator, while the lower order terms have an almost critical decay. We give explicit conditions on the size of the perturbation which prevent trapping.As an application, we prove smoothing estimates for the Schrödinger flow e itL and the wave flow e it √ L with variable coefficients on exterior domains and Dirichlet boundary conditions.
We consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain recently proved that there is initial data, in H s (R n ) with s < n 2(n+1) , for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.
We prove that the vortex structures of solutions to the 3D Navier-Stokes equations can change their topology without any loss of regularity. More precisely, we construct smooth high-frequency solutions to the Navier-Stokes equations where vortex lines and vortex tubes of arbitrarily complicated topologies are created and destroyed in arbitrarily small times. This instance of vortex reconnection is structurally stable and in perfect agreement with the existing computer simulations and experiments. We also provide a (non-structurally stable) scenario where the destruction of vortex structures is instantaneous.
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