We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriateâ€¦ (More)

Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete half-line SchrÃ¶dinger operators with slowly decaying potentials. Among our results weâ€¦ (More)

We derive a general upper bound on the spreading rate of wavepackets in the framework of SchrÃ¶dinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escapeâ€¦ (More)

We provide a general lower bound on the dynamics of one dimensional SchrÃ¶dinger operators in terms of transfer matrices. In particular it yields a non trivial lower bound on the transport exponentsâ€¦ (More)

The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finiteâ€¦ (More)

We prove that for any one-dimensional Schrr odinger operator with potential V (x) satisfying decay condition jV (x)j C x ?3=4? ; the absolutely continuous spectrum lls the whole positive semi-axis.â€¦ (More)

We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible scenario for the initial data that is compactly supported and the size ofâ€¦ (More)

We establish rigorous lower bounds on the speed of traveling fronts and on the bulk burning rate in reaction-diffusion equation with passive advection. The non-linearity is assumed to be of eitherâ€¦ (More)

We construct an initial data for the two-dimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known toâ€¦ (More)

We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasigeostrophic equation, the Burgersâ€¦ (More)