Time decay of scaling critical electromagnetic Schr\"odinger flows

We obtain a representation formula for solutions to Schr\"odinger equations with a class of homogeneous, scaling-critical electromagnetic potentials. As a consequence, we prove the sharp $L^{1}\to L^{\infty}$ time decay estimate for the 3D-inverse square and the 2D-Aharonov-Bohm potentials.Comment: 32 pages, 1 figur

On the mechanism of branching in negative ionization fronts

We explain a mechanism for branching of a planar negative front. Branching occurs as the result of a balance between the destabilizing effect of impact ionization and the stabilizing effect of electron diffusion on ionization fronts. The dispersion relation for transversal perturbation is obtained analytically and reads: $s = |k|/[2 (1 + |k|)] - D |k|^2$, where $D$, which is assumed to be small, is the ratio between the electron diffusion coefficient and the intensity of the externally imposed electric field. We estimate the spacing $\lambda$ between streamers in a planar discharge and deduce a scaling law $\lambda \sim D^{1/3}$

Ionization fronts in negative corona discharges

In this paper we use a hydrodynamic minimal streamer model to study negative corona discharge. By reformulating the model in terms of a quantity called shielding factor, we deduce laws for the evolution in time of both the radius and the intensity of ionization fronts. We also compute the evolution of the front thickness under the conditions for which it diffuses due to the geometry of the problem and show its self-similar character.Comment: 4 pages, 4 figure

Discrete Self-Similarity in Interfacial Hydrodynamics and the Formation of Iterated Structures

The formation of iterated structures, such as satellite and sub-satellite drops, filaments and bubbles, is a common feature in interfacial hydrodynamics. Here we undertake a computational and theoretical study of their origin in the case of thin films of viscous fluids that are destabilized by long-range molecular or other forces. We demonstrate that iterated structures appear as a consequence of discrete self-similarity, where certain patterns repeat themselves, subject to rescaling, periodically in a logarithmic time scale. The result is an infinite sequence of ridges and filaments with similarity properties. The character of these discretely self-similar solutions as the result of a Hopf bifurcation from ordinarily self-similar solutions is also described.Comment: LaTeX, 5 pages, replaced with minor changes, accepted for publication in Physical Review Letter

The Beads-on-String Structure of Viscoelastic Threads

Submitted to J. Fluid Mech.By adding minute concentrations of a high molecular weight polymer, liquid jets or bridges collapsing under the action of surface tension develop a characteristic shape of uniform threads connecting spherical uid drops. In this paper, high-precision measurements of this beads-on-string structure are combined with a theoretical analysis of the limiting case of large polymer relaxation times and high polymer extensibilities, for which the evolution can be divided into two distinct regimes. For times smaller than the polymer relaxation time, over which the beads-on-string structure develops, we give a simplfied local description, which still retains the essential physics of the problem. At times much larger than the relaxation time, we show that the solution consists of exponentially thinning threads connecting almost spherical drops. Both experiment and theoretical analysis of a one-dimensional model equation reveal a self-similar structure of the corner where a thread is attached to the neighbouring drops.NASA Microgravity Fluid Dynamic