87 research outputs found
Time decay of scaling invariant Schroedinger equations on the plane
We prove the sharp L^1-L^{\infty} time-decay estimate for the 2D-Schroedinger
equation with a general family of scaling critical electromagnetic potentials.Comment: 26 page
Stability of selfsimilar solutions to the fragmentation equation with polynomial daughter fragments distribution
We study fragmentation equations with power-law fragmentation rates and
polynomial daughter fragments distribution function . The corresponding
selfsimillar solutions are analysed and their exponentially decaying asymptotic
behaviour and regularity deduced. Stability of selfsimilar
solutions (under smooth exponentially decaying perturbations), with sharp
exponential decay rates in time are proved, as well as regularity
of solutions for . The results are based on explicit expansion in terms of
generalized Laguerre polynomials and the analysis of such expansions. For
perturbations with power-law decay at infinity stability is also proved.
Finally, we consider real analytic
Singularities on charged viscous droplets
We study the evolution of charged droplets of a conducting viscous liquid.
The flow is driven by electrostatic repulsion and capillarity. These droplets
are known to be linearly unstable when the electric charge is above the
Rayleigh critical value. Here we investigate the nonlinear evolution that
develops after the linear regime. Using a boundary elements method, we find
that a perturbed sphere with critical charge evolves into a fusiform shape with
conical tips at time , and that the velocity at the tips blows up as
, with close to -1/2. In the neighborhood of the
singularity, the shape of the surface is self-similar, and the asymptotic angle
of the tips is smaller than the opening angle in Taylor cones.Comment: 9 pages, 6 figure
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