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We establish vortex dynamics for the time-dependent Ginzburg–Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.
We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of
. This rate is due to competition between surface tension and gravitation at
wave numbers and is connected to the fact that, in the presence of surface tension, there is a so-called “slowest wave”. Additionally, we combine our dispersive estimates with
type energy bounds to prove a family of Strichartz estimates.
For external magnetic field hex ≤
Cε–α, we prove
that a Meissner state solution for the Chern-Simons-Higgs functional exists. Furthermore, if the solution
is stable among all vortexless solutions, then it is unique.
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