43,409 research outputs found
Strichartz type estimates for fractional heat equations
We obtain Strichartz estimates for the fractional heat equations by using
both the abstract Strichartz estimates of Keel-Tao and the
Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous
Strichartz estimate via replacing by
and a parabolic homogeneous Strichartz estimate.
Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue
spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the
Strichartz estimates for the fractional heat equations with a time dependent
potential of an appropriate integrability. As an application, we prove the
global existence and uniqueness of regular solutions in spatial variables for
the generalized Navier-Stokes system with data.Comment: 20 page
Spatially Modulated Interaction Induced Bound States and Scattering Resonances
We study the two-body problem with a spatially modulated interaction
potential using a two-channel model, in which the inter-channel coupling is
provided by an optical standing wave and its strength modulates periodically in
space. As the modulation amplitudes increases, there will appear a sequence of
bound states. Part of them will cause divergence of the effective scattering
length, defined through the phase shift in the asymptotic behavior of
scattering states. We also discuss how the local scattering length, defined
through short-range behavior of scattering states, modulates spatially in
different regimes. These results provide a theoretical guideline for new
control technique in cold atom toolbox, in particular, for alkali-earth-(like)
atoms where the inelastic loss is small.Comment: 5 pages, 5 figure
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