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We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i∂t u=−ε u+ 2u+ P (( ∂ x u ) |α| 2, ( ∂ x ū ) |α| 2 ) , t ∈R, x ∈Rn, where ε ∈ {−1,0,1}, n 2 denotes the spatial dimension and P(·) is a polynomial excluding constant and linear terms. © 2006 Elsevier Inc. All rights reserved.
The GPE (1.1) in physical dimensions (2 and 3 dimensions) is used in the meanfield quantum theory of Bose-Einstein condensate (BEC) formed by ultracold bosonic coherent atomic ensembles. Recently, several research groups [9, 12–14] have produced quantized vortices in trapped BECs, and a typical method they used is to impose a laser beam on the magnetic trap(More)
Global classical solutions near Maxwellians are constructed for the Boltzmann and Landau equations with soft potentials in the whole space. The construction of global solutions is based on refined energy analysis. Our results generalize the classical results in Ukai and Asano (Publ. Res. Inst. Math. Sci. 18 (1982), 477–519) to the very soft potentials for(More)
where the unknown are u, θ, π which denote, respectively, the velocity field, the scalar temperature and the scalar pressure. Data are the positive constants ν, χ, respectively, the viscosity and the thermal conductivity coefficients and the function f the external force field, and a(x), b(x), respectively, represent the initial velocity and initial(More)
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