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We prove the global well-posedness and scattering for the defocusing H 1 2-subcritical (that is, 2 < γ < 3) Hartree equation with low regularity data in R d , d ≥ 3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space H s R d with s > 4(γ − 2)/(3γ − 4), which also scatters in both time directions. This improves… (More)
We show a new Bernstein's inequality which generalizes the results of Cannone-Planchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness… (More)
This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation u t + (−△) α u = F (u) for initial data in the Lebesgue space L r (R n) with r ≥ r d nb/(2α − d) or the homogeneous Besov space ˙ B −σ p,∞ (R n) with σ = (2α − d)/b − n/p and 1 ≤ p ≤ ∞, where α > 0, F (u) = f (u) or Q(D)f (u) with Q(D) being a homogeneous… (More)
In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation u t + (−∂ 2 x) α/2 u + uu x = 0 with α ∈ (1, 2) is studied. It is shown that if the nondecreasing initial datum approaches the constant states u ± (u − < u +) as x → ±∞, respectively, then the corresponding solution converges toward the… (More)
In this paper, we obtain the global well-posedness and scattering result in the energy space for the Klein-Gordon-Hartree equation in the spatial dimension n 3. The result covers the special potential case |x| −γ , 2 < γ < min(4, n). The new ingredients are that: we first extend Menzala-Strauss's casuality for the nonlocal Klein-Gordon equations, which can… (More)
We obtain global well-posedness, scattering, uniform regularity, and global L 6 t L 6n 3n−8 x spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Hartree equation in R × R n , n ≥ 5.
In this paper we study the following modified quasi-geostrophic equation ∂ t θ + u · ∇θ + ν|D| α θ = 0, u = |D| α−1 R ⊥ θ with ν > 0 and 0 < α < 1. This equation was firstly introduced by Constantin-Iyer-Wu in . Here, we firstly prove the local existence result of smooth solutions by using the classical method, and then through constructing a suitable… (More)
In this paper, we prove the local well-posedness in critical Besov spaces for the com-pressible Navier-Stokes equations with density dependent viscosities under the assumption that the initial density is bounded away from zero.
In this paper, we establish the global well-posedness for the critical generalized KdV equation with the low regularity data. To be precise, we show that a unique and global solution exists for initial data in the Sobolev space H s ` R ´ with s > 1 2. Of course, we require that the mass is strictly less than that of the ground state in the focusing case.… (More)