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We show a new Bernstein’s inequality which generalizes the results of CannonePlanchon, 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 supercritical dissipation for the small initial data in the critical Besov space, and local wellposedness for… (More)

This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation ut + (−△) u = F (u) for initial data in the Lebesgue space L(R) with r ≥ rd , nb/(2α− d) or the homogeneous Besov space Ḃ p,∞(R ) 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 pseudo-differential… (More)

We obtain global well-posedness, scattering, uniform regularity, and global L t L 6n 3n−8 x spacetime bounds for energy-space solutions to the defocusing energycritical nonlinear Hartree equation in R× R, n ≥ 5.

Cannone, Meyer and Planchon [3] proved the global well-posedness of the incompressible Navier-Stokes equations for a class of highly oscillating data. In this paper, we prove the global well-posedness for the compressible NavierStokes equations in the critical functional framework with the initial data close to a stable equilibrium. Especially, this result… (More)

- Changxing Miao, Liutang Xue
- 2009

In this paper we consider the following 2D Boussinesq-Navier-Stokes systems ∂tu+ u · ∇u+∇p+ |D|u = θe2 ∂tθ + u · ∇θ + |D|θ = 0 with divu = 0 and 0 < β < α < 1. When 6− √ 6 4 < α < 1, 1−α < β ≤ f(α), where f(α) is an explicit function as a technical bound, we prove global well-posedness results for rough initial data. Mathematics Subject Classification… (More)

- Changxing Miao, Haigen Wu, Guixiang Xu, Junyong Zhang
- 2009

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)

- Changxing Miao, Liutang Xue
- 2009

In this paper we study the following modified quasi-geostrophic equation ∂tθ + u · ∇θ + ν|D| θ = 0, u = |D|Rθ with ν > 0 and 0 < α < 1. This equation was firstly introduced by Constantin-Iyer-Wu in [10]. Here, we firstly prove the local existence result of smooth solutions by using the classical method, and then through constructing a suitable modulus of… (More)

We establish global well-posedness and scattering for solutions to the masscritical nonlinear Hartree equation iut +∆u = ±(|x|−2 ∗ |u|2)u for large spherically symmetric L2x(R ) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state.

In this paper, we prove the local well-posedness in critical Besov spaces for the compressible Navier-Stokes equations with density dependent viscosities under the assumption that the initial density is bounded away from zero.

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order iut + ∆ u = |u| 8 d−4u. We prove that if a maximal-lifespan radial solution u : I × R → C obeys sup t∈I ‖∆u(t)‖2 < ‖∆W‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In… (More)