The weUposedness problem for an anisotropic incompressible viscous fluid in R3, ro- tating around a vector B（t, x） ：= （b1 （t, x）, b2 （t, x）, b3 （t, x））, is studied. The global wellposedness in the homogeneous case （B = e3） with sufficiently fast rotation in the space B0,1/2 is proved. In the inhomogeneous case （B = B（t, xh））, the global existence and uniqueness of the solution in B0,1/2 are obtained, provided that the initial data are sufficient small compared to the horizontal viscosity. Furthermore, we obtain uniform local existence and uniqueness of the solution in the x same function space. We also obtain propagation of the regularity in B2,11/2 under the additional assumption that B depends only on one horizontal space variable.
The scattering problem for the Klein–Gordon equation with cubic convolution nonlinearity is considered. Based on the Strichartz estimates for the inhomogeneous Klein–Gordon equation, we prove the existence of the scattering operator, which improves the known results in some sense.
In this paper, we shall prove that the Koch-Tataru solution u to the incompressible Navier-Stokes equations in Rd satisfies the decay estimates involving some borderline Besov norms with d ≥ 3. Moreover, u has a unique trajectory which is HSlder continuous with respect to the space variables.
We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to 2D compressible Navier-Stokes equations in a bounded domain. The initial vacuum is allowed. The proof is based on the new a priori estimate for 2D compressible Navier-Stokes equations and a logarithmic estimate for Lamé system.
The primary goal of this paper is to present a comprehensive study of the nonlinear Schrodinger equations with combined nonlinearities of the power-type and Hartreetype. Under certain structural conditions, the authors are able to provide a complete picture of how the nonlinear Schrodinger equations with combined nonlinearities interact in the given energy space. The method used in the paper is based upon the Morawetz estimates and perturbation principles.
The influence of the random perturbations on the fourth-order nonlinear SchrSdinger equations,iut＋△^2u＋ε△u＋λ｜u｜^p-1u=ξ,（t,x）∈R^＋×R^n,n≥1,ε∈{-1,0,＋1},is investigated in this paper. The local well-posedness in the energy space H^2（R^n） are proved for p 〉n＋4/n＋2,and p≤2^#-1 if n≥5.Global existence is also derived for either defocusing or focusing L^2-subcritical nonlinearities.
We consider the three-dimensional compressible Navier-Stokes-Poisson system where the electric field of the internal electrostatic potential force is governed by the self-consistent Poisson equation.If the Fourier modes of the initial data are degenerate at the low frequency or the initial data decay fast at spatial infinity,we show that the density converges to its equilibrium state at the L 2-rate (1+t)(-7/4) or L ∞-rate (1+t)(-5/2),and the momentum decays at the L 2-rate (1+t)(-5/4) or L ∞-rate (1+t)(-2).These convergence rates are shown to be optimal for the compressible Navier-Stokes-Poisson system.
The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schr6dinger equations. The local and global well-posedness are proved with values in the space ∑（R^n）={f E HI（R^n）, ｜·｜f ∈ L^2（R^n）}. When the nonlinearity is focusing and L2-supercritical, the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential. Especially for the repulsive case, the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time. Thus, compared with the deterministic equation for the repulsive case, the blow-up condition is stronger on average, and depends on the regularity of the noise. If φ = 0, our results coincide with the ones for the deterministic equation.
In this paper,we shall prove that the Koch-Tataru solution u to the incompressible Navier-Stokes equations in Rd satisfies the decay estimates involving some borderline Besov norms with d 3.Moreover,u has a unique trajectory which is Hlder continuous with respect to the space variables.
In this paper, for the full Euler system of the isothermal gas, we show that a globally stable supersonic conic shock wave solution does not exist when a uniform supersonic incoming flow hits an infinitely long and curved sharp conic body.
