We establish the global well-posedness and analyticity of mild solution to the generalized three-dimensional incompressible Navier–Stokes equations for rotating fluids if the initial data are in Fourier–Herz spaces ?q1-2α (R3) under appropriate conditions for α and q. As corollaries, we also give the corresponding conclusions of the generalized Navier–Stokes equation.
By the use of the weight functions, the transfer formula and the technique of real analysis, an extended multidimensional half-discrete Hardy–Hilbert-type inequality with a general homogeneous kernel and a best possible constant factor is given, which is an extension of a published result. Moreover, the equivalent forms, a few particular cases and the operator expressions with some examples are considered.
Let (X, d, μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let Mβ,ρ,q be the fractional type Marcinkiewicz integral operator on (X, d, μ). In this paper, for β ∈ [0, ∞), ρ ∈ (0, ∞) and q ∈ (1, ∞), under the assumption that Mβ,ρ,q is bounded on L2(μ), the authors prove that Mβ,ρ,q is bounded from the weighted Lebesgue space Lp(w) into the weighted weak Lebesgue space Lp, ∞(w) and from the weighted Morrey space Lp,κ,η(ω) into the weighted weak Morrey space WLp,κ,η(ω).
We introduce a new dimension for isometric linear actions of countable sofic groups on complex Banach spaces. This generalizes the Voiculescu dimension for isometric linear actions of countable amenable groups on complex Banach spaces, and answers a question of Gromov in the case of countable sofic groups.
How to find out the integral basis of the number field effectively is a problem that people have been thinking for a long time. This paper gives a simple method to find out the integral basis of the cubic field. In addition, people are interested in the existence of power integral basis in number field. There is a power integral basis in both the quadratic field and the cyclotomic field, but it is not clear for the cubic field. In this paper, we give the necessary and sufficient conditions for the existence of power integral basis in the cubic field, and give a complete answer for the case of the cubic field.
In this paper, we prove the global existence of weak solutions to a 3D Keller–Segel–Navier–Stokes system with logistic source. We also study the long time behavior of the solutions.
For positive integers n ≥ 2 and m ≥ 1, consider a function f satisfying the following: (1) the inhomogeneous biharmonic equation △(△f) = g (g ∈ C (Bn, Rm)) in Bn; (2) the boundary conditions f= ψ1 (ψ1 ∈ C (Sn-1, Rm)) on Sn-1 and ∂f/∂n = ψ2 (ψ2 ∈ C (Sn-1, Rm)) on Sn-1, where ∂/∂n stands for the inward normal derivative, Bn is the unit ball in Rn and Sn-1 is the unit sphere of Bn. The main aim of this paper is to discuss the Heinz–Schwarz type inequalities and the modulus of continuity of the solutions to the above inhomogeneous biharmonic Dirichlet problem.
The aims of this paper are to investigate the statistical convergence in TVScone metric spaces and to discuss statistical completeness of TVS-cone metric spaces. Let (X, E, P, d) be a TVS-cone metric space. By applying Minkowski function ρ in the ordered Hausdorff topological vector space E, we show that there exists a metric dρ (in usual sense) on X such that a sequence (xn) in X is statistically convergent to x ∈ X with respect to d if and only if it is statistically convergent to x with respect to dρ. We then show that every TVS-cone statistically Cauchy sequence is an almost usual TVS-cone Cauchy sequence, and every TVS-cone statistically convergent sequence is an almost usual TVS-cone convergent sequence. As a result, a TVS-cone metric space (X, d) is d-complete if and only if it is d-statistically complete. Based on the results obtained above, many properties of statistical convergence in the metric space can be generalized in parallel to the statistical convergence in the cone metric space.
In this paper, it studies the properties of dynamical systems about random iterated of function family based on the idea of Schwick and using the Zalcman lemma. It points out that the Fatou set's definition of random dynamics and the Fatou set's definitions of dynamics of semigroups of a function family are obviously different but equivalent. Furthermore, the following normal criterion is obtained. Let F= {fi|fi is a nonlinear analytic function on C (C), i ∈ M}, where M is non empty index set, ΣM = {(j1, j2, …, jn, …)|ji ∈ M, i ∈ N}, if for any index sequence σ = (j1, j2, …, jn, …) ∈ ΣM, the iterative sequence {Wσn = fjn º fjn-1 º … º fj1(z)|n ∈ N} is normal at point z, then the function family F is also normal at point z.
We mainly consider the relationship between quasihyperbolic uniform domains and quasisymmetric mappings in metric spaces, and show that quasihyperbolic uniform domains are invariance properties under quasisymmetric mapping of metric spaces.