Volume 40, Issue 2

We consider the three-dimensional Navier–Stokes equations whose initial data may have infinite kinetic energy. We establish unique existence of the mild solution to the Navier–Stokes equations for small initial data in the whole space ℝ3{\mathbb{R}^{3}} and a vertically periodic product space ℝ2×𝕋1{\mathbb{R}^{2}\times\mathbb{T}^{1}} which may be constant in vertical direction so that it includes the Oseen vortex. We further discuss its asymptotic stability under arbitrarily large three-dimensional perturbation in ℝ2×𝕋1{\mathbb{R}^{2}\times\mathbb{T}^{1}}.

Triple sequence convergence plays an extremely important role in the fundamental theory of mathematics. This paper contains four types of convergence concepts, namely, convergence almost surely, convergence incredibility, trust convergence in mean, and convergence in distribution, and discuss the relationship among them and some mathematical properties of those new convergence.

We investigate the relationship between the discrete fractional difference (Δγ∘Δβ∘Δα⁢f)⁢(t)(\Delta^{\gamma}\circ\Delta^{\beta}\circ\Delta^{\alpha}f)(t) and the positivity or monotonicity of the function f . Our approach relies on interpreting the fractional difference as an appropriate convolution operator. The results we provide demonstrate that when compared to the double sequential case, i.e., (Δβ∘Δα⁢f)⁢(t){(\Delta^{\beta}\circ\Delta^{\alpha}f)(t)}, there is relatively more complexity observed.

In this paper we will give q -analogues for the Pythagorean trigonometric identity sin2⁡z+cos2⁡z=1{\sin^{2}z+\cos^{2}z=1} in terms of Gosper’s q -trigonometry. We shall also give new q -analogues for the duplicate trigonometric identity sin⁡(x-y)⁢sin⁡(x+y)=sin2⁡x-sin2⁡y{\sin(x-y)\sin(x+y)=\sin^{2}x-\sin^{2}y}. Moreover, we shall give a short proof for an identity of Gosper, which was also established by Mező. The main argument of our proofs is the residue theorem applied to elliptic functions.