Global Well-Posedness and Analyticity of the Primitive Equations of Geophysics in Variable Exponent Fourier–Besov Spaces

We consider the Cauchy problem of the three-dimensional primitive equations of geophysics. By using the Littlewood–Paley decomposition theory and Fourier localization technique, we prove the global well-posedness for the Cauchy problem with the Prandtl number P=1 in variable exponent Fourier–Besov spaces for small initial data in these spaces. In addition, we prove the Gevrey class regularity of the solution. For the primitive equations of geophysics, our results can be considered as a symmetry in variable exponent Fourier–Besov spaces.

Dynamic Analysis of Software Systems with Aperiodic Impulse Rejuvenation

This paper aims to obtain the dynamical solution and instantaneous availability of software systems with aperiodic impulse rejuvenation. Firstly, we formulate the generic system with a group of coupled impulsive differential equations and transform it into an abstract Cauchy problem. Then we adopt a difference scheme and establish the convergence of this scheme by applying the Trotter–Kato theorem to obtain the system’s dynamical solution. Moreover, the instantaneous availability as an important evaluation index for software systems is derived, and its range is also estimated. At last, numerical examples are shown to illustrate the validity of theoretical results.

Trigonometric Fmn-Transform of Multi-Variable Functions and Its Application to the Partial Differential Equations and Image Processing

In this study, we focus on the extension of the trigonometric F m-transform technique for functions with one-variable in order to improve its approximation properties at the end points of [a,b] and then generalize the extended trigonometric Fm -transform technique to functions with more variables. The approximation and convergence properties of the direct and inverse multi-variable extended trigonometric Fm -transforms are discussed. The applicability of multi-variable trigonometric F m -transforms to approximate multi-variable functions are illustrated by some examples. Moreover, some direct formulas for the multi-variable extended trigonometric Fm -transforms of partial derivatives of multi-variable functions are obtained and they are applied to solving the Cauchy problem of the transport equation. Also, the application of multi-variable extended trigonometric Fm -transforms for image compression is described. Some examples for the validity of the obtained results about the partial differential equations and image compression are given. The results are compared with some existence ones in the literature.

Global well-posedness for the 3D magneto-micropolar equations with fractional dissipation

This paper focus on the Cauchy problem of the 3D incompressible magneto-micropolar equations with fractional dissipation in the Sobolev space. Liu, Sun and Xin obtained the global solutions to the 3D magneto-micropolar equations with $\alpha=\beta=\gamma=\frac{5}{4}$. Deng and Shang established the global well-posedness of the 3D magneto-micropolar equations in the case of $\alpha\geq\frac{5}{4}$, $\alpha+\beta\geq\frac{5}{2}$ and $\gamma\geq2-\alpha\geq\frac{3}{4}$. In this paper, we establish the global well-posedness of the 3D magneto-micropolar equations with $\alpha=\beta=\frac{5}{4}$ and $\gamma=\frac{1}{2}$, which improves the results of Liu-Sun-Xin and Deng-Shang by reducing the value of $\gamma$ to $\frac{1}{2}$.