Global existence of three-dimensional incompressible magneto-micropolar system with mixed partial dissipation,magnetic diffusion and angular viscosity

The magneto-micropolar fluid flows describe the motion of electrically conducting micropolar fluids in the presence of a magnetic field. The issue of whether the strong solution of magneto-micropolar equations in three-dimensional can exist globally in time with large initial data is still unknown. In this paper, we deal with the Cauchy problem of the three-dimensional magneto-micropolar system with mixed partial dissipation, magnetic diffusion and angular viscosity. More precisely, the global existence of smooth solutions to the three-dimensional incompressible magneto-micropolar fluid equations with mixed partial dissipation, magnetic diffusion and angular viscosity are obtained by energy method under the assumption that $H^1$-norm of the initial data $(u_0,b_0,w_0)$ sufficiently small, namely $6u_0,b_0,{\omega }_0{6}_{H^1\left({\mathbb{R}}^3\right)}^2\le \frac{\epsilon }{2}$, where $\epsilon$ is a sufficiently small positive number. This work follows the techniques in the paper of Cao and Wu (2011).     

Suslin’s algorithms for reduction of unimodular rows

A well-known lemma of Suslin says that for a commutative ring $A$ if $(v_1\left(X\right),\dots ,v_n\left(X\right))\in {\left(A\left[X\right]\right)}^n$ is unimodular where $v_1$ is monic and $n\ge 3$, then there exist ${\gamma }_1,\dots ,{\gamma }_{\ell }\in E_{n-1}\left(A\left[X\right]\right)$ such that the ideal generated by $\mathrm{Res}(v_1,e_1.{\gamma }_1{}^t(v_2,\dots ,v_n)),\dots ,\mathrm{Res}(v_1,e_1.{\gamma }_{\ell }{}^t(v_2,\dots ,v_n))$ equals $A$. This lemma played a central role in the resolution of Serre’s Conjecture. In the case where $A$ contains a set $E$ of cardinality greater than $deg{v}_1+1$ such that $y-y^{\prime }$ is invertible for each $y\ne y^{\prime }$ in $E$, we prove that the ${\gamma }_i$ can simply correspond to the elementary operations $L_1\to L_1+y_i{\sum }_{j=2}^{n-1}{u}_{j+1}{L}_j$, $1\le i\le \ell =deg{v}_1+1$, where $u_1{v}_1+\cdots +u_n{v}_n=1$. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in $K[X_1,\dots ,X_k]$ to ${}^t(1,0,\dots ,0)$ using elementary operations in the case where $K$ is an infinite field. Another feature of this paper is that it shows that the concrete local–global principles can produce competitive complexity bounds.     

Global bifurcation of positive radial solutions for an elliptic system in reactor dynamics

In this paper, we are concerned with the existence of positive radial solutions of the elliptic system $\{\begin{array}{c}?\Delta u=uv?\lambda u+f(\left|x\right|,u)\text{,}{R}_1<\left|x\right|<R_2\text{,}x\in {\mathbb{R}}^N,N\ge 1\text{,}\hfill \\ ?\Delta v=\mu u\text{,}{R}_1<\left|x\right|<R_2\text{,}x\in {\mathbb{R}}^N,N\ge 1\text{,}\hfill \\ u=v=0\text{,}\text{on }\left|x\right|=R_1\text{ and }\left|x\right|=R_2\text{,}\hfill \end{array}$ where $\left|x\right|={\left({\sum }_{i=1}^N{x}_i^2\right)}^{\frac{1}{2}}$, $\lambda >0$ is a constant, $\mu >0$ is a parameter and $0<R_1<R_2<\infty$, $f:[R_1,R_2]\times [0,\infty )\to [0,\infty )$ is continuous and $f(t,s)>0$ for all $(t,s)\in [R_1,R_2]\times (0,\infty )$. Under some appropriate conditions on the nonlinearity $f$, we show that the above system possesses at least one positive radial solution for any $\mu \in (0,\infty )$. The proof of our main results is based upon bifurcation techniques.