Search

This paper deals with stability and the large-time decay to any given global smooth solutions of the 3D density-dependent incompressible Boussinesq system. The decay rate for solutions of the corresponding Cauchy problem is obtained in this work. With the aid of this decay rate, it is shown that a small perturbation of initial data $(\overline{a}_0,\overline{\theta}_0, \overline{u}_0)$ still generates a global smooth solution to the density-dependent Boussinesq system, and this solution keeps close to the reference solution.

SnBiInZn based high entropy alloy (HEA) was studied as a low reflow temperature solder with melting point around 80 oC. The wetting angle is about 52o after reflow at 100 oC for 10 min. The interfacial intermetallic compound (IMC) growth kinetics was measured to be ripening-control with a low activation energy about 18.0 kJ/mol, however, the interfacial reaction rate is very slow, leading to the formation of a very thin IMC layer. The low melting point HEA solder has potential applications in advanced electronic packaging technology, especially for bio-medical devices., Comment: 20 pages, 7 figures

This paper is concerned with the Neumann initial-boundary value problem for the two-species chemotaxis system with consumption of chemoattractant \begin{equation*} u_t=\Delta u-\chi_1\nabla\cdot(u\nabla w), \end{equation*} \begin{equation*} v_t=\Delta v-\chi_2\nabla\cdot(v\nabla w), \end{equation*} \begin{equation*} w_t=\Delta w-(\alpha u+\beta v)w \end{equation*} in a smooth bounded domain $\Omega\subset\mathbb{R}^n$ ($n\geq2$), where the parameters $\chi_1$, $\chi_2$, $\alpha$ and $\beta$ are positive. It is proved that if \begin{equation*} \max\{\chi_1,\chi_2\}\|w(x,0)\|_{L^{\infty}(\Omega)}<\sqrt{\frac{2}{n}}\pi \end{equation*} the problem possesses a unique global classical solution that is uniformly bounded. Moreover, we prove that \begin{equation*} u(x,t)\to\frac{1}{|\Omega|}\int_{\Omega}u(x,0),\quad v(x,t)\to\frac{1}{|\Omega|}\int_{\Omega}v(x,0)\quad\mbox{and}\quad w(x,t)\to0\quad\mbox{as}\ t\to\infty \end{equation*} uniformly with respect $x\in\Omega$., Comment: 12 pages

This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla v\right)+f(u) ,\quad x\in\Omega,\ t>0$$ $$v_t=\Delta v-v+u,\quad x\in\Omega,\ t>0$$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n$ $(n\geq1)$, where the functions $D(u)$ and $S(u)$ are supposed to be smooth satisfying $D(u)\geq Mu^{-\alpha}$ and $S(u)\leq Mu^{\beta}$ with $M>0$, $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}$ for all $u\geq1$, and the logistic source $f(u)$ is smooth fulfilling $f(0)\geq0$ as well as $f(u)\leq a-\mu u^{\gamma}$ with $a\geq0$, $\mu>0$ and $\gamma\geq1$ for all $u\geq0$. It is shown that if $\alpha+2\beta<\gamma-1+\frac{2}{n}$, for $1\leq\gamma<2$ and $\alpha+2\beta<\gamma-1+\frac{4}{n+2}$, for $\gamma\geq2$, then for sufficiently smooth initial data the problem possesses a unique global classical solution which is uniformly bounded., Comment: 10 pages

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equations generalizing the porous-medium-type diffusion model $ \quad n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\chi(c)\nabla c), $ $ \quad c_t+u\cdot\nabla c=\Delta c-nf(c), $ $ \quad u_t+\kappa(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\Phi, $ $ \quad \nabla\cdot u=0, $ in a bounded convex domain $\Omega\subset\mathbb{R}^3$. It is proved that if $m\geq\frac{2}{3}$, $\kappa\in\mathbb{R}$, $0<\chi\in C^2([0,\infty))$, $0\leq f\in C^1([0,\infty))$ with $f(0)=0$ and $\Phi\in W^{1,\infty}(\Omega)$, then for sufficiently smooth initial data $(n_0, c_0, u_0)$ the model possesses at least one global weak solution.