In this paper, we consider the following system $$ \left \{ \textstyle\begin{array}{ll} n_{t}+u\cdot \nabla n&=\Delta n-\nabla \cdot (n\mathcal{S}(|\nabla c|^{2}) c)-nm, \\ c_{t}+u\cdot c&=\Delta c-c+m, m_{t}+u\cdot m&=\Delta m-mn, u_{t}&=\Delta u+\nabla P+(n+m)\nabla \Phi ,\qquad u=0 \end{array}\displaystyle \right . which models process of coral fertilization, in a smoothly three-dimensional ...

Abstract In this paper, we consider the small-convection limit of chemotaxis-Navier–Stokes system with logarithmic sensitivity and logistic-type source $$ \textstyle\begin{cases} n^{\kappa}_{t}+\boldsymbol{u}^{\kappa}\cdot \nabla{n}^{\kappa}= \Delta{n}^{\kappa}-\chi \nabla \cdot ({n^{\kappa}}\nabla \log{c^{\kappa}})+f(n^{\kappa}), &x\in \Omega , t>0, \\ c^{\kappa}_{t}+\boldsymbol{u}^{\ka...

In this article, we prove several new fractional nabla Bennett–Leindler dynamic inequalities with the help of a simple consequence Keller’s chain rule, integration by parts, mean and Hölder’s inequality for derivative on time scales. As result this, some classical are obtained as special cases, extended our to discrete continuous calculus. addition, when α=1, shall obtain well-known instances f...

We consider the following Keller-Segel system with gradient dependent chemotactic coefficient: \begin{document}$ \begin{equation*} \begin{cases} u_{t} = \Delta u- \chi \nabla\cdot (uf(|\nabla v|)\nabla v),\\ 0 v -v+g(u), \end{cases} \end{equation*} $\end{document} in smooth bounded domains $ \Omega \subset \mathbb{R}^{n}, \,n\geq 1 f(\xi) \big(\xi^{p-2}\big(1+\xi^{p}\big)^{\frac{q-p}{p}}\big), ...

Systems of the type $$\begin{cases} u_t = \nabla \cdot (D_1(u) u - S_1(u) v) + f_1(u, v),\\ v_t (D_2(v) v S_2(v) u) f_2(u, \end{cases} \qquad (\star)$$ can be used to model pursuit-evasion relationships between predators and prey. Apart from local kinetics given by $f_1$ $f_2$, key components in this system are taxis terms $-\nabla (S_1(u) v)$ $+\nabla (S_2(v) u)$; that is, species not only ass...

In this paper, we consider the homogenization problem for generalized elliptic systems$ \mathcal{L}_{ \varepsilon} = -\operatorname{div}[A(x/ \varepsilon)\nabla+V(x/ \varepsilon)]+B(x/ \varepsilon)\nabla+c(x/ \varepsilon)+\lambda I $with dimension two. Precisely, will establish $ W^{1, p} estimates, Hölder Lipschitz estimates and L^p convergence results with The operator has been studied by Qia...

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are H\"{o}lder continuous at $0$ provided $\int_{B_1}|u(x)|^3dx+\int_{B_1}|f(x)|^qdx$ or $\int_{B_1}|\nabla u(x)|^2dx$+$\int_{B_1}|\nabla u(x)|^2dx\left(\int_{B_1}|u(x)|dx\right)^2+\int_{B_1}|f(x)|^qdx$ with $q>3$ sufficiently small, which implies 2D Hausdorff measure of set singular points zero...