Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations
Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations
Consider the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations\begin{eqnarray*}\frac{\partial{\bf u}}{\partial t}-\alpha\triangle{\bf u}+({\bf u}\cdot\nabla){\bf u}+\nabla p={\bf f}({\bf x},t),\qquad{\bf u}({\bf x},0)={\bf u}_0({\bf x}).\end{eqnarray*}In this system, the dimension $n\geq 3$, ${\bf u}({\bf x},t)=(u_1({\bf x},t),u_2({\bf x},t),\cdots,u_n({\bf x},t))$ and${\bf f}({\bf x},t)=(f_1({\bf x},t),f_2({\bf x},t),\cdots,f_n({\bf x},t))$ are real vector valued functions of${\bf x}=(x_1,x_2,\cdots,x_n)$ and $t$. Additionally, …