Weighted estimates for nonstationary Navier-Stokes equations
The Cauchy problem for the Navier-Stokes equations is considered in the whole 3-D space \[ \frac{\partial v}{\partial t}-\nu\Delta v+(u\cdot\nabla)v + \nabla p=0, \qquad\text{div }v =0,\quad x\in \mathbb{R}^3,\;t>0 \] \[ v(x,0)=v_0(x)\quad x\in \mathbb{R}^3, \qquad v\to 0\;\text{as } | x| \to \infty \] Certain new results on the global solvability of the problem are obtained. Namely, if the given initial data \(v_0\) is sufficiently small then a global strong solution exists. This solution satisfies \[ v,t^{\frac 12}v\in L^\infty((0,\infty);L^2(\mathbb{R}^3)),\quad \nabla v,\;t^{\frac 12}Av\in L^2((0,\infty);L^2(\mathbb{R}^3)),\quad (1+| x| ^2)v\in L^\infty(\mathbb{R}^3\times (0,\infty)). \] Here \(A\) is the Stokes operator. Estimates of the solution are established in the corresponding functional spaces.
scientific article; zbMATH DE number 1213745
scientific article