Convergence of approximating solutions of the Navier–Stokes equations in higher ordered Sobolev norms

We show that the approximating solutions ${\left\{u_j\right\}}_{j=0}^{\infty }$ of the Navier–Stokes equations constructed by Kato with the initial data $u\left(0\right)\in L_{\sigma }^n\left(R^n\right)$ converge to the local strong solution $u$ in the topology of $W^{k,q}\left(R^n\right)$ for all $k\in N$ provided the convergence in the scaling invariant norm in $L^q\left(R^n\right)$ with the time weight holds. As an application of our convergence, it is clarified that the approximation of the pressure is established in $W^{k+1,q}\left(R^n\right)$.