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

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