On the Stationary Navier–Stokes Problem in $$\mathbb {R}^3$$ R 3 : An Approach in Weighted Sobolev Spaces

Abstract

In this paper, we study the steady-state Navier–Stokes equations in $$\mathbb {R}^3$$ . First, we establish the existence of very weak solution in $$\varvec{L}^p(\mathbb {R}^3)$$ with $$3/2< p < +\infty $$ under smallness conditions on the data. A uniqueness result is also given in case the data belong to $$\mathbb {L}^r(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)$$ with $$3/2<r<3$$ . We also discuss the case where the data are not necessarily small. In particular, these results enhance those obtained by Bjorland et al. (Commun Partial Differ Equ 26:216–246, 2011), and are in agreement with those obtained by Kim and Kozono (J Math Anal Appl 395(2):486–495, 2012). Second, we prove a result of existence and uniqueness of weak solution in the weighted Sobolev space $$\varvec{W}_0^{1,p}(\mathbb {R}^3)\cap \varvec{W}_0^{1,\,3/2}(\mathbb {R}^3)$$ in case of small external forces given by $$\mathrm{div}\mathbb {F}$$ with $$\mathbb {F} \in \mathbb {L}^p(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)$$ and $$1<p<3$$ .