On uniqueness of mild $$L^{3,\infty }$$-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains

Abstract

This paper is concerned with the uniqueness of bounded continuous $$L^{3,\infty }$$ L 3 , ∞ -solutions on the whole time axis $$\mathbb R$$ R or the half-line $$(-\infty ,T)$$ ( - ∞ , T ) to the Navier–Stokes equations in 3-dimensional unbounded domains. When $$\Omega $$ Ω is an unbounded domain, it is known that a small solution in $$BC(\mathbb R;L^{3,\infty })$$ B C ( R ; L 3 , ∞ ) is unique within the class of solutions which have sufficiently small $$L^{\infty }(\mathbb R; L^{3,\infty })$$ L ∞ ( R ; L 3 , ∞ ) -norm; i.e., if two solutions u and v exist for the same force f, bothu and v are small, then the two solutions coincide. There is another type of uniqueness theorem. Farwig et al. (Commun Partial Differ Equ 40:1884–1904, 2015) showed that if two solutions u and v exist for the same force f, u is small and if v has a precompact range $${\mathscr {R}}(v):=\{v(t); -\infty<t<T\}$$ R ( v ) : = { v ( t ) ; - ∞ < t < T } in $$L^{3,\infty },$$ L 3 , ∞ , then the two solutions coincide. However, there exist many solutions which do not have precompact range. In this paper, instead of the precompact range condition, by assuming some decay property of v(x, t) with respect to the spatial variable x near $$t=-\infty ,$$ t = - ∞ , we show a modified version of the above-mentioned uniqueness theorem. As a by-product, in the half-space$$\mathbb R^3_+,$$ R + 3 , we obtain a non-existence result of backward self-similar $$L^{3,\infty }$$ L 3 , ∞ -solutions sufficiently close to some homogeneous function Q(x/|x|)/|x| in a certain sense.