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We consider the stationary Navier–Stokes equations in the two‐dimensional torus T2$\mathbb {T}^2$. For any ε>0$\varepsilon >0$, we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces Ḃp+ε,q−1+2p(T2)$\dot{B}^{-1+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}^2)$ for given small external forces in Ḃp+ε,q−3+2p(T2)$\dot{B}^{-3+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}^2)$ when 1≤p<2$1\le p <2$. These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well‐posedness is proved by using the embedding property and the para‐product estimate in homogeneous Besov spaces. In addition, for the case (p,q)∈({2}×(2,∞])∪((2,∞]×[1,∞])$(p,q)\in (\lbrace 2\rbrace \times (2,\infty ])\cup ((2,\infty ]\times [1,\infty ])$, we can show the ill‐posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill‐posedness by showing the discontinuity of a certain solution map from Ḃp,q−3+2p(T2)$\dot{B}^{-3+\frac{2}{p}}_{p ,q}(\mathbb {T}^2)$ to Ḃp,q−1+2p(T2)$\dot{B}^{-1+\frac{2}{p}}_{p, q}(\mathbb {T}^2)$. [ABSTRACT FROM AUTHOR]

We prove in this paper that a sequence M : Z n → L(E) of bounded variation is a Fourier multiplier on the Besov space B s p,q(T n , E) for s ∈ R, 1 < p < ∞, 1 ≤ q ≤ ∞ and E a Banach space, if and only if E is a UMDspace. This extends the Theorem 4.2 in [3] to then−dimensional case. As illustration of the applicability of this results we study the solvability of two abstract Cauchy problems with periodic boundary conditions. [ABSTRACT FROM AUTHOR]