The two-dimensional stationary Navier-Stokes equations in toroidal Besov spaces

We consider the stationary Navier-Stokes equations in the two-dimensional torus T2$\mathbb {T}<^>2$. For any epsilon>0$\varepsilon >0$, we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces B?p+epsilon,q-1+2p(T2)$\dot{B}<^>{-1+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}<^>2)$ for given small external forces in B?p+epsilon,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 epsilon 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)is an element of({2}x(2,infinity])?((2,infinity]x[1,infinity])$(p,q)\in (\lbrace 2\rbrace \times (2,\infty ])\cup ((2,\infty ]\times [1,\infty ])$, 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 B?p,q-3+2p(T2)$\dot{B}<^>{-3+\frac{2}{p}}_{p ,q}(\mathbb {T}<^>2)$ to B?p,q-1+2p(T2)$\dot{B}<^>{-1+\frac{2}{p}}_{p, q}(\mathbb {T}<^>2)$.