Well-posedness and ill-posedness of the stationary Navier–Stokes equations in toroidal Besov spaces

Abstract

We consider the stationary Navier–Stokes equations in the n-dimensional torus for . We show the existence and uniqueness of solutions in homogeneous toroidal Besov spaces with for small external forces in . We can show its well-posedness by a similar method to that of Kaneko–Kozono–Shimizu (Indiana Univ. Math. J.), which has investigated the same problem in homogeneous Besov spaces on . Our advantage is to prove the ill-posedness in the critical exponents like p = n, and , . Indeed in such cases of p and q, there exists a sequence of external forces which converges to zero in and yields a sequence of solutions which does not converge to zero in . We can show this ill-posedness by constructing the sequence of external forces, as similar to those of initial data proposed by Yoneda (2010 J. Funct. Anal. 258 3376–87) in the non-stationary problem.