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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.