Well-posedness and ill-posedness of the stationary Navier–Stokes equations in toroidal Besov spaces. (5th September 2019)

• Record Type: Journal Article
• Authors: Tsurumi, Hiroyuki
• Is Part Of: Nonlinearity. Volume 32:Number 10(2019)
• Page Start: 3798
• Page End: 3819
• Publication Date: 2019-09-05
• Subjects: stationary Navier–Stokes equations -- toroidal Besov spaces -- well-posedness -- ill-posedness; 35Q30 -- 35R25 -- 42B37; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab22d3
• Languages: English
• ISSNs: 0951-7715

Abstract

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.