Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces

Abstract

This paper presents some progress toward an open question which proposed by Tsurumi (Arch. Ration. Mech. Anal. 234:2, 2019): whether or not the stationary Navier-Stokes equations in R d is well-posed from Ḃ−2 p,q to P Ḃ0 p,q with p = d and 1 ≤ q < 2. In this paper, we demonstrate that for the case 1 ≤ q < 2 the 4D stationary Navier-Stokes equations is ill-posed from Ḃ−2 4,q (R 4 ) to P Ḃ0 4,q (R 4 ) by showing that a sequence of external forces is constructed to show discontinuity of the solution map at zero. Indeed in such case of q, there exists a sequence of external forces which converges to zero in Ḃ−2 4,q and yields a sequence of solutions which does not converge to zero in Ḃ0 4,q .

Ill-posedness issue on a multidimensional chemotaxis equations in the critical Besov spaces

“…For more background of this model and related results, we refer to [12,13]. Many results with regard to the ill-posedness have been obtained for some important nonlinear PDEs including the incompressible Navier-Stokes equations [2,15,17], the stationary Navier-Stokes equations [11,14], the compressible Navier-Stokes equations [3,5] and so on.…”