Ill-posedness for the Navier-Stokes equations in critical Besov spaces B˙∞,q−1

We study the Cauchy problem for the incompressible Navier-Stokes equations in two and higher spatial dimensions(0.1)ut−Δu+u⋅∇u+∇p=0,divu=0,u(0,x)=δu0. For arbitrarily small δ>0, we show that the solution map δu0→u in critical Besov spaces B˙∞,q−1 (∀q∈[1,2]) is discontinuous at origin. It is known that the Navier-Stokes equation is globally well-posed for small data in BMO−1[20]. Taking notice of the embedding B˙∞,q−1⊂BMO−1 (q⩽2), we see that for sufficiently small δ>0, u0∈B˙∞,q−1 (q⩽2) can guarantee that (0.1) has a unique global solution in BMO−1, however, this solution is instable in B˙∞,q−1 and the solution can have an inflation in B˙∞,q−1 for certain initial data.