Global well-posedness for the incompressible Navier–Stokes equations in the critical Besov space under the Lagrangian coordinates

We consider global well-posedness of the Cauchy problem of the incompressible Navier–Stokes equations under the Lagrangian coordinates in scaling critical Besov spaces. We prove the system is globally well-posed in the homogeneous Besov space B˙_p,1^−1+n/p(Rⁿ) with 1≤p<∞. The former result was restricted for 1≤p<2n and the main reason why the well-posedness space is enlarged is that the quasi-linear part of the system has a special feature called a multiple divergence structure and the bilinear estimate for the nonlinear terms are improved by such a structure. Our result indicates that the Navier–Stokes equations can be transferred from the Eulerian coordinates to the Lagrangian coordinates even for the solution in the limiting critical Besov spaces.