Certain aspects of applying Newton's method to the Navier-Stokes equations

New effective method for inverting the Fŕechet derivative of nonlinear mapping corresponding to the initial-boundary value problem for Navier-Stokes equations has been demonstrated. The basic difficulty in the practical use of Newton's method is associated with the inversion of the Fŕechet derivative. The method of inverting Fŕechet derivative can be applied not only to the Navier-Stokes equations but also to a wide class of quasilinear evolution problems with analytical nonlinearities. The initial approximation of Newton method can be preferred on any prescribed time interval (O,T) without dividing (O,T) into small subintervals, since the division procedure leads to error accumulation for sufficiently large T. The efficiency of the Vanhorn regularization is explained by the inheritance of the analytical dependence of solutions to the Navier-Stokes equations on the data of the problem.