Well-posedness for the Navier–Stokes Equations

where u is the velocity and p is the pressure. It is well known that the NavierStokes equations are locally well-posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at ∞. For instance it is easy to see (see [9] for much more general results) that if s > n 2 then for any H initial data there exists a unique C([0, t];H(R)) local solution with a pressure p ∈ C([0, t];H(R)). In the sequel we consider solutions for less regular initial data. This has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces. The question we are interested in is the global well-posedness for small data and local well-posedness for large data, with respect to a certain space of