A priori estimates in terms of the maximum norm for the solutions of the Navier-Stokes equations (Q1881140)

The Cauchy problem for the incompressible Navier-Stokes equations \[ u_t+ u\cdot\nabla u+\nabla p=\Delta u,\quad\nabla\cdot u= 0, \] with initial condition \[ u(x,0)= f(x),\quad x\in\mathbb{R}^3, \] is considered. There is assumed that \[ f\in L^\infty\quad\text{and}\quad \nabla\cdot f= 0\quad(\text{in the sense of distributions}). \] For \(f\in L^\infty\) the existence of a regular solution is known. A priori estimates of the maximum norm of the derivatives of \(u\) in terms of the maximum norm of the initial function \(f\) are given, assuming that the solution exists and is \(C^\infty\) for \(0< t< T(f)\). First, for illustration, a priori estimates for certain parabolic systems are given.