Nonlinear systems of the type of the stationary Navier- Stokes system

The authors consider weak solutions u = (u1, u2, · · · , un) 2 H1( !Rn) of nonlinear equations in the following form: DA i (x, u,ru)+Bi(x, u,ru) = −(rp)i (i = 1, · · · , n) and r · u = g, where g is suitably smooth and p 2 L2( ! R). The matrix A i satisfies an ellipticity condition, and growth conditions for A i and Bi with respect to u and |ru| are given. Regularity is considered for the linear, semilinear, quasilinear, and nonlinear cases with decreasing results as the equation becomes more difficult. The known results for the Navier-Stokes equations when n >=4 are derived in a new fashion.