A priori estimates for quasilinear parabolic systems with quadratic nonlinearities in the gradient

summary:We derive local a priori estimates of the Hölder norm of solutions to quasilinear elliptic systems with quadratic nonlinearities in the gradient. We assume higher integrability of solutions and smallness of its BMO norm but the Hölder norm is estimated in terms of BMO norm of the solution under consideration, only

Solvability problem for strong-nonlinear nondiagonal parabolic system

summary:A class of $q$-nonlinear parabolic systems with a nondiagonal principal matrix and strong nonlinearities in the gradient is considered.We discuss the global in time solvability results of the classical initial boundary value problems in the case of two spatial variables. The systems with nonlinearities $q\in (1,2)$, $q=2$, $q>2$, are analyzed

Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices

summary:Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A(t)$-caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems