Hölder continuity of the solutions of a nondiagonal parabolic system of equations with \(p\)-Laplacians (Q610371)

The author deals with the following nonlinear parabolic system in divergence form with leading part modeled by \(p\)-Laplacians: \[ \begin{aligned} u_t &=a_1 \operatorname{div}\big(\big| \nabla u\big| ^{p-2}\nabla u\big) +b_1 \operatorname{div}\big(\big| \nabla v\big| ^{p-2}\nabla v\big) +f_1, \\ v_t &=a_2 \operatorname{div}\big(\big| \nabla u\big| ^{p-2}\nabla u\big) +b_2 \operatorname{div}\big(\big| \nabla v\big|^{p-2}\nabla v\big) +f_2, \end{aligned} \] \((t,x) \in(0,T] \times \Omega\), \(\Omega\) is a bounded domain in \(\mathbb R^n\) (\(n\) is an arbitrary positive integer), \(n>p\geq 2\). Such a system can describe a two-dimensional flow of a non-Newtonian fluid. For the solutions, the author proves the Hölder continuity. The method permits to estimate not only the Hölder norm but also the Hölder exponent.