Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy (Q2838080)

Let \(Q=\Omega\times (0,T)\), where \(\Omega\) is a bounded domain of \({\mathbb R}^n \), \(n \geq 1\). In this work, the author seek for everywhere regularity, in particular everywhere Hölder continuity, for systems of the type NEWLINE\[NEWLINE \;u_{t} =\mathrm{div} (A(x,t,u,Du)) + F(x,t,u,Du)\text{ in }Q,\tag{1} NEWLINE\]NEWLINE where the vector-valued unknown \(u\) take values in \({\mathbb R}^m \), \(m \geq 1\). They assume that \(m\) and \(n\) are arbitrary and study the case in which \(F\) depend only on \( u\), and \(Du\). They obtain the Hölder continuity for solutions of the system (1) by considering a family of nonlinear systems whose Hölder continuity of solutions is known such that is continuously homotopic to the system (1) and so carrying the same regularity over the system (1). In this homotopy, the main assumption is a BMO maximum principle, that instead of \( L^{\infty}\) maximum principle, the authors studied in the previous work.