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New a priori estimates for nondiagonal strongly nonlinear parabolic systems. (English) Zbl 1156.35336
Rencławowicz, Joanna (ed.) et al., Parabolic and Navier-Stokes equations. Part 1. Proceedings of the confererence, Bȩdlewo, Poland, September 10–17, 2006. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 81, Pt. 1, 13-30 (2008).
The author studies Cauchy-Dirichlet problem for nondiagonal parabolic systems in divergence form with quadratic growth in the gradient. The coefficients are assumed to be only continuous and strongly parabolic. Under such assumption it is known that in general the solution is only partially regular and the \(n-2\) Hausdorff measure of the set of the singularities is finite.
In this paper the author gives a new description of regular points of the solution in terms of the local smallness of the Campanato seminorm of the solution; this is a difference with respect to the usual procedure that relies on the smallness of the oscillation of the function as usual. An essential tool for such a result is represented by suitable quasireverse Hölder inequalities. The author estimates the local Hölder norm of the solution up to the boundary.
For the entire collection see [Zbl 1147.35005].

MSC:
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K55 Nonlinear parabolic equations
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