Well posedness of nonlinear parabolic systems beyond duality

Abstract

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial–boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system∂tu−div(ν(|∇u|)∇u)=−divf with a given strictly positive bounded function ν, such that limk→∞⁡ν(k)=ν∞ and f∈Lq with q∈(1,∞). The existence, uniqueness and regularity results for q≥2 are by now standard. However, even if a priori estimates are available, the existence in case q∈(1,2) was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range q∈(1,∞).Furthermore, our paper includes several new results that may be of independent interest and serve as the starting point for further analysis of more complicated problems. They include a parabolic Lipschitz approximation method in weighted spaces with fine control of the time derivative and a theory for linear parabolic systems with right hand sides belonging to Muckenhoupt weighted Lq spaces.