Generalized Gagliardo-Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems (Q2481085)

Let \(\Omega\) be an open bounded subset of \(\mathbb R^n\), \(n>2\) and \(Q=\Omega\times (-T,0)\), \(0<T<\infty.\) As in several previous articles of the authors, the paper is concerned with the regularity of weak, \(\mathbb R^{N}\)-valued solutions \(u\) to the parabolic system \[ -\sum_{i=1}^{n}D_ia^{i}(X,u,Du)+\dfrac{\partial u}{\partial t}=B^{o}(X,u,Du) \quad\text{in } Q. \] Here the coefficients \(a^{i}\) are assumed to be uniformly monotone in \(Du\), and the local regularity of \(u\) is studied. Let \(B(\alpha)\) denote the cube \(\{x\in \mathbb R^{n}\); \(| x_{i}-x_{i}^{o}| <\alpha\), \(i=1,\dots,n\}\), \(\alpha>0\). The main result asserts that \(u\in L^{2}(-a,0,H^{2} (B(\sigma),\mathbb R^{N}))\) for every \(a\in (0,T)\), and each cube \(B(\sigma)\subset B(3\sigma)\subset\subset\Omega.\) The proof is based on related results due to \textit{L. Fattorusso} [Rend. Circ. Mat. Palermo, (2) 39 No.~3, 412--426 (1990; Zbl 0733.35026) and Boll. Un. Mat. Ital. B (7) 1, No.~3, 741--764 (1987; Zbl 0656.35061)], and a Gagliardo-Nirenberg estimate.