Interior differentiability of weak solutions to parabolic systems with quadratic growth nonlinearities (Q1378848)

Let \(\Omega\subset \mathbb{R}^n\) \((n\geq 2)\) be a bounded open set, and \(0<T<\infty\). In the cylinder \(Q= \Omega\times (0,T)\), the authors consider the system of nonlinear PDE's: \[ u^i_t- \sum^n_{\alpha= 1} D_\alpha A^\alpha_i(\nabla u)= B_i(\nabla u)\quad (i= 1,\dots, N).\tag{1} \] The authors prove the theorem on interior differentiability of weak solutions of (1): Let \(u=(u^1,\dots, u^N)\), \(u^j\in W^{1,0}_2(Q)\cap C^\gamma(\overline Q)\) \((0<\gamma< 1; j=1,\dots, N)\) be a weak solution of (1). Then for any \(\Omega'\subset\subset\Omega\) and \(0< t_0< t_1<T: \nabla u^j\in (L^\sigma(\Omega'\times (t_0,t_1))^n\) \((\sigma\in[4,4(1+ {\gamma\over n})[, j=1,\dots, N)\); \(\nabla u^j\in (L^4(t_0, t_1; (L^s(\Omega))^N\) \((s\in [4,{4n\over n-2\gamma}[, j=1,\dots, N)\), where \(L^4(t_0, t_1; L^s(\Omega))\) denotes the vector space of all Bochner measurable functions \(v:(t_0, t_1)\to L^s(\Omega)\) such that \(\| v\|_{L^4(t_0, t_1; L^s(\Omega))}= \left(\int^{t_1}_{t_0}\| v(t)\|^4_{L^s(\Omega)}dt\right)^{1/4}< \infty\).