On limit the behavior of solutions of slightly compressible generalized Newtonian liquid equations with two small parameters (Q1382595)

The authors consider the system of nonlinear equations \[ \text{div }(\chi(| \nabla u|)\nabla u) + \nabla p = f,\qquad \text{div }u = 0,\quad u| _{\partial\Omega} = 0,\quad \Omega\subset \mathbb{R}^3,\tag{1} \] where \(\partial\Omega\) is a piece-wise Lipschitzian boundary, and the system \[ -\text{div }((1+k_\delta(| \nabla u_\delta^\varepsilon|))\nabla u_\delta^\varepsilon) + \nabla p_\delta^\varepsilon = f,\qquad \text{div }u_\delta^\varepsilon + \varepsilon p_\delta^\varepsilon = 0,\quad u_\delta^\varepsilon| _{\partial\Omega} = 0.\tag{2} \] Let \(u_\delta^\varepsilon\) be a generalized solution of problem (2) and \(u\) be a generalized solution of problem (1). The authors prove that \(\| u_\delta^\varepsilon - u\| \to 0\) as \(\delta\to 0\) and \(\varepsilon\to 0\).