The \(\mathcal{A}\)-Stokes approximation for non-stationary problems (Q2814320)

The paper is concerning the approximation theory for unsteady problems in fluid mechanics. The A-caloric approximation is used: in the heat equation, the operator \(\Delta u \) is replaced by \( \operatorname{div}( A \nabla u) \) where \(A\) is a tensor with good enough properties. The main point is to show that almost solutions to the A-caloric problem in a \(L^p\) type space can be approximated by functions in a \(L^s\) type space, with \(s<p\). The previous result [the author et al., J. Differ. Equations 253, No. 6, 1910--1942 (2012; Zbl 1245.35080)] is extended to the parabolic problems. An \(L^q\) theory for the non-stationary A-Stokes problem is given. Important tools are the parabolic maximal operators (with the inequality (2.4)), the Hardy-Littlewood maximal inequality and the Bogovskii operator (related with the differential equation \(\operatorname{div} u = f\)).