We study the prolongation of semibasic projectable tangent valued $k$-forms on fibered manifolds with respect to a bundle functor $F$ on local isomorphisms that is based on the flow prolongation of vector fields and uses an auxiliary linear $r$-th order connection on the base manifold, where $r$ is the base order of $F$. We find a general condition under which the Frölicher-Nijenhuis bracket is preserved. Special attention is paid to the curvature of connections. The first order jet functor and the tangent functor are discussed in detail. Next we clarify how this prolongation procedure can be extended to arbitrary projectable tangent valued $k$-forms in the case $F$ is a fiber product preserving bundle functor on the category of fibered manifolds with $m$-dimensional bases and local diffeomorphisms as base maps.
Steady-state system of equations for incompressible, possibly non-Newtonean of the p-power type, viscous flow coupled with the heat equation is considered in a smooth bounded domain Ω ⊂ ℝn, n = 2 or 3, with heat sources allowed to have a natural L1- structure and even to be measures. The existence of a distributional solution is shown by a fixed-point technique for sufficiently small data if p > 3/2 (for n = 2) or if p > 9/5 (for n = 3).
Some criteria for weak compactness of set valued integrals are given. Also we show some applications to the study of multimeasures on Banach spaces with the Radon-Nikodym property.