Weak Solutions for a Class of Non-Newtonian Fluids with Energy Transfer

On non-Newtonian Fluids with Energy Transfer

“…a flow around hot profiles, see experiments in [10]. This paper improves in several directions the recent results from [5], see Remark 2.1. Moreover, the technique used here to show the existence of a solution, based on Galerkin approximation and a subsequent limit passage through a "semiGalerkin" approximation (like in [4]), allows for weaker data qualification and yields a shorter and more constructive proof which avoids a Schauder fixed-point argument and which suggests conceptually a convergent numerical procedure.…”

“…A system (2.1) with a general heat flux j(θ, ∇θ) instead of κ(θ)∇θ was considered in [5]. Assuming monotonicity and q-polynomial structure of j(θ, ·) : R 3 → R 3 , existence of a weak solution was proved for p ≥ 5/2 and q ≥ 10p/(5p−1) > 2.…”

Planar flows of incompressible heat-conducting shear-thinning fluids—Existence analysis

“…In the second, called supercritical, the velocity cannot be used as a test function. While the subcritical cases were studied by many authors, we refer to [11], [10], and the existence theory can be established with help of the Minty method, the supercritical cases have been solved very recently. First such result for Newtonian fluid in spatially periodic setting was established in [12] and then extended to Navier's boundary conditions in [5].…”

On the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem

“…A similar modeling, using a variable exponent structure, is also proposed in [34]. Other interesting models are proposed, for instance, in [24] and [8]- [10].…”