Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids

摘要

The paper concerns existence of weak solutions to the equations describing a motion of some non-Newtonian fluids with non-standard growth conditions of the Cauchy stress tensor. Motivated by the fluids of strongly inhomogeneous behavior and having the property of rapid shear thickening, we observe that the L-p framework is not suitable to capture the described situation. We describe the growth conditions with the help of general x-dependent convex function. This formulation yields the existence of solutions in generalized Orlicz spaces. As examples of motivation for considering non-Newtonian fluids in such spaces, we recall the electrorheological fluids, magnetorheological fluids, and shear thickening fluids. The existence of solutions is established by the generalization of the classical Minty method to non-reflexive spaces. The result holds under the assumption that the lowest growth of the Cauchy stress is greater than the critical exponent q = (3d + 2)/(d + 2), where d is for space dimension. The restriction on the exponent q is forced by the convective term.