An application of Orlicz spaces in partial differential equations

Our purpose is to investigate mathematical properties of some systems of nonlinear partial differential equations where the nonlinear term is monotone and its behaviour - coercivity/growth conditions are given with the help of some general convex function defining Orlicz spaces. Our first result is the existence of weak solutions to unsteady flows of non-Newtonian incompressible nonhomogeneous (with non-constant density) fluids with nonstandard growth conditions of the stress tensor. We are motivated by the problem of anisotropic behaviour of fluids which are also characterised by rapid shear thickness. Since we are interested in flows with the rheology more general than power-law-type, we describe the growth conditions with the help of an x–dependent convex function and formulate our problem in generalized Orlicz (Musielak-Orlicz) spaces. As a second result we give a proof of the existence of weak solutions to the problem of the motion of one or several nonhomogenous rigid bodies immersed in a homogenous non-Newtonian fluid. The nonlinear viscous term in the equation is described with the help of a general convex function defining isotropic Orlicz spaces. The main ingredient of the proof is convergence of the nonlinear term achieved with the help of the pressure localisation method. The third result concerns the existence of weak solutions to the generalized Stokes system with the nonlinear term having growth conditions prescribed by an anisotropic N -function. Our main interest is directed to relaxing the assumptions on the N- function and in particular to capture the shear thinning fluids with rheology close to linear. Additionally, for the purpose of the existence proof, a version of the Sobolev-Korn inequality in Orlicz spaces is proved. Last but not least, we study also a general class of nonlinear elliptic problems, where the given right-hand side belongs only to the L1 space. Moreover the vector field is monotone with respect to the second variable and satisfies a non-standard growth condition described by an x-dependent convex function that generalizes both Lp(x) and classical Orlicz settings. Using truncation techniques and a generalized Minty method in the functional setting of non reflexive spaces we prove existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established. Sufficient conditions are specified which guarantee that the renormalized solution is already a weak solution to the problem.