Viscous Multipolar Fluids-Physical Background and Mathematical Theory-

Abstract

We introduce the theory of multipolar fluids in which constitutive laws depend linearly not only on the first spatial gradients of velocity as in classical Navier-Stokes theory of newtonian fluids but also on its higher order spatial gradients up to the order 2k − 1, k = 2, 3,… Such fluids are called k-polar fluids. A thermodynamic theory of the constitutive equations satisfying the second law of thermodynamics and the principle of material frame indifference is developed. Special thermodynamic processes as isothermal, barotropic, adiabatic and general heat-conductive motion for compressible multipolar fluids are studied. It is well known that there does not exist adequate existence theory for compressible newtonian fluids. We given a consistent theory for compressible multipolar fluids in two or three dimensions, i. e. we prove the global in time existence of weak solutions for the initial boundary value problems in bounded domains for the systems of partial differential equations describing isothermal, barotropic, adiabatic and general compressible motion. Under some assumptions on the regularity of the initial data and external forces, we prove existence of strong solutions, uniqueness and regularity. Some other properties as e. g. cavitation of density are discussed. We put stress on the lowest possible polarity of the fluid. In the isothermal case we consider the polarity k ≧ 2 and in barotropic and heat-conductive gas the polarity k ≧ 3.