We consider entire solutions u of the equations describing the stationary flow of a generalized Newtonian fluid in 2 D concentrating on the question, if a Liouville-type result holds in the sense that the boundedness ...We consider entire solutions u of the equations describing the stationary flow of a generalized Newtonian fluid in 2 D concentrating on the question, if a Liouville-type result holds in the sense that the boundedness of u implies its constancy. A positive answer is true for p-fluids in the case p > 1(including the classical Navier-Stokes system for the choice p = 2), and recently we established this Liouville property for the Prandtl-Eyring fluid model,for which the dissipative potential has nearly linear growth. Here we finally discuss the case of perfectly plastic fluids whose flow is governed by a von Mises-type stress-strain relation formally corresponding to the case p = 1. It turns out that, for dissipative potentials of linear growth, the condition of μ-ellipticity with exponent μ < 2 is sufficient for proving the Liouville theorem.展开更多
文摘We consider entire solutions u of the equations describing the stationary flow of a generalized Newtonian fluid in 2 D concentrating on the question, if a Liouville-type result holds in the sense that the boundedness of u implies its constancy. A positive answer is true for p-fluids in the case p > 1(including the classical Navier-Stokes system for the choice p = 2), and recently we established this Liouville property for the Prandtl-Eyring fluid model,for which the dissipative potential has nearly linear growth. Here we finally discuss the case of perfectly plastic fluids whose flow is governed by a von Mises-type stress-strain relation formally corresponding to the case p = 1. It turns out that, for dissipative potentials of linear growth, the condition of μ-ellipticity with exponent μ < 2 is sufficient for proving the Liouville theorem.
