A Liouville Theorem for Stationary Incompressible Fluids of Von Mises Type

Abstract

We consider entire solutions u of the equations describing the stationary flow of a generalized Newtonian fluid in 2D 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.

Liouville-type theorems for the stationary MHD equations in 2D

“…Finally, by taking R → ∞, we arrive at (7). Now we turn to the proof of (8). By Gagliardo-Nirenberg inequality, one notices that…”

“…Other types of Liouville properties for the stationary NS on the plane were also extensively studied, such as under the growth condition lim sup |x| −α |u(x)| < ∞ as |x| → ∞ for some α > 0, see [2,10]; existence and asymptotic behavior of solutions in an exterior domain, see [5,14,15,18,22,24,25]. For more references on Liouville theorems of (3), we refer to [6,8,11,16,28] and the references therein.…”

Liouville-type results in two dimensions for stationary points of functionals with linear growth

“…The validity of Liouville theorems has been established in the 2-D-case, i.e. for n = 2, for instance in the papers [3], [8], [9], [10], [11], [15], [21], [23], [29] and [30]. We like to mention that the case of potentials F satisfying (1.2) and (1.3) is treated in [10] assuming µ < 2.…”

Liouville-type results in two dimensions for stationary points of functionals with linear growth

“…The validity of Liouville theorems has been established in the 2-D-case, i.e. for n = 2, for instance in the papers [16], [17], [18], [19], [20], [21], [22], [23], [24] and [25]. We like to mention that the case of potentials F satisfying (1.2) and (1.3) is treated in [19] assuming µ < 2.…”