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.
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Fuchs, M., Müller, J. A Liouville Theorem for Stationary Incompressible Fluids of Von Mises Type. Acta Math Sci 39, 1–10 (2019). https://doi.org/10.1007/s10473-019-0101-1
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DOI: https://doi.org/10.1007/s10473-019-0101-1