Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (1): 1-10.doi: 10.1007/s10473-019-0101-1

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A LIOUVILLE THEOREM FOR STATIONARY INCOMPRESSIBLE FLUIDS OF VON MISES TYPE

Martin FUCHS, Jan MÜLLER   

  1. Fachbereich 6.1 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken, Germany
  • Received:2018-05-26 Online:2019-02-25 Published:2019-03-13
  • Contact: Martin FUCHS E-mail:fuchs@math.uni-sb.de

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.

Key words: generalized Newtonian fluids, perfectly plastic fluids, von Mises flow, Liouville theorem

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