Proceedings of the Royal Society of Edinburgh, 115A, 39-59, 1990 Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid John A. Nohel 1 Robert L. Pego 2 and Athanasios E. Tzavaras 3 Center for the Mathematical Sciences, University of Wisconsin-Madison, Madison, WI 53705, U.S.A. (MS received 28 July 1989) Dedicated to Professor Jack K. Hale on the occasion of his 60th birthday Synopsis We study the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain-rate that results in steady states having, in general, discontinuities in the strain rate. We show that every solution tends to a steady state as <—»°°, and we identify steady states that are stable. 1. Formulation and discussion of model problem We study the quasilinear system v, = S x , (1.1) o, + o = g(v x ), (1.2) on [0, l ] x [ 0 , oo), where S:=T+fx, T:=a + v x , (1.3) with / a fixed positive constant throughout. We impose the boundary conditions 5(0,0 = 0 and v(l,t) = 0, fSO, (1.4) and the initial conditions v(x,0) = v o (x), o(x,0) = o 0 (x), Oijtgl; (1.5) accordingly, S(x, 0) = 5 0 (JC) : = o 0 (x) + v Ox (x) +fx. (1.6) The function g: U -»IR is assumed to be smooth, odd, and §g(§) > 0, for £ ¥= 0. 1 Also Department of Mathematics. 2 Department of Mathematics, University of Michigan. 3 Also Department of Mathematics.