Type: Article
Publication Date: 1982-01-01
Citations: 51
DOI: https://doi.org/10.1090/qam/666668
In this paper we study the nonexistence of global smooth solutions of one-dimensional motions for nonlinear viscoelastic fluids and solids by the method of Rozhdestvenskii [1], This method has been applied to prove the nonexistence of global smooth solutions for the shearing motions in an elastic circular tube in [2].It is well known that the quasilinear hyperbolic equationexhibits the breakdown of smooth solutions in finite time for a certain class of initial data of arbitrary smoothness, no matter how small.This breakdown of smooth solutions is usually associated with the formation of a propagating singular surface often called a Shockwave.The absence of some dissipative or damping mechanism in the above equation causes this rather unrealistic result.Nishida [3] and Slemrod [4] have studied the equation v" = o<vx)x -ocv, (0.2)which includes the effect of first-order linear damping which is not present in (0.1).For (0.2) Nishida showed the existence of a global smooth solution for the small initial data.Slemrod showed the breakdown of smooth solutions for large initial data.His motivation for studying (0.2) was based on his model equation for shearing perturbations of steady shearing flows in a nonlinear, isotropic, incompressible, viscoelastic fluid, in the absence of an applied driving force.In experiments the analysis of the plane Poiseuille flow is more common.In Sec.11 shall discuss the plane Poiseuille flow of the above fluid.MacCamy [6] considered the equation v" = a(0)ff(vx)x + a(t -z)a(vx)x dz+f (0.3)showed the existence of a global smooth solution for small initial data, and conjectured the breakdown of smooth solutions for large initial data.The effect of fading memory for elastic materials causing a dissipative mechanism is included in this model as the stress functional in the stress-strain relation.I shall show the breakdown of smooth solutions in this problem in Sec. 2.