Global existence of smooth shearing motions of nonlinear viscoelastic fluid

Weak solutions of hyperbolic-parabolic Volterra equations

“…The same results hold for ( 1 ) at least in the case where k' is locally of bounded variation on R+, see [27] for the global existence of a weak solution and see [2,9,15,26] for the breakdown of smooth solutions. The diffusion equation (that corresponds to the completely viscous case) has always (under reasonable assumptions on a ) a strong solution and using the maximum principle it is easy to show that ux remains bounded.…”

“…The purpose of this paper is to establish the existence of a global weak solution of the equation (1) ut(t,x)= Í k(t-s)(o(ux)) (s,x)ds + f(t, (where the subscripts denote differentiation with respect to the corresponding variables), and to show that this weak solution satisfies certain a priori L°°-bounds. Equation (1) is of interest because it appears in mathematical models of viscoelasticity (see [18 and 31]), but also because it is of a form somewhere between a parabolic and a hyperbolic equation.…”

“…In [5 and 22] weak solutions of equations that contain (1) as a special case are studied. For equation (1) it is required that k(0) = 1 and k'(0+) = -co and less is assumed of the nonlinearity a than in this paper. In [23] a weak solution is found when k satisfies a (rather complicated but not very restrictive) frequency domain condition and there is no requirement that k(0) = 1.…”

“…There is a wealth of results in this direction, see, e.g., [1,3,17,20,21,24,25,29,30,32]. In many of these papers the equation studied is u"(t,x) = tj)(ux(t,x))x+ / a(t-s)y/(ux(s,x))xds + g(t,x), which is the same as (1) if </> = y/ = o, k(0) = 1, and a = k'.…”

Some Mathematical Problems in Continuum Mechanics

“…We treat the case when the body is bounded with pinned and insulated ends as well as the case when the body occupies the entire real line. 4. By adapting some of the ideas developed in Brandon's thesis, we establish existence of smooth shearing flows of viscoelastic fluids of the K-BKZ type with smooth and small initial data.…”