A hereditary partial differential equation with applications in the theory of simple fluids

Abstract

The linearized boundary-initial history value problem for simple fluids obeying the Coleman-Noll constitutive equation

$$S + p\delta = 2\int\limits_0^\infty {m(s)(E(t - s} ) - E(t))ds$$

is considered. Here S is the stress tensor, δ the Kronecker delta, p the constitutively indeterminate mean normal stress, E the infinitesimal strain tensor, and m(s) a material function. The shear relaxation modulus G is defined as

$$G(s) = \int\limits_\infty ^s {m(\xi )d\xi .}$$

((i))

In this paper it is shown that if G satisfies the assumptions

$$G \in C^2 [0,\infty ),{\text{ }}G(s) \to 0{\text{ as }}s \to \infty,$$

((i))

$$( - 1)^k \frac{{d^k G(s)}}{{ds^k }} > 0,{\text{ }}k = 0,1,$$

((ii))

$$G''(s) \geqq 0,$$

((iii))

then the rest state of the fluid is stable in an appropriate “fading memory” norm. The additional assumption

$$ - \int\limits_0^\infty {G'} (s)s^2 ds < \infty$$

((iv))

yields asymptotic stability.