Global existence of smooth shearing motions of a nonlinear viscoelastic fluid
The aim is to establish global existence of smooth rectilinear shearing motions of incompressible nonlinear viscoelastic fluids of K-BKZ type. The equation of motion that we shall consider is \(u_{tt} (x, t)= \int^\infty_0 a'(s) g(u_x(x, t)- u_x(x, t- s))_x ds+ f(x, t)\), \(x\in B\), \(t\geq 0\), where subscripts \(x\) and \(t\) indicate partial derivatives and \(a'\) denotes the derivative of \(a\). Here the unknown \(u\) is a component of the displacement, \(f\) is a forcing function, \(a: [0, \infty)\to {\mathbf R}\) and \(g: {\mathbf R} \to {\mathbf R}\) are smooth constitutive functions, and \(B\subset {\mathbf R}\) is an interval.
scientific article; zbMATH DE number 784743