On the Oldroyd model of a viscoelastic fluid
The author studies the following initial-boundary value problem: \(v_t(t, x)+ D(v)- \gamma_1\Delta v(t,x)- \gamma_2\Delta \int^t_0\exp(\alpha(s- t)) v(s, x) ds+ \nabla p(t,x)= f(t,x)\), \(\text{div }v(t, x)= 0\), \((t,x)\in Q\), \(\int_\Omega p(t,x) dx= 0\), \(t\geq 0\), \(v(0,x)= v_0(x)\), \(x\in\Omega\), \(v(t,x)= 0\), \(t\geq 0\), \(x\in\partial\Omega\), where \(v(t,x)\) is the velocity, \(p(t,x)\) is the pressure, \(x\in\mathbb{R}^n\), \(\gamma_1> 0\), \(\gamma_2\), \(\alpha\in\mathbb{R}\), \(D(v)= \sum^n_{k=1} v_k(\partial v/\partial x_k)\), \(Q= [0,\infty)\times \Omega\), \(\Omega\in\mathbb{R}^n\) is a bounded domain, \(n\geq 2\), and \(\partial\Omega\) is the boundary of \(\Omega\) of class \(C^2\). This problem describes the motion of viscoelastic Oldroyd fluid. The author studies the solvability in Sobolev classes under necessary conditions for small initial and boundary data for arbitrary \(n\geq 2\). Thus the paper generalizes some previous results obtained for \(n= 2,3\). The main result of the paper establishes conditions under which the problem has a unique solution. The proof consists in reducing the problem to a sequence of auxiliary problems, the solutions of which can be easily found. Additionally, the author describes some properties of the solution obtained.
scientific article; zbMATH DE number 1407620