Attractors for a model of polymer motion with objective derivative in the rheological relation
The author sudies the problem on weak solutions of the system \[ \frac{\partial v}{\partial t}-\nu \Delta v+\sum\limits_{i=1}^nv_i\frac{\partial v}{\partial x_i}-\kappa \frac{\partial \triangle v}{\partial t}- 2\kappa \text{ div }(\sum\limits_{i=1}^nv_i\frac{\partial \mathcal{E}(v)}{\partial x_i})-2\kappa \text{ div }(\mathcal{E}(v)W_{\rho}(v) -W_{\rho}(v)\mathcal{E}(v))+ \text{grad }p=f,\tag{1} \] \((x,t)\in \Omega\times (0, \infty); \text{div} v=0, (x,t)\in \Omega\times (0, \infty); v(x,t)=0 (x,t)\in \Omega\times (0, \infty); v(x,0)=a, (x,t)\in \Omega,\) describing the motion of a homogeneous incompressible polymer of constant density in the rheological relation with objective drivative. The main results are contained in the following theorems: {Theorem 1.} For any \(a\in V^1\) system (1) has a solution on \(\mathbb{R}_+\) that satisfies the inequality \[ \|v(t)\|_1+\|v'(t)\|_{-1}\leq R(1+\|a\|^2_1e^{-at})\;\; \text{for}\; \text{a.e.}\; t\geq 0 \] with constants \(R > 0\) and \(\alpha > 0\) independent of \(v\). {Theorem 2. } Let system (1) be autonomous, \(f \in L_2(\Omega)^n\), and \(0 < \delta < 1\). Then the trajectory space \(\mathcal{H}^+\) has a minimal trajectory attractor \(\mathcal{U}\). The attractor is bounded in \(L_{\infty}(\mathbb{R}^+;V^1)\) and compact in \(C(\mathbb{R}^+;V^{1-\delta})\). In the topology of the space \(C(\mathbb{R}^+;V^{1-\delta})\), it attracts families of trajectories that are bounded in \(L_{\infty}(\mathbb{R}^+;V^1)\). {Theorem 3.} Let the conditions of Theorem 2 hold. Then the trajectory space \(\mathcal{H}^+\) has a global attractor \(\mathcal{A}\). The attractor is bounded in \(V^1\) and compact in \(V^{1-\delta}\). In the topology of the space \(V^{1-\delta}\) it attracts families of trajectories that are bounded in \(L_{\infty}(\mathbb{R}^+;V^1)\).
scientific article; zbMATH DE number 6336247