A Cahn-Hilliard-Navier-Stokes model with delays (Q325783)

A 2D non-autonomous Cahn-Hilliard-Navier-Stokes system is studied in the paper. NEWLINE\[NEWLINE\begin{cases} \frac{\partial v}{\partial t}-\nu\Delta v+(v\cdot\nabla) v+\nabla p-K\mu\nabla\phi= Q(t-\tau(t),(v,\phi)(t-\tau(t))),\\ \operatorname{div} v=0,\\ \frac{\partial \phi}{\partial t}+v\cdot\nabla\phi-\Delta\mu=0,\quad \mu=-\epsilon\Delta\phi+\alpha f(\phi) \end{cases}\tag{1}NEWLINE\]NEWLINE Here, \(x\in\Omega\subset\mathbb{R}^2\), \(t\in(0,\infty+)\), \(\Omega\) is a bounded domain occupied by the fluid, \(v=(v_1,v_2)\) is the velocity of the fluid, \(p\) is the pressure, \(\phi\) is the order (phase) parameter. \(v,p,\phi\) are unknown functions. \(\nu,\epsilon,K,\alpha\) are positive constants. \(\nu\) is the kinematic viscosity coefficient, \(K\) is the capillary coefficient, \(\epsilon\) and \(\alpha\) are two physical parameters. \(f\), \(Q\) and \(\tau\) are given functions. The differentiable function \(\tau\) describes the delay. NEWLINE\[NEWLINE \tau:[0,\infty)\;\rightarrow\;[0,r]\quad \text{for}\;r=\mathrm{const}>0. NEWLINE\]NEWLINENEWLINENEWLINEThe system (1) is added by the initial and boundary conditions NEWLINE\[NEWLINE(v,\phi)(s)=\theta(s)=(\theta_1,\theta_2)(s),\quad s\in[-r,0],\tag{2} NEWLINE\]NEWLINE NEWLINE\[NEWLINEv=0,\quad \frac{\partial \phi}{\partial n}=\frac{\partial \mu}{\partial n}=0 \quad \text{on}\;\partial\Omega,\tag{3}NEWLINE\]NEWLINE where \(n\) is the outward normal to \(\partial\Omega\).NEWLINENEWLINEIt is proved that the problem (1), (2), (3) has a unique weak solution on a finite time interval. This solution is strong for the smooth initial data. The exponential behavior of weak solution is discussed. The stability of stationary solutions of the problem is studied in the last part of the article.NEWLINENEWLINEI must remark that the strange numbering of the formulas hampers reading.