Weak solutions for an incompressible, generalized Newtonian fluid interacting with a linearly elastic Koiter type shell (Q2927864)

The mathematical model of the motion of a non-Newtonian fluid in a 3D domain bounded by a elastic shell is studied in the paper. The stress tensor of a fluid has a form NEWLINE\[NEWLINE S(D)=\mu (\delta+| D|)^{q-2}D,\quad D(v)=\frac{1}{2}(\nabla v+(\nabla v)^T), NEWLINE\]NEWLINE where \(v\) is the velocity of a fluid, \(\mu>0\), \(\delta\geq 0\), \(6/5<q<\infty\). A fluid occupies the time-depended domain \(\Omega_\eta\in\mathbb{R}^3\). The function \(\eta(x,t)\) defines a boundary of \(\Omega_\eta\). Let \(\partial\Omega\) is the middle surface of the elastic shell. It is proposed that the middle surface \(\partial\Omega\) consists of the fixed part \(\Gamma\) and variable part \(M=\partial\Omega\setminus\Gamma\). \(\eta:\, M\rightarrow\mathbb{R}\). The functions \(v\) and \(\eta\) are unknown and satisfy to the equationsNEWLINENEWLINENEWLINE\[NEWLINE \begin{cases} \frac{\partial v}{\partial t}+v\cdot\nabla v-\text{div}\,S(D) +\nabla p=f,\quad \text{div}\,v=0,\quad x\in\Omega_\eta,\;0<t<T, \\ v(x+\eta\nu,t)=\frac{\partial \eta}{\partial t}\nu,\quad x\in M,\;0<t<T, \\ v=0,\quad x\in \Gamma,\;0<t<T, \\ \eta(x,0)=\eta_0,\quad \frac{\partial \eta}{\partial t}(x,0)=\eta_1,\quad x\in M,\quad v(x,0)=v_0\quad x\in\Omega_{\eta_0}. \end{cases} NEWLINE\]NEWLINE Here, \(f\), \(v_0\), \(\eta_0\), \(\eta_1\) are the given functions, \(\nu\) is the unit normal vector. The function \(\eta\) satisfies to a special conditions of elasticity additionally.NEWLINENEWLINEIt is proved that the problem has a weak solution \((v,\eta)\) on the time interval \((0,T)\). \(T=\infty\) if the data of the problem are sufficiently small.