Existence of solutions with the prescribed flux of the Navier-Stokes system in an infinite cylinder
The nonstationary analog to the Poiseuille flow is studied in the infinite cylinder \(\Pi=\{x\in \mathbb R^3:\, x'=(x_1,x_2)\in \sigma\subset \mathbb R^2,\,x_3\in \mathbb R\}\). The vector of velocity \(v(x,t)\) and the pressure \(p(x,t)\) satisfy the equations \[ \begin{aligned} &\frac{\partial v}{\partial t}-\nu\Delta v+(v\cdot\nabla)v+\nabla p=f, \quad \text{div}\,v=0,\\ &v| _{t=0}=v_0(x),\quad v(x,t)| _{\partial\sigma}=0,\\ &\int\limits_{\sigma}v_3(x,t)\,dx'=F(t). \end{aligned} \] Here \(F\) is the given flux. It is proposed that the data \(f\) and \(v_0\) do not depend on \(x_3\). The author proves that the problem has a unique solution with the same property. The Galerkin approximations are used for the construction of the solution.
scientific article; zbMATH DE number 5126602