On a \(\nu\)-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-type boundary condition
The viscosity limit of the Navier-Stokes equations is studied. Let \((u,p)\) be a solution to the problem \[ \begin{aligned} &\frac{\partial u}{\partial t}+(u\cdot\nabla)u-\nu\Delta u+\nabla p=f, \quad \text{div}\,u=0,\quad x\in \Omega,\;0<t<T\\ &u(x,0)=u_0(x)\quad x\in \Omega,\\ &u\cdot n=0,\quad \text{curl}\,u \times n=\text{curl}\,(v+\nu w) \times n, \quad x\in \partial\Omega,\;0<t<T, \end{aligned}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb R^3\) with a smooth boundary. Let \((v,q)\) be a solution to the initial-boundary value problem for Euler equations \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla q=f, \quad \text{div}\,v=0,\quad x\in \Omega,\;0<t<T\\ &v(x,0)=u_0(x),\quad x\in \Omega,\\ &v\cdot n=0,\quad x\in \partial\Omega,\;0<t<T. \end{aligned}\tag{2} \] Let \((w,r)\) be a solution to the linear problem \[ \begin{aligned} &\frac{\partial w}{\partial t}+(v\cdot\nabla)w+(w\cdot\nabla)v+\nabla r=\Delta v, \quad \text{div}\,w=0,\quad x\in \Omega,\;0<t<T\\ &w(x,0)=0,\quad x\in \Omega,\\ &w\cdot n=0,\quad x\in \partial\Omega,\;0<t<T. \end{aligned}\tag{3} \] The authors construct a family of strong solutions \(\{u^\nu\}\) of the problem (1) on the time interval \((0,T)\), depending continuously on the viscosity coefficient \(\nu\) for \(0\leq \nu<\nu^*\). \(\{u^\nu\}\) tends to the solution \(v\) of the problem (2) for \(\nu\rightarrow 0+\).
scientific article; zbMATH DE number 5725438