Global solutions of the Navier-Stokes equations in thin three-dimensional domains
The Navier-Stokes equations \[ \frac{\partial v}{\partial t}-\nu\Delta v+v\cdot\nabla v +\nabla p=f,\quad \nabla\cdot v=0 \tag{1} \] are considered in 3D-domain \(\Omega_\varepsilon=(0,l_1)\times (0,l_2) \times (0,\varepsilon)\). The equations (1) are associated with the periodic boundary conditions \[ v(t,0,x_2,x_3)=v(t,l_1,x_2,x_3),\quad v(t,x_1,0,x_3)=v(t,x_1,l_2,x_3),\quad v(t,x_1,x_2,0)=v(t,l_1,x_2,\varepsilon) \tag{2} \] and initial condition \[ v(0,x)=v_0(x).\tag{3} \] The additional average condition \[ \int_{\Omega_\varepsilon}v(t,x) dx=0 \tag{4} \] is required. The author describes the set of data \((v_0,f)\) for which the problem (1), (2), (3), (4) has a global regular solution. The global existence conditions intend in particular that \(\varepsilon\) is small enough.
scientific article; zbMATH DE number 1295272