Structure of solutions of 3D axisymmetric Navier-Stokes equations near maximal points
The authors consider 3D incompressible Navier-Stokes equations: NEWLINE\[NEWLINE \partial_t v +v\cdot \nabla v+\nabla p=\rho \Delta v,\;\;\nabla\cdot v=0 NEWLINE\]NEWLINE for \(t\geq 0, x\in\mathbb{R}^3\). Introducing a cylindrical coordinate system \(x=(x_1 ,x_2 ,x_3 )=(r\cos\theta ,r\sin\theta ,z)\), they consider a \(z\)-axis symmetric solution of the form NEWLINE\[NEWLINE v (x)=v^r (r,z,t)e_r +v^{\theta} (r,z,t)e_{r,z,t} + v^z (r,z,t)e_z . NEWLINE\]NEWLINE Here \(e_r ={}^t (\cos\theta ,\sin\theta ,0),\;e_{\theta} ={}^t (-\sin\theta ,\cos\theta ,0),\;e_z =(0,0,1)\). They assume that at some point \((x_0 ,t_0 )\) the flow speed \(Q=|v|\) takes an almost maximum value in the following sense: \(4\max{x\in\mathbb{R}^3 ,\;t\leq t_0} Q(x ,t )\leq Q(x_0 ,t_0 )\). Then, they prove that the solution is close to a constant function on some cubic domain of the half region \(\{t\leq t_0 \}\) near \((x_0 ,t_0 )\).
scientific article; zbMATH DE number 6038887