Boundary characteristic point regularity for Navier-Stokes equations: blow-up scaling and petrovskii-type criterion (a formal approach) (Q435877)

The 3D Navier-Stokes equations NEWLINE\[NEWLINE \frac{\partial u}{\partial t}+(u\cdot\nabla)u-\Delta u+\nabla p=0,\quad \text{div}\,u=0 NEWLINE\]NEWLINE are considered in a non-cylindrical domain \(Q\subset \mathbb{R}^3\times [-1,0)\). \(Q\) is a smooth domain of a backward parabolic shape with vertex \((0,0)\). The vertex is the only characteristic point. It means that the plane \(\{t=0\}\) is tangent to \(\partial Q\) at the origin and other characteristics for \(t\in[-1,0)\) intersect \(\partial Q\) transversely. The equations are added by initial and boundary conditions NEWLINE\[NEWLINE\begin{aligned} & u(x,-1)=u_0(x)\;\text{on}\;Q\cap\{t=-1\},\\ & u=0\;\text{on}\;\partial Q. \end{aligned}NEWLINE\]NEWLINE The problem of regularity (in Wiener's sense, see [the second author, Appl. Anal. 71, No. 1--4, 149--165 (1999; Zbl 1034.35048)]) of the vertex \((0,0)\) is studied. The vertex \((0,0)\) of the backward paraboloid \(Q\) is regular according to Wiener if for any bounded data \(u_0(x)\) the solution \(u\) satisfies to the condition \(u(0,0^-)=0\). The authors use a blow-up scaling and a special matching with a boundary layer near \(\partial Q\). They prove that the regularity of the vertex does not depend of a convection term. The similar regularity analysis of the well-posed Burnett equations NEWLINE\[NEWLINE\begin{aligned} \frac{\partial u}{\partial t}+(u\cdot\nabla)u+\Delta^2 u+\nabla & p=0,\quad \text{div}\,u=0\;\text{in}\;Q,\\ & u=\nabla u\cdot n=0\;\text{on}\;\partial Q,\\ & u(x,-1)=u_0\end{aligned} NEWLINE\]NEWLINE is presented too.