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

The three-dimensional (3D) Navier-Stokes equations (0.1)ut + (u {dot operator} ) u = - p + u, div u = 0 in Q0, where u = [u, v, w]T is the vector field and p is the pressure, are considered. Here, Q0 R3 [- 1, 0) is a smooth domain of a typical backward paraboloid shape, with the vertex (0, 0) being its only characteristic point: the plane {t = 0} is tangent to Q0 at the origin, and other characteristics for t [0, - 1) intersect Q0 transversely. Dirichlet boundary conditions on the lateral boundary Q0 and smooth initial data are prescribed: (0.2)u = 0 on Q0, and u (x, - 1) = u0 (x) in Q0 {t = - 1} (div u0 = 0) . Existence, uniqueness, and regularity studies of (0.1) in non-cylindrical domains were initiated in the 1960s in pioneering works by Lions, Sather, Ladyzhenskaya, and Fujita-Sauer. However, the problem of a characteristic vertex regularity remained open. In this paper, the classic problem of regularity (in Wiener's sense) of the vertex (0, 0) for (0.1), (0.2) is considered. Petrovskii's famous "2 sqrt(log log)-criterion" of boundary regularity for the heat equation (1934) is shown to apply. Namely, after a blow-up scaling and a special matching with a boundary layer near Q0, the regularity problem reduces to a 3D perturbed nonlinear dynamical system for the first Fourier-type coefficients of the solutions expanded using solenoidal Hermite polynomials. Finally, this confirms that the nonlinear convection term gets an exponentially decaying factor and is then negligible. Therefore, the regularity of the vertex is entirely dependent on the linear terms and hence remains the same for Stokes' and purely parabolic problems. Well-posed Burnett equations with the minus bi-Laplacian in (0.1) are also discussed. 2011 Elsevier Ltd. All rights reserved.