Finite speed axially symmetric Navier-Stokes flows passing a cone

Abstract

Let D be the exterior of a cone inside a ball, with its altitude angle at most π/6 in R3, which touches the x3 axis at the origin. For any initial value v0=v0,rer+v0,θeθ+v0,3e3 in a C2(D‾) class, which has the usual even-odd-odd symmetry in the x3 variable and has the partial smallness only in the swirl direction: |rv0,θ|≤1100, the axially symmetric Navier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary condition have a finite-energy solution that stays bounded for all time. In particular, no finite-time blowup of the fluid velocity occurs. Compared with standard smallness assumptions on the initial velocity, no size restriction is made on the components v0,r and v0,3. In a broad sense, this result appears to solve 2/3 of the regularity problem of ASNS in such domains in the class of solutions with the above symmetry. Equivalently, this result is connected to the general open question which asks that if an absolute smallness of one component of the initial velocity implies the global smoothness, see e.g. page 873 in Chemin et al. (2017) [6]. Our result seems to give a positive answer in a special setting.As a byproduct, we also construct an unbounded solution of the forced Navier Stokes equation in a special cusp domain that has finite energy. The forcing term, with the scaling factor of −1, is in the standard regularity class, and it can be generated by an electric current in a long and straight wire (i.e. Ampère force). This result confirms the intuition that if the channel of a fluid is very thin, arbitrarily high speed in the classical sense can be attained under a mildly singular, physically reasonable force.