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 R 3 , which touches the x 3 axis at the origin. For any initial value v 0 = v 0,r e r + v 0,θ e θ + v 0,3 e 3 in a C 2 (D) class, which has the usual even-odd-odd symmetry in the x 3 variable and has the partial smallness only in the swirl direction: |rv 0,θ | ≤ 1 100 , the axially symmetric Navier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary condition has 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 v 0,r and v 0,3 . 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. Contents 2020 Mathematics Subject Classification. 35Q30, 76N10.