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.

The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond, Math. Ann., 21(1):109-117, 1883). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky, Crelle, 80:1-32, 1875) that a solution to the heat equation may not be time-analytic at t = 0 even if the initial function is real analytic. Recently, it was shown in [6,8,26] that solutions to the heat equation in the whole space, or in the half space with zero boundary value, are analytic in time under essentially optimal conditions. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any δ > 0, we find a solution to the heat equation on the whole plane, with exponential growth of order 2 + δ, which is nowhere analytic in time.