The regularity problem for axially symmetric Navier-Stokes equations on some bounded regions

The regularity problem of the Navier-Stokes equations (NS) in $\mathbb{R}^3$ asks whether a global smooth solution exists for any initial velocity $v_0$ that is divergence free and lies in the Schwartz class $\mathcal{S}(\mathbb{R}^3)$. This problem is still wide open in general for large initial values, and one of the essential barriers is the supercriticality of the (NS). If confining attention to axially symmetric vector fields, it was observed that the axially symmetric Navier-Stokes equations (ASNS) are critical after some proper transformations, which raises some hope to settle the regularity problem for (ASNS). Despite this problem on $\mathbb{R}^3$ remains open as well, it was solved recently on some particular bounded cusp domains with a Navier-slip boundary condition. Motivated by this work, we continue to study the regularity problem for (ASNS) on more regular and more realistic bounded regions than those cusp domains under the Navier-Hodge-Lions (NHL) boundary condition. As a byproduct, for some very thin cusp domains with the NHL boundary condition, we construct unbounded solutions to the forced (ASNS) that has finite energy. The forcing term, with the scaling factor of $−1$, is in the standard regularity class.