One open question in the study of the steady incompressible three-dimensional Navier-Stokes equations is if the only solution with finite Dirichlet integral and vanishing condition at infinity is the trivial solution. Several partial results have been proven by requiring certain integral or decay conditions on the solution. We will explore a certain class of solutions, called axially-symmetric D-solutions, and discuss some results about these solutions. In this thesis, we will prove that certain axially-symmetric periodic D-solutions are identicially zero.

This thesis contains two main results: a Li-Yau type gradient estimate 3.3.1, and

a Zhong-Yang type eigenvalue estimate 4.3.1. The classical version of these results is formulated in the setting of manifolds with nonnegative Ricci curvature. Here we present

proofs of analogous results under integral curvature assumptions, which are more general

and apply in many more settings than pointwise lower bounds. Although the totality of this

work has not been published, part of it was published in [RO19] or appears in the preprint

[ROSWZ18].

The Li-Yau gradient estimate that we prove is an inequality satisfied by the gradient of the Neumann heat kernel. We restrict our attention to compact domains within

an ambient space manifold, and assume that the amount of negative Ricci curvature of the

manifold is small in an Lp average sense. The domains are not necessarily convex, but must

satisfy an interior rolling R-ball condition 1.3.4. As a corollary of this theorem, we derive

a parabolic Harnack inequality 3.4.1 and a mean value inequality 3.4.2, as well as a lower

bound for the first nontrivial Neumann eigenvalue on this class of domains 3.4.3.

The Zhong-Yang type estimate that we present is a lower bound for the first

nonzero eigenvalue of the drift Laplacian in the setting of closed smooth metric measure

spaces. It is derived assuming that the amount of negative Bakry-Émery Ricci curvature

of the manifold is small in an Lp average sense. The estimate is sharp, since it recovers the

classical result in the limit where the Ricci tensor is nonnegative. Moreover, we show that

the smallness of the curvature assumption is necessary in example 4.4.2.

Relevant results and theory in the Axially Symmetric Navier Stokes Equations are reviewed. Then we obtain pointwise, a priori bounds for the $r$, $\theta$ and $z$ components of the vorticity of axially symmetric solutions to the three-dimensional Navier-Stokes equations, which improves on an earlier bound in \cite{BZ:1}. Finally, we show that, for any Leray-Hopf solution, $v$, we can use the $\theta$ component of vorticity to bound the velocity and derive

\begin{align*}

|v(x,t)|\leq\frac{C|\ln{r}|^{1/2}}{r^2},\qquad 0 < r \leq 1/2,

\end{align*}

where $r$ is the distance from the $z$ axis.