## Analysis

### Department of Mathematics and Statistics

### Texas Tech University

Consider any Leray-Hopf weak solution of the three-dimensional Navier-Stokes equations for incompressible, viscous fluid flows. We prove that any Lagrangian trajectory associated with such a velocity field has an asymptotic expansion, as time tends to infinity, which describes its long-time behavior very precisely.According to the Ahlfors-Gehring theorem, a simply connected domain $\Omega$ in the extended complex plain is a quasidisk if and only if there exists a sufficient condition for the injectivity of a holomorphic function in $\Omega$ in relation to the growth of its Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also give sufficient conditions for the existence of homeomorphic and quasiconformal extensions to $\overline{\mathbb{C}}$ for harmonic mappings defined on quasidisks.
The Ahlfors-Gehring theorem has been extended to finitely connected domains $\Omega$ by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in $\Omega$ if and only if the components of $\partial\Omega$ are either points or quasicircles. We generalize this theorem to harmonic mappings.

We will introduce a pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in ${\mathbb C}^n$ and prove some basic theorems for these operators. For example, we will characterize the case when the pre-Schwarzian derivative is holomorphic and, also, show that the pre-Schwarzian is stable only with respect to rotations of the identity, among other theorems.

Along the way we will make some observations related to the dilatation of pluriharmonic mappings and to the dilatation of their affine transformations. These will reveal some differences between the theories in the plane and in higher dimensions.

The celebrated Riemann hypothesis can be reformulated as a simply-stated criterion concerning least-squares approximation. While carrying out computations related to this criterion, we have observed a curious phenomenon: for no apparent reason, at least the first billion entries of a certain infinite triangular matrix associated to the Riemann zeta function are all positive. In this talk, I shall explain the background leading to this observation, and make a conjecture. (Joint work with Hugues Bellemare et Yves Langlois.)We will first introduce some basic facts about the energy of a condenser $(D,K)$ in $\mathbb{R}^{n}$. Then we will consider a variation of $D$ described via a family of smooth domains $D_{t}$, $t\in(0,1)$, whose boundaries $\partial D_{t}$ are level sets of a $C^{2}$ function $V$. We will show that, if $V$ is subharmonic, then the energy of the condenser $(D_{t},K)$ is a concave function of $t$, and we will characterize the cases where this function is affine.In 1950s, J.-L. Lions generalized the deterministic Navier-Stokes equations when the space is n-dimensional by adding a diffusive term with a fractional Laplacian and claimed the uniqueness of its solution when the exponent of the fractional Laplacian is equal to or bigger than 1/2 + n/4. E.g., it has to be no smaller than 5/4 when n = 3. In 2020, Luo and Titi (also Buckmaster, Colombo and Vicol in a pre-print) proved the non-uniqueness of weak solution to such generalized Navier-Stokes equations when this exponent is strictly smaller than 5/4, complementing the work of Lions, by relying on the convex integration technique due to [Buckmaster and Vicol, 2019]. We prove an analogous result in the stochastic case; i.e., non-uniqueness in law of three-dimensional generalized Navier-Stokes equations with an exponent strictly less than 5/4 that is forced by random noise. An analogous result holds in the two-dimensional case as well. A circle domain $\Omega$ in the Riemann sphere is a domain each of whose boundary components is either a circle or a point. A circle domain $\Omega$ is called conformally rigid if every conformal map from $\Omega$ onto another circle domain is the restriction of a Möbius transformation. In this talk I will present some new rigidity theorems for circle domains satisfying a certain quasihyperbolic condition. As a corollary, John and Hölder circle domains are rigid. This provides new evidence for a conjecture of He and Schramm, relating rigidity and conformal removability. This talk is based on joint work with Malik Younsi.