During the past year a group of 12 graduate students, a visiting researcher and a postdoctoral fellow, supported in part by this contract, have studied various nonlinear partial differential equations. Seminars were held twice weekly on the topic of soliton theory. The graduate students involved have been studying the problem of singularity that arises in nonlinear equations.

In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting $\varepsilon >0$ to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders $O(1)$ and $O(\varepsilon^\alpha)$, respectively, with $\alpha>2$, for which the limiting system is the primitive equations with only horizontal viscosity as $\varepsilon$ tends to zero. In particular we show that for ``well prepared" initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as $\varepsilon$ tends to zero, and that the convergence rate is of order $O\left(\varepsilon^\frac\beta2\right)$, where $\beta=\min\{\alpha-2,2\}$. Note that this result is different from the case $\alpha=2$ studied in [Li, J.; Titi, E.S.: \emph{The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation}, J. Math. Pures Appl., \textbf{124} \rm(2019), 30--58], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order $O\left(\varepsilon\right)$.

We show that a family $\mathcal{F}$ of analytic functions in the unit disk ${\mathbb{D}}$ all of whose zeros have multiplicity at least$k$and which satisfy a condition of the form $$f^n(z)f^{(k)}(xz)\ne1$$ for all $z\in{\mathbb{D}}$ and $f\in\mathcal{F}$ (where$n$≥3,$k$≥1 and 0<|$x$|≤1) is normal at the origin. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman–Pang rescaling method. Furthermore we prove the corresponding Picard-type theorem for entire functions and some generalizations.

An elliptic equation arising from the study o fstatic solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field, is considered. We classify the solutions and establish the uniqueness of radially symmetric solutions. We also complete a classification of symmetric solutions of an elliptic equation on the sphere.

We consider a system coupling the incompressible Navier–Stokes equations to the Vlasov–Fokker–Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme is asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has a numerical stability condition controlled by the non-stiff convection operator, with an implicit treatment of the stiff drag term and the Fokker–Planck operator. Yet, consistent to a standard asymptotic-preserving Fokker–Planck solver or an incompressible Navier–Stokes solver, only the conjugate–gradient method and fast Poisson and Helmholtz solvers are needed. Numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the scheme, with several interesting applications.