Zhen Lei

Here we establish a global well-posedness of \textit{mild} solutions to the three-dimensional incompressible Navier-Stokes equations if the initial data are in the space $\mathcal{X}^{-1}$ defined by $(1.3)$ and if the norms of the initial data in $\mathcal{X}^{-1}$ are bounded exactly by the viscosity coefficient $\mu$.

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space $\mathbb{R}^{1 + n}$ to the unit sphere $\mathbb{S}^2$, which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.

This paper is concerned with the 1-D compressible Euler-Poisson equations with moving physical vacuum boundary condition. It is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local well-posedness of classical solutions is established in the case of the adiabatic index $1<\gamma<3$.

We establish a global well‐posedness of mild solutions to the three‐dimensional, incompressible Navier‐Stokes equations if the initial data are in the space ${\cal{X}}^{-1}$ defined by \input amssym ${\cal{X}}^{‐1} = \{f \in {\cal{D}}^\prime(R^3): \int_{{\Bbb{R}}^3}|\xi|^{‐1}|\widehat{f}|d\xi < \infty\}$ and if the norms of the initial data in ${\cal{X}}^{-1}$ are bounded exactly by the viscosity coefficient μ.

Let $v(x, t)= v^r e_r + v^\theta e_\theta + v^z e_z$ be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by $b = v^r e_r + v^z e_z$ the radial-axial vector field. Under a general scaling invariant condition on $b$, we prove that the quantity $\Gamma = r v^\theta$ is H\"older continuous at $r = 0$, $t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \cite{KNSS} and Seregin-Sverak in \cite{SS}. As another application, we prove that if $b \in L^\infty([0, T], BMO^{-1})$, then $v$ is regular. This provides an answer to an open question raised by Koch and Tataru in \cite{KochTataru} about the uniqueness and regularity of Navier-Stokes equations in the axially-symmetric case.

Let $v$ be a solution of the axially symmetric Navier-Stokes equation. We determine the structure of certain (possible) maximal singularity of $v$ in the following sense. Let $(x_0, t_0)$ be a point where the flow speed $Q_0 = |v(x_0, t_0)|$ is comparable with the maximum flow speed at and before time $t_0$. We show after a space-time scaling with the factor $Q_0$ and the center $(x_0, t_0)$, the solution is arbitrarily close in $C^{2, 1, \alpha}_{{\rm local}}$ norm to a nonzero constant vector in a fixed parabolic cube, provided that $r_0 Q_0$ is sufficiently large. Here $r_0$ is the distance from $x_0$ to the $z$ axis. Similar results are also shown to be valid if $|r_0v(x_0, t_0)|$ is comparable with the maximum of $|rv(x, t)|$ at and before time $t_0$.