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Zhen Lei

Publications and source records attributed to Zhen Lei.

288 records · Page 16Linked to original sources
arXiv2012

Singularities of solutions to compressible Euler equations with vacuum

Presented are two results on the formation of finite time singularities of solutions to the compressible Euler equations in two and three space dimensions for isentropic, polytropic, ideal fluid flows. The initial velocity is assumed to be symmetric and the initial sound speed is required to vanish at the origin. They are smooth in Sobolev space $H^3$, but not required to have a compact support. It is shown that the $H^3$ norm of the velocity field and the sound speed will blow up in a finite time.

math.AP↗
arXiv2012

Global Mild Solutions of the Navier-Stokes Equations

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 $μ$.

math.AP↗
arXiv2012

Global Solutions of Evolutionary Faddeev Model With Small Initial Data

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.

math.AP↗
arXiv2012

Global solutions for micro-macro models of polymeric fluids

We provide a new proof for the global well-posedness of systems coupling fluids and polymers in two space dimensions. Compared to the well-known existing method based on the losing a priori estimates, our method is more direct and much simpler. The co-rotational FENE dumbbell model and the coupling Smoluchowski and Navier-Stokes equations are studied as examples to illustrate our main ideas.

math.AP↗
arXiv2011

Well-posedness of 1-D compressible Euler-Poisson equations with physical vacuum

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 $1<γ<3$.

math.AP↗
arXiv2010

A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations

Let $v(x, t)= v^r e_r + v^θe_θ+ v^z e_z$ 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 $Γ= r v^θ$ is Hölder 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.

math.AP↗
arXiv2010

Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions

We derive rigorous one- and two-dimensional mean-field equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. We show how the dipolar interaction modifies the contact interaction of the strongly confined atoms. In addition, our equations introduce a nonlocal potential, which is anisotropic for pancake-shaped condensates. We propose to observe this anisotropy via measurement of the condensate aspect ratio. We also derive analytically approximate density profiles from our equations. Both the numerical solutions of our reduced mean-field equations and the analytical density profiles agree well with numerical solutions of the full Gross-Pitaevskii equation while being more efficient to compute.

cond-mat.quant-gas↗
arXiv2010

Structure of Singularities of 3D Axi-symmetric Navier-Stokes Equations

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 $C^{2, 1, α}_{\rm local}$ 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 $t_0$.

math.AP↗
arXiv2010

New Conserved Quantities of the Incompressible Euler Equations

We show two new conserved quantities for the three-dimensional incompressible Euler equations. Due to professor Lin's comments, these quantities are deeply related to the concept of topological degree and not new. I will post a new version soon.

math.AP↗
arXiv2009

Remarks on the Blowup Criteria for Oldroyd Models

We provide a new method to prove and improve the Chemin-Masmoudi criterion for viscoelastic systems of Oldroyd type in \cite{CM} in two space dimensions. Our method is much easier than the one based on the well-known \textit{losing a priori estimate} and is expected to be easily adopted to other problems involving the losing \textit{a priori} estimate.

math.AP↗
arXiv2009

On 2D Viscoelasticity with Small Strain

An exact two-dimensional rotation-strain model describing the motion of Hookean incompressible viscoelastic materials is constructed by the polar decomposition of the deformation tensor. The global existence of classical solutions is proved under the smallness assumptions only on the size of initial strain tensor. The proof of global existence utilizes the weak dissipative mechanism of motion, which is revealed by passing the partial dissipation to the whole system.

math.AP↗
arXiv2009

Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data

In this paper, we study the 3D axi-symmetric Navier-Stokes Equations with swirl. We prove the global regularity of the 3D Navier-Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some $L^p$ norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction between the angular velocity and the angular vorticity fields.

math.AP↗
arXiv2009

A Complete Global Solution to the Pressure Gradient Equation

We study the domain of existence of a solution to a Riemann problem for the pressure gradient equation in two space dimensions. The Riemann problem is the expansion of a quadrant of gas of constant state into the other three vacuum quadrants. The global existence of a smooth solution was established in Dai and Zhang [Arch. Rational Mech. Anal., {\bf 155}(2000), 277-298] up to the free boundary of vacuum. We prove that the vacuum boundary where the system is degenerate is the trivial coordinate axes.

math.AP↗
arXiv2009

Global Solutions for Incompressible Viscoelastic Fluids

We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near equilibrium initial data. The results hold in both two and three dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals and mixture problems.

math.AP↗
arXiv2009

BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity

In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.

math.AP↗
arXiv2007

Extensions of McCoy Rings

A ring $R$ is said to be right McCoy if the equation $f(x)g(x)=0,$ where $f(x)$ and $g(x)$ are nonzero polynomials of $R[x],$ implies that there exists nonzero $s \in R$ such that $f(x)s = 0$. It is proven that no proper (triangular) matrix ring is one-sided McCoy. If there exists the classical right quotient ring $Q$ of a ring $R$, then $R$ is right McCoy if and only if $Q$ is right McCoy. It is shown that for many polynomial extensions, a ring $R$ is right McCoy if and only if the polynomial extension over $R$ is right McCoy. Other basic extensions of right McCoy rings are also studied.\leftskip0truemm \rightskip0truemm \{\it Keywords}: matrix ring, McCoy ring, polynomial ring, upper triangular matrix ring.

math.RA↗
arXiv2007

Global Low Regularity Solutions of Quasi-linear Wave Equations

In this paper we prove the global existence and uniqueness of the low regularity solutions to the Cauchy problem of quasi-linear wave equations with radial symmetric initial data in three space dimensions. The results are based on the end-point Strichartz estimate together with the characteristic method.

math.AP↗