arXiv subjects

Dongyi Wei

Publications and source records attributed to Dongyi Wei.

At least 55 records · Page 3Linked to original sources

arXiv2020

Transition threshold for the 3D Couette flow in a finite channel

In this paper, we study nonlinear stability of the 3D plane Couette flow $(y,0,0)$ at high Reynolds number ${Re}$ in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gg Re^{\frac 13}$ due to the mixing-enhanced dissipation effect. This result confirms the transition threshold conjecture proposed by Trefethen et al.(Science, 261(1993), 578-584). To this end, we develop the resolvent estimate method to establish the space-time estimates for the full linearized Navier-Stokes system around the flow $(V(t,y,z), 0,0)$, where $V(t,y,z)$ is a small perturbation(but independent of $Re$) of the Couette flow $y$.

math.AP↗

arXiv2020

On the 3D Relativistic Vlasov-Maxwell System with large Maxwell field

This paper is devoted to the study of relativistic Vlasov-Maxwell system in three space dimension. For a class of large initial data, we prove the global existence of classical solution with sharp decay estimate. The initial Maxwell field is allowed to be arbitrarily large and the initial density distribution is assumed to be small and decay with rate $(1+|x|+|v|)^{-9-}$. In particular, there is no restriction on the support of the initial data.

math.AP↗

arXiv2020

Asymptotic decay for defocusing semilinear wave equations in $\mathbb{R}^{1+1}$

This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equation. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions enclosed by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality.

math.AP↗

arXiv2020

On the global behaviors for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$

In this paper, we study the asymptotic decay properties for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$ with pure power nonlinearity. By applying new vector fields to null hyperplane, we derive improved time decay of the potential energy, with a consequence that the solution scatters both in the critical Sobolev space and energy space for all $p>1+\sqrt{8}$. Moreover combined with Brézis-Gallouet-Wainger type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate $t^{-\frac{1}{2}}$ when $p>\frac{11}{3}$ and with rate $t^{ -\frac{p-1}{8}+ε}$ for all $1 2\sqrt{5}-1$.

math.AP↗

arXiv2019

Linear stability of pipe Poiseuille flow at high Reynolds number regime

In this paper, we prove the linear stability of the pipe Poiseuille flow for general perturbations at high Reynolds number regime. This is a long-standing problem since the experiments of Reynolds in 1883. Our work lays a foundation for the theoretical analysis of hydrodynamic stability of pipe flow, which is one of the oldest yet unsolved problems of fundamental fluid dynamics.

math.AP↗

arXiv2019

Small gaps of GOE

In this article, we study the smallest gaps of the Gaussian orthogonal ensemble. The main result is that the smallest gaps, after normalized by $n$, will tend to a Poisson distribution, and the limiting density of the $k$-th normalized smallest gaps is $ 2{}x^{2k-1}e^{-x^{2}}/(k-1)!$.

math.PR↗

arXiv2018

Diffusion and mixing in fluid flow via the resolvent estimate

In this paper, we first present a Gearhardt-Prüss type theorem with a sharp bound for m-accretive operators. Then we give two applications: (1) give a simple proof of the result proved by Constantin et al. on relaxation enhancement induced by incompressible flows; (2) show that shear flows with a class of Weierstrass functions obey logarithmically fast dissipation time-scales.

math.AP↗

arXiv2018

Non-crossing chords of a polygon with forbidden positions

In this paper, we systematically study non-crossing chords of simple polygons in the plane. We first introduce the reduced Euler characteristic of a family of line-segments, and subsequently investigate the structure of the diagonals and epigonals of a polygon. Interestingly enough, the reduced Euler characteristic of a subfamily of diagonals and epigonals characterizes the geometric convexity of polygons. In particular, an alternative and complete answer is given for a problem proposed by G. C. Shephard. Meanwhile, we extend such research to non-crossing diagonals and epigonals with forbidden positions in some appropriate sense. We prove that the reduced Euler characteristic of diagonals with forbidden positions only depends on the information involving convex partitions by those forbidden diagonals, and it determines the shapes of polygons in a surprising way. Incidentally, some kinds of generalized Catalan's numbers naturally arise.

math.CO↗

arXiv2018

Linear inviscid damping for the $β$-plane equation

In this paper, we study the linear inviscid damping for the linearized $β$-plane equation around shear flows. We develop a new method to give the explicit decay rate of the velocity for a class of monotone shear flows. This method is based on the space-time estimate and the vector field method in sprit of the wave equation. For general shear flows including the Sinus flow, we also prove the linear damping by establishing the limiting absorption principle, which is based on the compactness method introduced by Wei-Zhang-Zhao in \cite{WZZ2}. The main difficulty is that the Rayleigh-Kuo equation has more singular points due to the Coriolis effects so that the compactness argument becomes more involved and delicate.

math.AP↗

arXiv2018

Transition threshold for the 2-D Couette flow in a finite channel

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Couette flow $(y,0)$ at large Reynolds number $Re$ in a finite channel. We develop a systematic method to establish the resolvent estimates of the linearized operator and space-time estimates of the linearized Navier-Stokes equations. In particular, three kinds of important effects: enhanced dissipation, inviscid damping and boundary layer, are integrated into the space-time estimates in a sharp form. As an application, we prove that if the initial velocity $v_0$ satisfies $\|v_0-(y, 0)\|_{H^2}\le cRe^{-\frac 12}$ for some small $c$ independent of $Re$, then the solution of the 2-D Navier-Stokes equations remains within $O(Re^{-\frac 12})$ of the Couette flow for any time.

math.AP↗

arXiv2018

Spectrum of SYK model

This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.

math-ph↗

arXiv2018

Spectrum of SYK model III: Large deviations and concentration of measures

In \cite{FTD1}, we proved the almost sure convergence of eigenvalues of the SYK model, which can be viewed as a type of \emph{law of large numbers} in probability theory; in \cite{FTD2}, we proved that the linear statistic of eigenvalues satisfies the \emph{central limit theorem}. In this article, we continue to study another important theorem in probability theory\,-- the \emph{concentration of measure theorem}, especially for the Gaussian SYK model. We will prove a \emph{large deviation principle} (LDP) for the normalized empirical measure of eigenvalues when $q_n=2$, in which case the eigenvalues can be expressed in term of these of Gaussian random antisymmetric matrices. Such LDP result has its own independent interest in random matrix theory. For general $q_n\geq 3$, we can not prove the LDP, we will prove a concentration of measure theorem by estimating the Lipschitz norm of the Gaussian SYK model.

math-ph↗

arXiv2018

Spectrum of SYK model II: Central limit theorem

In our previous paper \cite{FTD1}, we derived the almost sure convergence of the global density of eigenvalues of random matrices of the SYK model. In this paper, we will prove the central limit theorem for the linear statistic of eigenvalues and compute its variance.

math-ph↗

arXiv2018

A sufficient condition for a hypersurface to be isoparametric

Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose $\mathfrak{a}$ is a symmetric $(0,2)$ tensor field whose dual $(1,1)$ tensor $\mathcal{A}$ has $n$ distinct eigenvalues, and $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=1,\cdots, n-1$. We show that all the eigenvalues of $\mathcal{A}$ are constants, generalizing a theorem of de Almeida and Brito \cite{dB90} to higher dimensions. As a consequence, a closed hypersurface $M^n$ in $S^{n+1}$ is isoparametric if one takes $\mathfrak{a}$ above to be the second fundamental form, giving affirmative evidence to Chern's conjecture.

math.DG↗

arXiv2018

Transition threshold for the 3D Couette flow in Sobolev space

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number $\text{Re}$. It was proved that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0\text{Re}^{-1}$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture in physical literatures.

math.AP↗

arXiv2018

Pseudospectral bound and transition threshold for the 3D Kolmogorov flow

In this paper, we establish the pseudospectral bound for the linearized operator of the Navier-Stokes equations around the 3D Kolmogorov flow. Using the pseudospectral bound and the wave operator method introduced in [LWZ], we prove the sharp enhanced dissipation rate for the linearized Navier-Stokes equations. As an application, we prove that if the initial velocity satisfies $\big\| U_0-\big(k_f^{-2}\sin(k_fy),0,0\big)\big\|_{H^2}\le cν^{\frac{7}{4}}$ ($ν$ the viscosity coefficient) and $k_f\in (0,1)$, then the solution does not transition away from the Kolmogorov flow.

math.AP↗

arXiv2017

Linear inviscid damping and enhanced dissipation for the Kolmogorov flow

In this paper, we prove the linear inviscid damping and voticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of the Kolmogorov flow with the size of $ν^{\frac 23+}$, here $ν$ is the viscosity coefficient.

math.AP↗

arXiv2017

Uniqueness of the Mean Field Equation and Rigidity of Hawking Mass

In this paper, we prove that the even solution of the mean field equation $Δu=λ(1-e^u) $ on $S^2$ must be axially symmetric when $4<λ\leq 8$. In particular, zero is the only even solution for $λ=6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.

math.AP↗