Wei Dongyi Ab

• Peking University

This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets after the formation of curvature singularities. The most challenging part of this paper is to handle the Cau...

In this paper, we prove the existence of self-similar algebraic spiral solutions of the 2-D incompressible Euler equations for the initial vorticity of the form \(|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)\) with \(\mu > \frac12\) and \(\mathring{\omega}\in L^1({\mathbb{T}})\), satisfying m-fold symmetry (\(m\ge 2\)) and a dominant condition. As a...

In this paper, we investigate the stability of the 2D Couette flow (y,0)T under the influence of a uniform magnetic field (β,0)T. Our focus is on the magnetohydrodynamic (MHD) equations on T×R, characterized by distinct viscosity coefficient ν and magnetic diffusion coefficient µ. We derive space-time estimates for the linearized equations for all...

In this paper, we consider the defocusing nonlinear wave equation $-\partial _t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb {R}\times \mathbb {R}^d$ . Building on our companion work ( Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403.11471), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$ , there exists a smooth co...

We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on \mathbb{T}\times \mathbb{R} . More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible...

In this paper, we prove the blow-up of the $3$-D isentropic compressible Navier-Stokes equations for the adiabatic exponent $\gamma=5/3$, which corresponds to the law of monatomic gas. This is the degenerate case in the sense of [Merle, Rapha\"el, Rodnianski and Szeftel, Ann. of Math. (2), 196 (2022), 567-778; Ann. of Math. (2), 196 (2022), 779-889...

In this paper, we study the long-time dynamics of solutions to the defocusing semilinear wave equation \(\Box _g\phi =|\phi |^{p-1}\phi \) on the Schwarzschild black hole spacetimes. For \(\frac{1+\sqrt{17}}{2}<p<5\) and compactly supported initial data, we show that the solution decays like \(|\phi |\lesssim t^{-1+\epsilon }\) in the domain of out...

In this paper, we study the long time asymptotic behaviors for solutions to the Chern-Simons-Higgs equations with a pure power defocusing nonlinearity. We obtain quantitative inverse polynomial time decay for the potential energy for all data with finite conformal energy. Consequently, the solution decays in time in the pointwise sense for all powe...

In this paper, we study the global well-posedness of the 3-D inhomogeneous incompressible Navier-Stokes system (INS in short) with initial density $\rho_0$ being discontinuous and initial velocity $u_0$ belonging to some critical space. Firstly, if $\rho_0u_0$ is sufficiently small in the space $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$ an...

This paper investigates nonlinear Landau damping in the 3D Vlasov-Poisson (VP) system. We study the asymptotic stability of the Poisson equilibrium $\mu(v)=\frac{1}{\pi^2(1+|v|^2)^2}$ under small perturbations. Building on the foundational work of Ionescu, Pausader, Wang, and Widmayer \cite{AIonescu2022}, we provide a streamlined proof of nonlinear...

We study the asymptotic behavior of small data solutions to the screened Vlasov Poisson(i.e. Vlasov-Yukawa) equation on ${\mathbb R}\times{\mathbb R}$ near vacuum. We show that for initial data small in Gevrey-2 regularity, the derivative of the density of order $n$ decays like $(t+1)^{-n-1}$.

The 2-D Peskin problem describes a 1-D closed elastic string immersed and moving in a 2-D Stokes flow that is induced by its own elastic force. The geometric shape of the string and its internal stretching configuration evolve in a coupled way, and they combined govern the dynamics of the system. In this paper, we show that certain geometric quanti...

The dynamics of probability density functions has been extensively studied in science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics that can be formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as...

We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics, i.e., asymptotic compactness in dynamical systems, global stability of evolution equations, and localized cont...

In this paper, we solve Lions' open problem: {\it the uniqueness of weak solutions for the 2-D inhomogeneous Navier-Stokes equations (INS)}. We first prove the global existence of weak solutions to 2-D (INS) with bounded initial density and initial velocity in $L^2(\mathbb R^2)$. Moreover, if the initial density is bounded away from zero, then our...

In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubble...

In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb R\times \mathbb R^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler equations}), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued s...

In this paper, we study nonlinear stability of the 3D plane Couette flow ( y , 0 , 0 ) (y,0,0) at high Reynolds number R e {Re} in a finite channel T × [ − 1 , 1 ] × T \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 an...

We show that the a-parameterized family of the generalized Constantin–Lax–Majda model, also known as the Okamoto–Sakajo–Wunsch model, admits exact self-similar finite-time blowup solutions with interiorly smooth profiles for all a≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \us...

We study solutions to the Yang–Mills–Higgs equations on the maximal Cauchy development of the data given on a ball of radius R in \(\mathbb {R}^3\). The energy of the data could be infinite and the solution grows at most inverse polynomially in \(R-t\) as \(t\rightarrow R\). As applications, we derive pointwise decay estimates for Yang–Mills–Higgs...

We will prove the Berry-Esseen theorem for the number counting function of the circular $\beta$-ensemble (C$\beta$E), which will imply the central limit theorem for the number of points in arcs of the unit circle in mesoscopic and macroscopic scales. We will prove the main result by estimating the characteristic functions of the Pr\"{u}fer phases a...

We show that the 1D Hou-Luo model on the real line admits exact self-similar finite-time blowup solutions with smooth self-similar profiles. The existence of these profiles is established via a fixed-point method that is purely analytic. We also prove that the profiles satisfy some monotonicity and convexity properties that are unknown before, and...

We show that the De Gregorio model on the real line admits infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions fall into two classes: the basic class and the general class. The basic class consists of countably infinite solutions that are eigenfu...

In this paper, we study the long time dynamics of solutions to the defocusing semilinear wave equation $\Box_g\phi=|\phi|^{p-1}\phi$ on the Schwarzschild black hole spacetimes. For $\frac{1+\sqrt{17}}{2}<p<5$ and sufficiently smooth and localized initial data, we show that the solution decays like $|\phi|\lesssim t^{-1+\epsilon}$ in the domain of o...

We show that the $a$-parameterized family of the generalized Constantin-Lax-Majda model, also known as the Okamoto-Sakajo-Wensch model, admits exact self-similar finite-time blowup solutions with interiorly smooth profiles for all $a\leq 1$. Depending on the value of $a$, these self-similar profiles are either smooth on the whole real line or compa...

In this paper, we prove the existence of self-similar algebraic spiral solutions for 2-D incompressible Euler equations for the initial vorticity of the form $|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)$ with $\mu>\frac12$ and $\mathring{\omega}\in L^1(\mathbb T)$ satisfying $m$-fold symmetry ($m\geq 2$) and a dominant condition. As an important ap...

The 2-D Peskin problem describes a 1-D closed elastic string immersed and moving in a 2-D Stokes flow that is induced by its own elastic force. The geometric shape of the string and its internal stretching configuration evolve in a coupled way, and they combined govern the dynamics of the system. In this paper, we show that certain geometric quanti...

We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressibl...

It has been shown in [Yang-Yu 2019] that general large solutions to the Cauchy problem for the Maxwell-Klein-Gordon system (MKG) in the Minkowski space $\mathbb{R}^{1+3}$ decay like linear solutions. One hence can define the associated radiation field on the future null infinity as the limit of $(r\underline{\alpha}, r\phi)$ along the out going nul...

In this paper, we study the linearized Navier–Stokes system around monotone shear flows in a finite channel with non-slip boundary condition. We prove that if the flow is linearly stable for the Euler equations, then it is also linearly stable for the Navier–Stokes equations at high Reynolds number. More importantly, we establish the inviscid dampi...

The Chaplygin gas model is both interesting and important in the theory of gas dynamics and conservation laws, all the characteristic families of which are linearly degenerate. Majda conjectured that the shock formation never happens for smooth data. In this article, we prove the conjecture for the two space dimensional axisymmetric case. Different...

In this paper, we study the number of traveling wave families near a shear flow under the influence of Coriolis force, where the traveling speeds lie outside the range of the flow u. Under the \(\beta \)-plane approximation, if the flow u has a critical point at which u attains its minimal (resp. maximal) value, then a unique transitional \(\beta \...

For discretely observed functional data, estimating eigenfunctions with diverging index is essential in nearly all methods based on functional principal components analysis. In this paper, we propose a new approach to handle each term appeared in the perturbation series and overcome the summability issue caused by the estimation bias. We obtain the...

We show that the De Gregorio model on the real line admits infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions fall into two classes: the basic class and the general class. The basic class consists of countably infinite solutions that are eigenfu...

In this paper, we prove that the fluctuation of the extreme process of the maxima of all the largest eigenvalues of $m\times m$ principal minors (with fixed $m$) of the classical Gaussian orthogonal ensemble (GOE) of size $n\times n$ is given by the Gumbel distribution as $n$ tends to infinity. We also derive the joint distribution of such maximal...

In this paper, we prove the linear stability of the pipe Poiseuille flow for general perturbations at high Reynolds number regime. This has been 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 pr...

We study solutions to the Yang-Mills-Higgs equations on the maximal Cauchy development of the data given on a ball of radius $R$ in $\mathbb{R}^3$. The energy of the data could be infinite and the solution grows at most inverse polynomially in $R-t$ as $t\rightarrow R$. As applications, we derive pointwise decay estimates for Yang-Mills-Higgs field...

By introducing new weighted vector fields as multipliers, we derive quantitative pointwise estimates for solutions of defocusing semilinear wave equation in $\mathbb {R}^{1+3}$ with pure power nonlinearity for all $1<p\leq 2$. Consequently, the solution vanishes on the future null infinity and decays in time polynomially for all $\sqrt {2}<p\leq 2$...

This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equations. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao [4]. Moreover, for sufficiently localized data belonging to some weighted energy space,...

This paper is devoted to the study of relativistic Vlasov-Maxwell system in space dimension three. For a class of large initial data, we prove the global existence of classical solution with sharp decay estimates. The initial Maxwell field is allowed to be arbitrarily large and the initial density distribution is assumed to be small and decay with...

By introducing new weighted vector fields as multipliers, we derive quantitative pointwise estimates for solutions of defocusing semilinear wave equation in $\mathbb{R}^{1+3}$ with pure power nonlinearity for all $1<p\leq 2$. Consequently, the solution vanishes on the future null infinity and decays in time polynomially for all $\sqrt{2}<p\leq 2$....

In this paper, we study the transition threshold problem for the 2-D Navier–Stokes equations around the Couette flow (y, 0) at high 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,...

In this paper, we study the number of traveling wave families near a shear flow $(u,0)$ under the influence of Coriolis force, where the traveling speeds lie outside the range of $u$. Let $\beta$ be the Rossby number. If the flow $u$ has at least one critical point at which $u$ attains its minimal (resp. maximal) value, then a unique transitional $...

In our previous paper [R. Feng, G. Tian and D. Wei, Spectrum of SYK model, Peking Math. J. 2 (2019) 41–70], 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 statistics of eigenvalues of the SYK model and compute its...

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 smal...

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 r...

In [Spectrum of SYK model, preprint (2018), arXiv:1801.10073], we proved the almost sure convergence of eigenvalues of the SYK model, which can be viewed as a type of law of large numbers in probability theory; in [Spectrum of SYK model II: Central limit theorem, preprint (2018), arXiv:1806.05714], we proved that the linear statistic of eigenvalues...

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...

In this paper, we study the linear inviscid damping for the linearized \(\beta \)-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 the sprit of the wave equation. For general s...

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 spac...

In this article, we study the smallest gaps between eigenvalues of the Gaussian orthogonal ensemble (GOE). The main result is that the smallest gaps, after being normalized by n, will converge to a Poisson distribution, and the limiting density of the kth normalized smallest gap is \(2{}x^{2k-1}e^{-x^{2}}/(k-1)!\). The proof is based on the method...

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...

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

We will prove the Berry-Esseen theorem for the number counting function of the circular $\beta$-ensemble (C$\beta$E), which will imply the central limit theorem for the number of points in arcs. We will prove the main result by estimating the characteristic functions of the Pr\"ufer phases and the number counting function, which will imply the the...

In this paper, 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 the Kolmogorov flow by developing the hypocoercivity method introduced by Villani (2009).

In this paper, we prove that the even solution of the mean field equation Δu=λ(1-eu)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta u=\lambda (1-e^u) $$\end{docu...

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena...

In this paper, we first present a Gearhardt-Pr\"uss 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 logarithmicall...

In this paper, we give the asymptotic expansion of \(n_{0,d}\) and \(n_{1,d}\), where \((3d-1+g)!\, n_{g,d}\) counts the number of genus g curves in \({\mathbb {C}}P^2\) through \(3d-1+g\) points in general position and can be identified with certain Gromov–Witten invariants.

In this paper, we study the linear inviscid damping for the linearized $\beta$-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 fl...

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 particul...

In this article, we study the largest gaps of the classical random matrices of CUE and GUE, and we will derive the rescaling limit of the $k$-th largest gap, which is given by the Gumbel distribution.

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number Re. It was proved that if the initial velocity v0 satisfies for some c0 > 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. This result confirms...

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 flo...

We will study the spectral properties of the random matrix of SYK model, which is a simple model of the black hole in physics literature. Our primary interest is the distribution of the eigenvalues as the number of Majorana fermions tends to infinity. We will first prove the almost surely convergence of the global density of the eigenvalues. Then w...

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.

In this paper, we study pseudospectral bounds 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 [22], we prove the sharp enhanced dissipation rate for the linearized Navier‐Stokes equations. As an application, we prove that if the initi...

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 t...

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...

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...

In this paper, we prove that for every nonnegative initial data in L1(R2), the Patlak–Keller–Segel equation is globally well-posed if and only if the total mass M≤8π. Our proof is based on some monotonicity formulas of nonnegative mild solutions.

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

In this paper, we prove the global well-posedness of the incompressible MHD equations near a homogeneous equilibrium in the domain $R^k\times T^{d-k}, d\geq2,k\geq1$ by using the comparison principle and constructing the comparison function.

In this paper, we prove the global well-posedness of the incompressible MHD equations near a homogeneous equilibrium in the domain $R^k\times T^{d-k}, d\geq2,k\geq1$ by using the comparison principle and constructing the comparison function.

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena...

In this paper, we solve Gallay's conjecture on the spectral lower bound and pseudospecrtal bound for the linearized operator of the Navier-Stokes equation in $R^2$ around rapidly rotating Oseen vortices.

In this paper, we solve Gallay's conjecture on the spectral lower bound and pseudospecrtal bound for the linearized operator of the Navier-Stokes equation in $R^2$ around rapidly rotating Oseen vortices.

In this paper we investigate non-crossing chords of simple polygons in the plane systematically. We first develop the Euler characteristic of a family of line-segments, and subsequently study the structure of the diagonals and epigonals of a polygon. A special phenomenon is that the Euler characteristic of a set of diagonals (or epigonals) characte...

In this paper, we give the asymptotic expansion of $n_{0,d}$ and $n_{1,d}$.

In this paper, we study the MHD equations with small viscosity and resistivity coefficients, which may be different. This is a typical setting in high temperature plasmas. It was proved that the MHD equations are globally well-posed if the initial velocity is close to 0 and the initial magnetic field is close to a homogeneous magnetic field in the...

In this paper, we study the MHD equations with small viscosity and resistivity coefficients, which may be different. This is a typical setting in high temperature plasmas. It was proved that the MHD equations are globally well-posed if the initial velocity is close to 0 and the initial magnetic field is close to a homogeneous magnetic field in the...

Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^{p}(M)}\big)\leq \Lambda\quad \text{for some}\,\,p>1, \eeno where $\tau(u_n)$ is the tension field of $u_n$, then there hold the so called energy identity and n...

Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^{p}(M)}\big)\leq \Lambda\quad \text{for some}\,\,p>1, \eeno where $\tau(u_n)$ is the tension field of $u_n$, then there hold the so called energy identity and n...

In this paper, we prove the decay estimates of the velocity and $H^1$ scattering for the 2D linearized Euler equations around a class of monotone shear flow in a finite channel. Our result is consistent with the decay rate predicted by Case in 1960.

In this paper, we prove the decay estimates of the velocity and $H^1$ scattering for the 2D linearized Euler equations around a class of monotone shear flow in a finite channel. Our result is consistent with the decay rate predicted by Case in 1960.

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_\theta(r,z,t)\|_{L^\infty}\leq\|ru_\theta(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if $\|ru_\theta(r,z,0)\|_{L^\infty}$ or $ \|ru_\theta(r,z,t)\|_{L^\infty(r\leq r_0)}$ is small compared with certain dimensionl...