The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

三维可压缩Euler方程球对称解的生命区间

对于可压缩的三维Ｅｕｌｅｒ方程，当其初值为振幅ε的小扰动且具有球对称性质时，我们研究了经典解的生命区间．并证明无论初值的扰动多么小，经典解都在有限时间内爆破．     

Shock Formation for 2D Isentropic Euler Equations with Self-similar Variables

The author studies the 2D isentropic Euler equations with the ideal gas law. He exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time 1 3-H¨older bound. Moreover, these point shocks are of self-similar type and share the same profile, which is a solution to the 2D self-similar Burgers equation. The proof of the solutions, following the 3D construction of Buckmaster, Shkoller and Vicol (in 2023), is based on the stable 2D self-similar Burgers profile and the modulation method.     

Controllability of the 3D Compressible Euler System

The paper is devoted to the controllability problem for 3D compressible Euler system. The control is a finite-dimensional external force acting only on the velocity equation. We show that the velocity and density of the fluid are simultaneously controllable. In particular, the system is approximately controllable and exactly controllable in projections.     

KAM theory and the 3D Euler equation

We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold M   is not mixing in the C^k$C^k$ topology (k>4$k>4$ and non-integer) for any prescribed value of helicity and sufficiently large values of energy. This can be regarded as a 3D version of Nadirashvili's and Shnirelman's theorems showing the existence of wandering solutions for the 2D Euler equation. Moreover, we obtain an obstruction for the mixing under the Euler flow of C^k$C^k$-neighborhoods of divergence-free vectorfields on M  . On the way we construct a family of functionals on the space of divergence-free C¹$C^1$ vectorfields on the manifold, which are integrals of motion of the 3D Euler equation. Given a vectorfield these functionals measure the part of the manifold foliated by ergodic invariant tori of fixed isotopy types. We use the KAM theory to establish some continuity properties of these functionals in the C^k$C^k$-topology. This allows one to get a lower bound for the C^k$C^k$-distance between a divergence-free vectorfield (in particular, a steady solution) and a trajectory of the Euler flow.     

三维不可压缩Euler方程的H1(R~3)轴对称解

在适当的初值条件下，对三维不可压缩Ｅｕｌｅｒ方程组，我们证明了Ｈ１（Ｒ３）轴对称解的存在性．     

2D Euler equation on the strip: Stability of a rectangular patch

We consider the 2D Euler equation of incompressible fluids on a strip ?×𝕋 and prove the stability of the rectangular stationary state χ_|x|<L for large enough L.     

一般无界区域中带有阻尼的三维可压缩欧拉方程

考虑在一般的三维无界区域中的具有滑移边界条件的带有阻尼的可压缩欧拉方程.当初始值接近平衡态时,获得了全局存在性和唯一性.同时,研究了在半空间情形下系统的衰减率.证明了经典解的L~2范数以(1+t)~(-3/4)衰减到常值背景解.     

The 3D Non-isentropic Compressible Euler Equations with Damping in a Bounded Domain

The authors investigate the global existence and asymptotic behavior of classical solutions to the 3D non-isentropic compressible Euler equations with damping on a bounded domain with slip boundary condition. The global existence and uniqueness of classical solutions are obtained when the initial data are near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.     

关于高阶Euler数的同余

给出了高阶Euler数的一些同余式.     

Growth of solutions for QG and 2D Euler equations

We study the rate of growth of sharp fronts of the Quasi-geostrophic equation and 2D incompressible Euler equations. The development of sharp fronts are due to a mechanism that piles up level sets very fast. Under a semi-uniform collapse, we obtain a lower bound on the minimum distance between the level sets.

     

Euler number of the moduli space of sheaves on a rational nodal curve

In this paper, we use finite group actions to compute the Euler number of the moduli space of rank 2 stable sheaves on a rational nodal curve.

     

Euler数和高阶Euler数的推广

The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.     

Viscous Approximations and Decay Rate of Maximal Vorticity Function for 3-D Axisymmetric Euler Equations

In this paper, we are first concerned with viscous approximations for the three-dimensional axisymmetric incompressible Euler equations. It is proved that the viscous approximations, which are the solutions of the corresponding Navier-Stokes equations, converge strongly in L^2([0, T]; Lloc^2(R^3)) provided that they have strong convergence in the region away from the symmetry axis. This result has been proved by the authors for the approximate solutions generated by smoothing the initial data,with no restriction of the sign of the initial data. Then we discuss the decay rate for maximal vorticity function, which is established for both approximate solutions generated by smoothing the initial data and viscous approximations respectively. One sufficient condition to guarantee the strong convergence in the region away from the symmetry axis is given, and a decay rate for maximal vorticity function in the region away from the symmetry axis is obtained for non-negative initial vorticity.     

Spanning Euler tours and spanning Euler families in hypergraphs with particular vertex cuts

An Euler tour in a hypergraph is a closed walk that traverses each edge of the hypergraph exactly once, while an Euler family, first defined by Bahmanian and ?ajna, is a family of closed walks that jointly traverse each edge exactly once and cannot be concatenated. In this paper, we study the notions of a spanning Euler tour and a spanning Euler family, that is, an Euler tour (family) that also traverses each vertex of the hypergraph at least once. We examine necessary and sufficient conditions for a hypergraph to admit a spanning Euler family, most notably when the hypergraph possesses a vertex cut consisting of vertices of degree two. Moreover, we characterise hypergraphs with a vertex cut of cardinality at most two that admit a spanning Euler tour (family). This result enables us to reduce the problem of existence of a spanning Euler tour (which is NP-complete), as well as the problem of a spanning Euler family, to smaller hypergraphs.     

Solutions to the 2D Euler equations with velocity unbounded at infinity

We prove existence of solutions to the two-dimensional Euler equations with vorticity bounded and with velocity locally bounded but growing at infinity at a rate slower than a power of the logarithmic function. We place no integrability conditions on the initial vorticity. This result improves upon a result of Serfati which gives existence of a solution to the two-dimensional Euler equations with bounded velocity and vorticity.