雷震
博士 教授 博士生导师
复旦大学 数学科学学院
主要研究领域是流体力学中的偏微分方程
个性化签名
- 姓名:雷震
- 目前身份:在职研究人员
- 担任导师情况:博士生导师
- 学位:博士
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学术头衔:
博士生导师, 国家杰出青年科学基金获得者
- 职称:高级-教授
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学科领域:
偏微分方程
- 研究兴趣:主要研究领域是流体力学中的偏微分方程
雷震,复旦大学数学科学学院教授、博士生导师、副院长,主要研究领域是流体力学中的偏微分方程。2006年博士毕业于复旦大学,2011年起任复旦大学数学科学学院教授、博士生导师。2007年在美国加州理工学院做博士后,2010年冬季在纽约大学做访问学者,2012年在哈佛大学做高级研究学者,2014年春季为美国普林斯顿高级研究院member。先后荣获全国百篇优秀博士学位论文(2008年)、教育部自然科学一等奖(2011年)、国家自然科学基金委优秀青年基金(2012年)、教育部新世纪优秀人才(2012年)、上海市自然科学牡丹奖(2014年)等,2017年获得国家杰出青年科学基金。在Communications on Pure and Applied Mathematics,Communications in Mathematical Physics,Transactions of the American Mathematical Society,Journal des Mathématiques Pures et Appliquées,Journal of Functional Analysis,Journal of Differential Equations,Communications in Partial Differential Equations等国际知名学术期刊上发表论文50余篇。
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主页访问
224
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关注数
0
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成果阅读
855
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成果数
12
【期刊论文】A priori bound on the velocity in axially symmetric Navier-Stokes equations
arXiv,-0001,():
-1年11月30日
Let v be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound |v(x,t)|≤Cr2|lnr|1/2, where r∈(0,1/2) is the distance from x to the z axis, and C is a constant depending only on the initial value. This provides a pointwise upper bound (worst case scenario) for possible singularities while the recent papers \cite{CSTY2} and \cite{KNSS} gave a lower bound. The gap is polynomial order 1 modulo a half log term.
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【期刊论文】Uniform Bound of the Highest Energy for the 3D Incompressible Elastodynamics
arXiv,-0001,():
-1年11月30日
This article concerns the time growth of Sobolev norms of classical solutions to the 3D incompressible isotropic elastodynamics with small initial displacements.
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arXiv,-0001,():
-1年11月30日
This paper is concerned with the three dimensional compressible Euler--Poisson equations with moving physical vacuum boundary condition. This fluid system is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local existence of classical solutions for initial data in certain weighted Sobolev spaces is established in the case that the adiabatic index satisfies 1<γ<3.
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【期刊论文】Global Well-posedness of Incompressible Elastodynamics in Two Dimensions
arXiv,-0001,():
-1年11月30日
We prove that for sufficiently small initial displacements in some weighted Sobolev space, the Cauchy problem of the systems of incompressible isotropic elastodynamics in two space dimensions admits a uniqueness global classical solution.
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【期刊论文】On Axially Symmetric Incompressible Magnetohydrodynamics in Three Dimensions
arXiv,-0001,():
-1年11月30日
The global regularity for the incompressible magnetohydrodynamic equations (MHD) in three dimensions is a long standing open problem of fluid dynamics and PDE theory. The Navier-Stokes equations can be viewed as a special case of MHD with a constant magnetic field, whose global regularity problem is known as a Clay Millennium Prize Problem. In this article, we prove the global regularity of axially symmetric solutions to the ideal MHD in three dimensions for a family of non-trivial magnetic fields. The proofs are based on the special structures of MHD and can of course also applied to the resistive MHD. Our result might indicate that there are richer fantastic research topics in MHD than Navier-Stokes equations.
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【期刊论文】Structure of Helicity and Global Solutions of Incompressible Navier-Stokes Equation
arXiv,-0001,():
-1年11月30日
In this paper we derive a new energy identity for the three-dimensional incompressible Navier-Stokes equations by a special structure of helicity. The new energy functional is critical with respect to the natural scalings of the Navier-Stokes equations. Moreover, it is conditionally coercive. As an application we construct a family of finite energy smooth solutions to the Navier-Stokes equations whose critical norms can be arbitrarily large.
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【期刊论文】Almost Global Existence for 2-D Incompressible Isotropic Elastodynamics
arXiv,-0001,():
-1年11月30日
We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval [0,T/ϵ], for initial data of the form ϵU0, where T depends only on some Sobolev norm of U0. We show that for such data there exists a unique solution on a time interval [0,expT/ϵ], provided that ϵ is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted L2 estimates and the ghost weight introduced by Alinhac.
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【期刊论文】Rotation-Strain Decomposition for the Incompressible Viscoelasticity in Two Dimensions
arXiv,-0001,():
-1年11月30日
In \cite{Lei}, the author derived an exact rotation-strain model in two dimensions for the motion of incompressible viscoelastic materials via the polar decomposition of the deformation tensor. Based on the rotation-strain model, the author constructed a family of large global classical solutions for the 2D incompressible viscoelasticity. To get such a global well-posedness result, the equation for the rotation angle was essential to explore the underlying weak dissipative structure of the whole viscoelastic system even though the momentum equation for the velocity field and the transport equation for the strain tensor have already formed a closed subsystem. In this paper, we revisit such a result without making use of the equation of the rotation angle. The proof relies on a new identity satisfied by the strain matrix. The smallness assumptions are only imposed on the H2 norm of initial velocity field and the initial strain matrix, which implies that the deformation tensor is allowed being away from the equilibrium of 2 in the maximum norm.
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arXiv,-0001,():
-1年11月30日
We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In \cite{HL09}, Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares many properties of the 3D incompressible Euler and Navier-Stokes equations. The main difference between the 3D model of Hou and Lei and the reformulated 3D Euler and Navier-Stokes equations is that the convection term is neglected in the 3D model. In \cite{HSW09}, the authors proved that the 3D inviscid model can develop a finite time singularity starting from smooth initial data on a rectangular domain. A global well-posedness result was also proved for a class of smooth initial data under some smallness condition. The analysis in \cite{HSW09} does not apply to the case when the domain is axisymmetric and unbounded in the radial direction. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin boundary condition will develop a finite time singularity starting from smooth initial data in an axisymmetric domain. Moreover, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition.
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arXiv,-0001,():
-1年11月30日
We consider the Cauchy problem to the two-dimensional incompressible liquid crystal equation and the heat flows of harmonic maps equation. Under a natural geometric angle condition, we give a new proof of the global well-posedness of smooth solutions for a class of large initial data in energy space. This result was originally obtained by Ding-Lin in \cite{DingLin} and Lin-Lin-Wang in \cite{LinLinWang}. Our main technical tool is a rigidity theorem which gives the coercivity of the harmonic energy under certain angle condition. Our proof is based on a frequency localization argument combined with the concentration-compactness approach which can be of independent interest.
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