Absolute continuity of spectral shift

In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectral measure of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.     

The Cauchy problem for the dissipative Boussinesq equation

This work is devoted to the solvability and finite time blow-up of solutions of the Cauchy problem for the dissipative Boussinesq equation in all space dimension. We prove the existence and uniqueness of local mild solutions in the phase space by means of the contraction mapping principle. By establishing the time-space estimates of the corresponding Green operators, we obtain the continuation principle. Under some restriction on the initial data, we further study the results on existence and uniqueness of global solutions and finite time blow-up of solutions with the initial energy at three different level. Moreover, the sufficient and necessary conditions of finite time blow-up of solutions are given.     

Construction of Inertial Manifolds by the Hadamard Method

ＣｏｎｓｔｒｕｃｔｉｏｎｏｆＩｎｅｒｔｉａｌＭａｎｉｆｏｌｄｓｂｙｔｈｅＨａｄａｍａｒｄＭｅｔｈｏｄＬｉｕＱｉｎｇｍｉｎ（刘青民）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＬａｎｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，Ｌａｎｚｈｏｕ，７３００００）Ａｂ...     

GLOBAL DYNAMICS OF DISSIPATIVE GENERALIZED KORTEWEG-DE VRIES EQUATIONS

§1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　　ＳｉｎｃｅｔｈｅｄｉｓｃｏｖｅｒｙｂｙＫｏｒｔｅｗｅｇａｎｄｄｅＶｒｉｅｓ(1895)ａｎｄｔｈｅｗｏｒｋｏｆＺａｂｕｓｋｙａｎｄＫｒｕｓｔａｌ(1965),ｔｈｅｒｅｈａｖｅｂｅｅｎｎｕｍｅｒｏｕｓｓｉｇｎｉｆｉｃａｎｔｃｏｎｔｒｉｂｕｔｉｏｎｓｔｏｔｈｅＫｏｒｔｅｗｅｇｄｅＶｒｉｅｓ(ＫｄＶ)ｅｑｕａｔｉｏｎｓａｎｄｔｈｅｓｏｌｉｔｏｎｓｏｌｕｔｉｏｎｓ,ｅｓｐｅｃｉａｌｌｙｔｈｅｍｅｔｈｏｄｏｆＩｎｖｅｒｓｅＳｃａｔｔｅｒｉｎｇＴｒａｎｓｆｏｒｍａｎｄｔｈｅｍｅｔｈｏｄｏｆ…     

Boundary conditions changing operators in non-conformal theories

Boundary conditions changing operators have played an important role in conformal field theory. Here, we study their equivalent in the case where a mass scale is introduced, in an integrable way, either in the bulk or at the boundary. More precisely, we propose an axiomatic approach to determine the general scalar products _bθ₁, … ,θ_mθ₁′, … ,θ_n′_a, between asymptotic states in the Hilbert spaces with a and b boundary conditions respectively, and compute these scalar products explicitly in the case of the Ising and sinh-Gordon models with a mass and a boundary interaction. These quantities can be used to study statistical systems with inhomogeneous boundary conditions, and, more interestingly maybe, dynamical problems in quantum impurity problems. As an example, we obtain a series of new exact results for the transition probability in the double-well problem of dissipative quantum mechanics.     

EXACT ENVELOPE WAVE SOLUTION TO NONLINEAR SCHRODINGER EQUatION WITH DISSIPATIVE TERM

ＥＸＡＣＴＥＮＶＥＬＯＰＥＷＡＶＥＳＯＬＵＴＩＯＮＴＯＮＯＮＬＩＮＥＡＲＳＣＨＲ■ＤＩＮＧＥＲＥＱＵａｔＩＯＮＷＩＴＨＤＩＳＳＩＰＡＴＩＶＥＴＥＲＭＰＡＮＺＵＬＩＡＮＧ；ＺＨＥＮＧＫＥＪＩＥＡＮＤＹＡＮＧＱＩＮＧＪＩＡＮＡｂｓｔｒａｃｔ：Ｉｎｔｈｉｓ...     

Characteristic boundary layers in real vanishing viscosity limits

In this paper, we study the vanishing viscosity limit of initial boundary value problems for one-dimensional mixed hyperbolic-parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems: in particular, we assume that an eigenvalue of the inviscid system vanishes uniformly. We prove the stability of boundary layers expansions in small time (i.e before shocks for the inviscid system) as long as the amplitude of the boundary layers remains sufficiently small. In particular, by using Lagrangian coordinates, we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous Dirichlet condition for the velocity at the boundary.     

Real-time evolution of strongly coupled fermions driven by dissipation

We consider the real-time evolution of a strongly coupled system of lattice fermions whose dynamics is driven entirely by dissipative Lindblad processes, with linear or quadratic quantum jump operators. The fermion 2-point functions obey a closed set of differential equations, which can be solved with linear algebra methods. The staggered occupation order parameter of the t$t$-V$V$ model decreases exponentially during the dissipative time evolution. The structure factor associated with the various Fourier modes shows the slowing down of low-momentum modes, which is due to particle number conservation. The processes with nearest-neighbor-dependent Lindblad operators have a decay rate that is proportional to the coordination number of the spatial lattice.     

用耗散粒子动力学方法研究高分子链取向对形变液滴回缩法测定界面张力值的影响

在利用形变液滴回缩法(DDRM)测量了分子共混体系界面张力的过程中, 要求椭球液滴内高分子链应力松弛速度远快于椭球的回缩速度. 我们建立高分子链取向模型, 用耗散粒子动力学研究高分子链的取向及应力松弛对界面张力测量值的影响. 结果表明, 当高分子链沿着流场方向取向时, 应力是否完全松弛对界面张力测量值的影响较大, 当高分子链取向方向垂直于流场方向时, 应力是否松弛对测量值影响较小.     

Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces

In this paper we study the Cauchy problem for the semilinear fractional power dissipative equation u_t+(−Δ)^αu=F(u) for the initial data u₀ in critical Besov spaces with , where α>0, F(u)=P(D)u^b+1 with P(D) being a homogeneous pseudo-differential operator of order d[0,2α) and b>0 being an integer. Making use of some estimates of the corresponding linear equation in the frame of mixed time–space spaces, the so-called “mono-norm method” which is different from the Kato's “double-norm method,” Fourier localization technique and Littlewood–Paley theory, we get the well-posedness result in the case .