Asymptotic expansion of stochastic flows

Summary We study the asymptotic expansion in small time of the solution of a stochastic differential equation. We obtain a universal and explicit formula in terms of Lie brackets and iterated stochastic Stratonovich integrals. This formula contains the results of Doss [6], Sussmann [15], Fliess and Normand-Cyrot [7], Krener and Lobry [10], Yamato [17] and Kunita [11] in the nilpotent case, and extends to general diffusions the representation given by Ben Arous [3] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by Strichartz [14].     

On the semilinear Schrödinger equation with time dependent coefficients

We consider nonlinear Schrödinger equation with time dependent coefficients. Fanelli [5] found a transformation between solutions of the original equation and of the usual Schrödinger equation with power nonlinearity involving time dependent coefficients in some Lorentz spaces. In this paper we extend the results in [5] in space‐time integrability properties of solutions. Particularly, we prove that the existence and uniqueness of solutions can be described exclusively in terms of Lebesgue spaces (not Lorentz spaces as in [5]) as far as the space integrability of solutions. We also discuss the equation with coefficient of an explicit homogeneous function and describe the associated Strichartz estimate and contraction mapping argument.     

A new equation on the Calabi-Yau metrics in low dimensions

We obtain an equation on the metrics of compact Kähler manifolds in dimensions 2 and 3, whose solutions are Calabi-Yau metrics. This equation differs from the Monge-Ampère equation considered by Calabi [1] and Yau [2].     

Random data Cauchy theory for supercritical wave equations II: a global existence result

We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in \(\bigcap_{s<1/2} (H^s(\Theta)\times H^{s-1}(\Theta))\). We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations, inspired by earlier works by Bourgain [2, 3] on the non linear Schrödinger equation, which allow us to obtain also precise large time dynamical informations on our solutions.     

正定可对称化矩阵与预对称迭代算法

１．问题的提出 我们引入正定可对称化矩阵定义的背景是为了研究求解二阶椭圆型非自共轭方程的离散迭代有效算法、这类方程的椭圆型是本质的分析性质。是由二阶项决定的,在离散方程中表现为正定性;非自共轭性则是由方程中的一阶项引起的,在相当广泛一类问题中可通过变量代换化为自共轭。因此,我们称这类问题为正定可对称化问题。 例１．高维二阶常系数椭圆型方程其中 Ａ为常系数正定对称（ｓ．ｐ．ｄ）阵, 为正交阵, Ｄ是对角元素为正的对角阵。 先作变量代换,通过演算,偏微分方程对于新变量变成这里进而令可将原非自共轭偏微分算子…     

Diffusion-wave phenomena

We announce the forthcoming papers [4] and [5] in which we prove the existence and the uniqueness, find properties, asymptotic and regularity of the solution to diffusion-wave phenomena. We give explicit solutions for 1, 2, …, n -term time fractional diffusion-wave equations. For the distributed order equation and for the equation with n-term time fractional derivatives, we prove that the first fundamental solution is always a probability density for x ∈ R . (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

THE PERIODIC HARRY-DYM EQUATION

The development of the inverse scattering transform(I.S.T)has made it possible tosolve certain physically significant nonlinear evolution equations with periodic boundaryconditions.Date and Tanaka have considered kdv equation;Ma and Ablowitz havediscussed the cubic Schrodinger equation.In this paper,following closely the analysis in[2,3]the author considers Harry-Dym eqution(q~2)_t=-2r_(xxx)(Ⅰ)where q(x,t)is periodic in x with period π for all time q(x,t)=q(x π,t),q(x,t)=r~(-1)(x,t)>0     

On a nonlinear system consisting of three different types of differential equations

We consider a system consisting of a first order differential equation, a parabolic and an elliptic equation. Existence of weak solutions is proved by using the Schauder fixed point theorem. The paper improves some results of [3, 6] which is illustrated by example     

About Kacʼs program in kinetic theory

In this Note we present the main results from the recent work of Mischler and Mouhot (2011) [15], which answers several conjectures raised fifty years ago by Kac (1956) [10]. There Kac introduced a many-particle stochastic process (now denoted as Kac?s master equation) which, for chaotic data, converges to the spatially homogeneous Boltzmann equation. We answer the three following questions raised in Kac (1956) [10]: (1) prove the propagation of chaos for realistic microscopic interactions (i.e. in our results: hard spheres and true Maxwell molecules); (2) relate the time scales of relaxation of the stochastic process and of the limit equation by obtaining rates independent of the number of particles; (3) prove the convergence of the many-particle entropy towards the Boltzmann entropy of the solution to the limit equation (microscopic justification of the H-theorem of Boltzmann in this context). These results crucially rely on a new theory of quantitative uniform in time estimates of propagation of chaos.     

A boundary integral equation method for the transmission eigenvalue problem

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation.     

Comment on “Group analysis of KdV equation with time dependent coefficients”

We show that the group classification results of the article entitled “Group analysis of KdV equation with time dependent coefficients” which appeared in [A.G. Johnpillai, M.C. Khalique, Group analysis of KdV equation with time dependent coefficients, Appl. Math. Comput. 216 (2010) 3761-3771] can be obtained from those of a more general class by a change of variables.     

Self-adjointness of a generalized Camassa-Holm equation

It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, it is shown in [1] and [2] that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization [3] of the Camassa-Holm equation [4], we prove that the Camassa-Holm equation is self-adjoint. This property is important, e.g. for constructing conservation laws associated with symmetries of the equation in question. Accordingly, we construct conservation laws for the generalized Camassa-Holm equation using its symmetries.     

THE SUPER-BIHAMILTONIAN REDUCTION ON C~∞(S~1, OSP(1|2))

In this article, we will show that the super-bihamiltonian structures of the Kuper- KdV equation in [3], the Kuper-CH equation in [17, 18] and the super-HS equation in [11, 16, 19] can be obtained by applying a super-bihamiltonian reduction of different super-Poisson pairs defined on the loop algebra of osp（1｜2）.     

Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary

The optimal control of a class of stochastic parabolic systems is studied. This class includes systems with noise depending on spatial derivatives of the state, Neumann boundary control, and Dirichlet boundary observation, and extends a class of stochastic systems with distributed control studied by Da Prato [3] and Da Prato and Ichikawa [4]. The work is based on the direct study of the Riccati equation arising in the optimal control problem over finite time horizon. The problem over infinite time horizon and the corresponding algebraic Riccati equation are also considered.     

Stopping Distance from Observations by Stepwise Regression

In a regression equation with many terms stepwise regression allows to pick out significant terms and to exclude insignificant ones. This can be used to solve rank deficient equation systems. Yet significance is not condition. There is a rather simple method for stepwise regression starting from extended normal equations proposed by Efroymson 1960 [1] and improved by Breaux 1968 [2]. Numerically more stable methods apply orthogonal transformations (Eldén 1972 [3], Gragg-LeVeque-Trangenstein 1979 [4]). If symmetry is properly made use of [2, 5] the method of Efroymson is the most efficient one with respect to computing time. But reasonable results are obtained only if the terms in the regression equation are physically justified. This illustrates the search for a formula of the stopping distance derived from observations, an exercise given in [9], a book widely used on some universities. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Boltzmann equation with frictional force

The Euler equations with frictional force have been extensively studied. Since the Boltzmann equation is closely related to the equations of gas dynamics, we study, in this paper, the Boltzmann equation with frictional force when the external force is proportional to the macroscopic velocity. It is shown that smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution which approaches to the global Maxwellian time asymptotically. The analysis is based on the macro-micro decomposition for the Boltzmann equation introduced in Liu et al. [Energy method for the Boltzmann equation, Physica D 188 (3-4) (2004) 178-192] and Liu and Yu [Boltzmann equation: micro-macro-decompositions and positivity of shock profiles, Comm. Math. Phys. 246(1) (2004) 133-179] through energy estimates.     

The super-bihamiltonian reduction on C(, OSP(|))

In this article, we will show that the super-bihamiltonian structures of the Kuper-KdV equation in [3], the Kuper-CH equation in [17, 18] and the super-HS equation in [11, 16,19] can be obtained by applying a super-bihamiltonian reduction of different super-Poisson pairs defined on the loop algebra of osp(1|2).     

On the weak control of slowly damped systems : PMM vol. 42, no. 6, 1978. pp. 1000–1005

Asymptotic formulas are obtained which make it possible to derive the first approximation solution of the Riccati matrix algebraic equation of special form. Method is based on Bass' formulas [1] and the theory of perturbations [2], The problem of control of a slowly damped oscillator is investigated in detail. Formulation of the problem in this paper differs substantially from that in [3] (no assumption is made about single-frequency oscillations, and only a stationary system is considered over an infinite time interval).     

Arrival times for the wave equation

We establish a definition of arrival time of a wavefront for a propagating wave in anisotropic media that is initially at rest and where the governing partial differential equation is the anisotropic wave equation. This definition of arrival time is not the same as the one in [8, 12]; it eliminates pathological discontinuities that can occur with the older definition and is still consistent with physical intuition. What is substantively new here is that we show that the newly defined arrival time is locally Lipschitz‐continuous. Then following the method in [8, 12] we establish that it satisfies the eikonal equation. Furthermore, in the isotropic case we establish that the arrival time, as defined here, is the unique viscosity solution of the eikonal equation. Our motivation for this work is to use this arrival time at points in the interior of a physical or biological material, which is estimated from displacement measurements, to determine properties of the medium that are represented as functions in the eikonal equation; see [8, 10, 11, 12]. © 2010 Wiley Periodicals, Inc.     

BLOW-UP OF THE SOLUTION FOR A KIND OF HIGHER ORDER HYPERBOLIC EVOLUTION SYSTEMS

In this paper, we give some results on the blow-up behaviors of the solution to the mixed problem for some higher nonlinear hyperbolic evolution equation in finite time. By introducing the "blow-up factor K(u,ut)" we get some new results, which generalize the conclusions of [3] and [4].