Probability Distribution Function of Passive Scalars in Shell Models

A shell-model version of passive scalar problem is introduced, which is inspired by the model of K. Ohkitani and M. Yakhot [K. Ohkitani and M. Yakhot, Phys. Rev. Lett. 60 (1988) 983; K. Ohkitani and M. Yakhot, Prog. Theor. Phys. 81 (1988) 329]. As in the original problem, the prescribed random velocity field is Gaussian and δ correlated in time. Deterministic differential equations are regarded as nonlinear Langevin equation. Then, the Fokker-Planck equations of PDF for passive scalars are obtained and solved numerically. In energy input range (n<5, n is the shell number.), the probability distribution function (PDF) of passive scalars is near the Gaussian distribution. In inertial range (5≤n≤16) and dissipation range (n≥17), the probability distribution function (PDF) of passive scalars has obvious intermittence. And the scaling power of passive scalar is anomalous. The results of numerical simulations are compared with experimental measurements.     

Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation

We consider a space-continuous and time-discrete polymer model for positive temperature and the associated zero temperature model of last passage percolation type. In our previous work, we constructed and studied infinite-volume polymer measures and one-sided infinite minimizers for the associated variational principle, and used these objects for the study of global stationary solutions of the Burgers equation with positive or zero viscosity and random kick forcing, on the entire real line. In this paper, we prove that in the zero temperature limit, the infinite-volume polymer measures concentrate on the one-sided minimizers and that the associated global solutions of the viscous Burgers equation with random kick forcing converge to the global solutions of the inviscid equation.     

Remarks about the Inviscid Limit of the Navier–Stokes System

In this paper we prove two results about the inviscid limit of the Navier-Stokes system. The first one concerns the convergence in H ^s of a sequence of solutions to the Navier-Stokes system when the viscosity goes to zero and the initial data is in H ^s . The second result deals with the best rate of convergence for vortex patch initial data in 2 and 3 dimensions. We present here a simple proof which also works in the 3D case. The 3D case is new.     

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.     

Dimension Theory for Invariant Measures of Endomorphisms

We establish the exact dimensional property of an ergodic hyperbolic measure for a C ² non-invertible but non-degenerate endomorphism on a compact Riemannian manifold without boundary. Based on this, we give a new formula of Lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms. Our results extend several well known theorems of Barreira et al. (Ann Math 149:755–783, 1999) and Ledrappier and Young [Commun Math Phys 117(4):529–548, 1988] for diffeomorphisms to the case of endomorphisms.     

Multiscale Expansion of Invariant Measures for SPDEs

We derive the first two terms in an -expansion for the invariant measure of a class of semilinear parabolic SPDEs near a change of stability, when the noise strength and the linear instability are of comparable order ². This result gives insight into the stochastic bifurcation and allows to rigorously approximate correlation functions. The error between the approximate and the true invariant measure is bounded in both the Wasserstein and the total variation distance.Acknowledgements The work of D.B. was supported by DFG-Forschungsstipendium BL535/5-1. The work of M.H. was supported by the Fonds National Suisse. Both authors would like to thank the MRC at the University of Warwick and especially David Elworthy for their warm hospitality.     

Influence of Velocity Shell Correlations on Anomalous Scaling Exponents of Passive Scalars in Shell Models

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of passive scalars of turbulence. Different to the original problem, the distribution function of the prescribed random velocity field is multi-dimensional normal and delta-correlated in time. Here, our random velocity field is spatially correlative. For comparison, we also give the result obtained by the Gaussian random velocity field without spatial correlation. The anomalous scaling exponents H（p） of passive scalar advected by two kinds of random velocity above are determined for structure function up to p= 15 by numerical simulations of the random shell model with Runge-Kutta methods to solve the stochastic differential equations. We observed that the H（p） ＇s obtained by the multi-dimeasional normal distribution random velocity are more anomalous than those obtained by the independent Gaussian random velocity.     

A Kato Type Theorem for the Inviscid Limit of the Navier-Stokes Equations with a Moving Rigid Body

The issue of the inviscid limit for the incompressible Navier-Stokes equations when a no-slip condition is prescribed on the boundary is a famous open problem. A result by Kato (Math Sci Res Inst Publ 2:85?C98, 1984) says that convergence to the Euler equations holds true in the energy space if and only if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes. Of course, if one considers the motion of a solid body in an incompressible fluid, with a no-slip condition at the interface, the issue of the inviscid limit is as least as difficult. However it is not clear if the additional difficulties linked to the body??s dynamic make this issue more difficult or not. In this paper we consider the motion of a rigid body in an incompressible fluid occupying the complementary set in the space and we prove that a Kato type condition implies the convergence of the fluid velocity and of the body velocity as well, which seems to indicate that an answer in the case of a fixed boundary could also bring an answer to the case where there is a moving body in the fluid.     

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in $$\mathbb R^2$$

We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in . We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstrophy balance.     

旋转不变的非局域均值算法在磁共振图像去噪中的应用

磁共振成像（MRI）实验时常采用多次扫描累加平均提高图像信噪比（SNR）,但当扫描过程中运动引起图像变形时,简单地累加平均就无法奏效.为此,本研究组曾提出一种匹配加权平均方法（MWA）提高图像的信噪比.在此基础上,该文提出一种旋转不变的非局域均值算法（RINLM）,即选取圆形邻域区域并将其划分为一系列以中心像素为圆心的等面积圆环,再计算模式的相似性.RINLM算法可以更好地利用图像中旋转的冗余信息、找到更多的相似结构,提高算法的去噪性能.我们把该方法应用于低信噪比图像序列的平均和去噪中,可以更好地处理旋转的局部运动.与非局域均值算法（NLM）相比,RINLM算法可以进一步提高图像的信噪比;与MWA方法相比,其与RINLM算法的结合可以进一步提高磁共振图像序列信噪比,更好的保持图像边缘信息.     

Absolute Continuity of the Invariant Measures for Some Stochastic PDEs

We consider a free system and an interacting systems having invariant measures μ and ν respectively. Under suitable assumptions we prove an explicit formula relating ν with μ and implying the absolute continuity of ν with respect to μ. We apply our result to a reaction-diffusion equation and to the Burgers equation.     

Influence of Velocity Shell Correlations on Anomalous Scaling Exponents of Passive Scalars in Shell Models

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of passive scalars of turbulence. Different to the original problem, the distribution function of the prescribed random velocity field is multi-dimensional normal and delta-correlated in time. Here, our random velocity field is spatially correlative. For comparison, we also give the result obtained by the Gaussian random velocity field without spatial correlation. The anomalous scaling exponents H(p) of passive scalar advected by two kinds of random velocity above are determined for structure function up to p=15 by numerical simulations of the random shell model with Runge-Kutta methods to solve the stochastic differential equations. We observed that the H(p)'s obtained by the multi-dimensional normal distribution random velocity are more anomalous than those obtained by the independent Gaussian random velocity.     

Invariant Probability Measures and Non-wandering Sets for Impulsive Semiflows

We consider impulsive dynamical systems defined on compact metric spaces and their respective impulsive semiflows. We establish sufficient conditions for the existence of probability measures which are invariant under such impulsive semiflows. Under these conditions we also deduce the forward invariance of their non-wandering sets except the discontinuity points.     

Invariant Measures of Gaussian Type for 2D Turbulence

Gaussian measures ?? ^??,?? are associated to some stochastic 2D models of turbulence. They are Gibbs measures constructed by means of an invariant quantity of the system depending on some parameter ?? (related to the 2D nature of the fluid) and the viscosity???. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure ?? ^??,?? is invariant for this flow and is the unique invariant measure. Finally, we prove that the deterministic inviscid equation has a ?? ^??,?? -stationary solution (for any ??>0).     

Gibbsian Dynamics and Invariant Measures for Stochastic Dissipative PDEs

We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg–Landau, Kuramoto–Sivashinsky and Cahn–Hilliard equations with stochastic forcing.