Deautonomisation of differential-difference equations and discrete Painlevé equations

We examine a family of integrable differential-difference equations and obtain their non-autonomous extensions using a discrete/continuous integrability criterion.     

From quantum Euler–Maxwell equations to incompressible Euler equations

The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained.     

p-Euler equations and p-Navier–Stokes equations

We propose in this work new systems of equations which we call p-Euler equations and p-Navier–Stokes equations. p-Euler equations are derived as the Euler–Lagrange equations for the action represented by the Benamou–Brenier characterization of Wasserstein-p distances, with incompressibility constraint. p-Euler equations have similar structures with the usual Euler equations but the ‘momentum’ is the signed ($p?1$)-th power of the velocity. In the 2D case, the p-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the p-Laplacian of the streamfunction. By adding diffusion presented by γ-Laplacian of the velocity, we obtain what we call p-Navier–Stokes equations. If $\gamma =p$, the a priori energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the p-Navier–Stokes equations in ${\mathbb{R}}^d$ for $\gamma =p$ and $p\ge d\ge 2$ through a compactness criterion.     

Boundary layer problem: Navier–Stokes equations and Euler equations

This work is concerned with the boundary layer turbulence, which is an outstanding problem in fluid mechanics. We consider an incompressible viscous fluid in 2D domains with permeable walls. The permeability is described by the Yudovich condition. The goal of the article is to study the fluid behavior at vanishing viscosity (large Reynold’s numbers). We show that the vanishing viscous limit is a solution of the Euler equations with the Yudovich condition on the inflow region of the boundary.     

Nonlinear pseudoparabolic equations as singular limit of reaction–diffusion equations

In this article, a solution of a nonlinear pseudoparabolic equation is constructed as a singular limit of a sequence of solutions of quasilinear hyperbolic equations. If a system with cross diffusion, modelling the reaction and diffusion of two biological, chemical, or physical substances, is reduced then such an hyperbolic equation is obtained. For regular solutions even uniqueness can be shown, although the needed regularity can only be proved in two dimensions.     

Reflectionless Sturm–Liouville equations

We consider compactly supported perturbations of periodic Sturm–Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g., bounds and asymptotics for the relevant m$m$-functions.     

Matrix Painlevé II equations

Theoretical and Mathematical Physics - We use the Painlevé–Kovalevskaya test to find three matrix versions of the Painlevé II equation. We interpret all these equations as...     

A note on stochastic semilinear equations and their associated Fokker–Planck equations

The main purpose of this paper is to prove existence and uniqueness of (probabilistically weak and strong) solutions to stochastic differential equations (SDE) on Hilbert spaces under a new approximation condition on the drift, recently proposed in [6] to solve Fokker–Planck equations (FPE), extended in this paper to a considerably larger class of drifts. As a consequence we prove existence of martingale solutions to the SDE (whose time marginals then solve the corresponding FPE). Applications include stochastic semilinear partial differential equations with white noise and a non-linear drift part which is the sum of a Burgers-type part and a reaction diffusion part. The main novelty is that the latter is no longer assumed to be of at most linear, but of at most polynomial growth. This case so far had not been covered by the existing literature. We also give a direct and more analytic proof for existence of solutions to the corresponding FPE, extending the technique from [6] to our more general framework, which in turn requires to work on a suitable Gelfand triple rather than just the Hilbert state space.     

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ ⁿ . We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].     

About inclusion of Maxwell’s equations in systems relativistic of the invariant equations

Maxwell’s equations, relativistic invariant equations, foundations of difference schemes.     

Existence and concentration behavior of sign-changing solutions for quasilinear Schr?dinger equations equations

We consider the quasilinear Schrdinger equations of the form-ε~2?u + V(x)u- ε~2?(u2)u = g(u), x ∈ R~N,where ε 0 is a small parameter, the nonlinearity g(u) ∈ C~1(R) is an odd function with subcritical growth and V(x) is a positive Hlder continuous function which is bounded from below, away from zero, and infΛV(x) inf ?ΛV(x) for some open bounded subset Λ of RN. We prove that there is an ε0 0 such that for all ε∈(0, ε0],the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε→ 0~+.     

Vanishing viscosity limit for Navier–Stokes equations with general viscosity to Euler equations

In this paper, we study vanishing viscosity limit of 1-D isentropic compressible Navier–Stokes equations with general viscosity to isentropic Euler equations. Firstly, we improve estimates of the entropy flux, then we obtain that the weak solution of the isentropic Euler equations is the inviscid limit of the isentropic compressible Navier–Stokes equations with general viscosity using the compensated compactness frame recently established by G.-Q. Chen and M. Perepelitsa.     

Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.     

A numerical study of variable depth KdV equations and generalizations of Camassa–Holm-like equations

In this paper we numerically study the KdV-top equation and compare it with the Boussinesq equations over uneven bottoms. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa–Holm equation, we find several finite difference schemes that conserve a discrete energy for the fully discrete scheme. Because of its accuracy for the conservation of energy, our numerical scheme is also of interest even in the simple case of flat bottoms. We compare this approach with the discontinuous Galerkin method.     

Studies on the Painlevé equations

Summary In this series of papers, we study birational canonical transformations of the Painlevé system , that is, the Hamiltonian system associated with the Painlevé differential equations. We consider also -function related to and particular solutions of . The present article concerns the sixth Painlevé equation. By giving the explicit forms of the canonical transformations of associated with the affine transformations of the space of parameters of , we obtain the non-linear representation: GG_*, of the affine Weyl group of the exceptional root system of the type F₄ A canonical transformation of G_* can extend to the correspondence of the -functions related to . We show the certain sequence of -functions satisfies the equation of the Toda lattice. Solutions of , which can be written by the use of the hypergeometric functions, are studied in details.     

评《Stochastic differential equations》

吴让泉同志的专著《Stochastic differential equations》于1985年由Pitman出版公司出版。该书除预备知识这一章外,共分三章.第一章的主要部分介绍了Emeyy关于半鞅随机微分方程解的存在性及唯一性结果(1978年),此外介绍了作者与毛学荣对这一结果的一个推广.第二章详细讨论了线性随机微分方程解的表示,部分结果是经典的(例如见 Arnold:“SDE”,John Wiley＆Sons,1974),部分结果(随机 Liouville公式)是作者新近的工作(但Jacod在“Sem.Prob.XVI,1982”中讨论了非连续半鞅的线性随机微分方程,所得结果更一     

Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations

In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward–backward stochastic differential equations (FBSDEs) is established, which can be regarded as an extension of the Feynman–Kac formula to the non-Markovian framework.     

Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations

We consider some reaction–diffusion equations perturbed by white noise and prove Schauder estimates for the elliptic problem associated with the generator of the corresponding transition semigroup, defined in the Banach space of continuous functions. This requires the proof of some new interpolation result.     

Doubly nonlocal Cahn–Hilliard equations

We consider a doubly nonlocal nonlinear parabolic equation which describes phase-segregation of a two-component material in a bounded domain. This model is a more general version than the recent nonlocal Cahn–Hilliard equation proposed by Giacomin and Lebowitz [26], such that it reduces to the latter under certain conditions. We establish well-posedness results along with regularity and long-time results in the case when the interaction between the two levels of nonlocality is strong-to-weak.