Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhge?m–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.     

On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials

We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator systems. We prove the existence of the solution of the symmetrized KS equation for manybody interaction potentials which are either attractive (nonpositive) and finite-range or infinite-range and repulsive (positive). The proposed procedure of symmetrization of the KS equation is new and based on the superstability of many-body potentials.     

Stability of solitary waves and weak rotation limit for the Ostrovsky equation

Ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation. In this model dispersion is taken into account while dissipation is neglected. Existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter (which can be either positive or negative). It is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations. The issue of passing to the limit as the rotation parameter tends to zero for solutions of the Cauchy problem is investigated on a bounded time interval.     

Influence of cubic nonlinearity on the dispersion relation for long waves on a water surface

The influence of cubic nonlinearity on the dispersion relation for long waves on a water surface is analyzed. In the long wavelength limit, it is shown that the dispersion relation is not affected by the cubic terms. The results are compared with the dispersion relation for stationary solutions to the Korteweg—de Vries equation.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated with a metric g of class C² and we prove dispersion and Strichartz estimates for the solution of this equation in terms of geodesics associated with g. We introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the Euclidean case holds. To deal with the case of non-trapped long range perturbation of the Euclidean metric, we prove a global velocity moments effect on the solution. In particular, we obtain global in time Strichartz estimates for metrics such that the dispersion estimate is not satisfied.     

Notes on a finite differencing scheme for the dispersion equation

Finite difference techniques applied to atmospheric dispersion problems often encounter time step limitations due to the variance in the characteristic length scales (horizontal to vertical) of both the field variables and the computational region. Methods to maximize the integration time step are explored and techniques are described which ensure numerical accuracy and stability of these optimized time step techniques.

To circumvent time step limitations arising from consideration of the vertical diffusion term in the dispersion equation, a column implicitization technique is suggested which, through correction terms added to the differencing equation to compensate for truncation errors, provides an efficient and economical atmospheric dispersion solver which is insensitive to the common time step limitations of explicit schemes when large aspect ratio computational volumes are required. Further, it is shown that a relaxed stability criteria proposed by Leonard and Clancy for explicit differencing of the horizontal terms in the dispersion equation, presents a further saving in computational time provided correction terms to the differencing equation are included to eliminate phase and amplitude errors resulting from the larger time steps employed.     

Breaking Homoclinic Connections for a Singularly Perturbed Differential Equation and the Stokes Phenomenon

Behavior of the separatrix solution y ( t )=−(3/2)/cosh²( t /2) (homoclinic connection) of the second order equation y "= y + y ² that undergoes the singular perturbation ɛ² y ""+ y "= y + y ², where ɛ>0 is a small parameter, is considered. This equation arises in the theory of traveling water waves in the presence of surface tension. It has been demonstrated both rigorously [1, 2] and using formal asymptotic arguments [3, 4] that the above-mentioned solution could not survive the perturbation.The latter papers were based on the Kruskal–Segur method (KS method), originally developed for the equation of crystal growth [5]. In fact, the key point of this method is the reduction of the original problem to the Stokes phenomenon of a certain parameterless "leading-order" equation. The main purpose of this article is further development of the KS method to study the breaking of homoclinic connections. In particular: (1) a rigorous basis for the KS method in the case of the above-mentioned perturbed problem is provided; and (2) it is demonstrated that breaking of a homoclinic connection is reducible to a monodromy problem for coalescing (as ɛ→0) regular singular points, where the Stokes phenomenon plays the role of the leading-order approximation.     

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

A two-dimensional model for extensional motion of a pre-stressedincompressible elastic layer near its cut-off frequencies isderived. Leading-order solutions for displacement and pressureare obtained in terms of the long wave amplitude by direct asymptoticintegration. A governing equation, together with correctionsfor displacement and pressure, is derived from the second-orderproblem. A novel feature of this (two-dimensional) hyperbolicgoverning equation is that, for certain pre-stressed states,time and one of the two (in-plane) spatial variables can changeroles. Although whenever this phenomenon occurs the equationstill remains hyperbolic, it is clearly not wave-like. The second-ordersolution is completed by deriving a refined governing equationfrom the third-order problem. Asymptotic consistency, in thesense that the dispersion relation associated with the two-dimensionalmodel concurs with the appropriate order expansion of the three-dimensionalrelation at each order, is verified. The model has particularapplication to stationary thickness vibration of, or transientresponse to high frequency shock loading in, thin walled bodies.     

Three-Dimensional Water Waves

We study the evolution of small-amplitude water waves when the fluid motion is three dimensional. An isotropic pseudodifferential equation that governs the evolution of the free surface of a fluid with arbitrary, uniform depth is derived. It is shown to reduce to the Benney-Luke equation, the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and to the nonlinear shallow water theory in the appropriate limits. We compute, numerically, doubly periodic solutions to this equation. In the weakly two-dimensional long wave limit, the computed patterns and nonlinear dispersion relations agree well with those of the doubly periodic theta function solutions to the KP equation. These solutions correspond to traveling hexagonal wave patterns, and they have been compared with experimental measurements by Hammack, Scheffner, and Segur. In the fully two-dimensional long wave case, the solutions deviate considerably from those of KP, indicating the limitation of that equation. In the finite depth case, both resonant and nonresonant traveling wave patterns are obtained.     

Error analysis and applications of the Fourier-Galerkin Runge-Kutta schemes for high-order stiff PDEs

An integrating factor mixed with Runge-Kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. In this paper, the novel hybrid Fourier-Galerkin Runge-Kutta scheme, with the aid of an integrating factor, is proposed to solve nonlinear high-order stiff PDEs. Error analysis and properties of the scheme are provided. Application to the approximate solution of the nonlinear stiff Korteweg-de Vries (the 3rd order PDE, dispersive equation), Kuramoto-Sivashinsky (the 4th order PDE, dissipative equation) and Kawahara (the 5th order PDE) equations are presented. Comparisons are made between this proposed scheme and the competing method given by Kassam and Trefethen. It is found that for KdV, KS and Kawahara equations, the proposed method is the best.     

New exact solutions and conservation laws of a class of Kuramoto Sivashinsky (KS) equations

Lie symmetry method is used to perform detailed analysis on a class of KS equations. It is shown that the Lie algebra of the equation spanned by the vector fields of dilations in time and space are lost as a result of the linearity of the equation when n = 1. Symmetry reductions are carried out using each member of the optimal system. The reduced equations are further studied to obtain certain general solutions. Moreover, the conserved vectors are obtained through the application of Noether's theorem.     

一类多维广义KS型方程解的BLOW-UP

考虑一类高维广义KS型方程解的Blow-up,利用特征函数和凸性方法,证明了在某些条件下,问题的光滑解只能在一个有界区间中存在,并在有限时间内爆破.     

Optimized composite finite difference schemes for atmospheric flow modeling

In this paper, we use some finite difference methods in order to solve an atmospheric flow problem described by an advection–diffusion equation. This flow problem was solved by Clancy using forward‐time central space (FTCS) scheme and is challenging to simulate due to large errors in phase and amplitude which are generated especially over long propagation times. Clancy also derived stability limits for FTCS scheme. We use Von Neumann stability analysis and the approach of Hindmarsch et al. which is an improved technique over that of Clancy in order to obtain the region of stability of some methods such as FTCS, Lax–Wendroff (LW), Crank–Nicolson. We also construct a nonstandard finite difference (NSFD) scheme. Properties like stability and consistency are studied. To improve the results due to significant numerical dispersion or numerical dissipation, we derive a new composite scheme consisting of three applications of LW followed by one application of NSFD. The latter acts like a filter to remove the dispersive oscillations from LW. We further improve the composite scheme by computing the optimal temporal step size at a given spatial step size using two techniques namely; by minimizing the square of dispersion error and by minimizing the sum of squares of dispersion and dissipation errors.     

Approximate Analytical and Numerical Solutions of the Stationary Ostrovsky Equation

Approximate stationary solutions of the Ostrovsky equation describing long weakly nonlinear waves in a rotating liquid are constructed. These solutions may be regarded as a periodic sequence of arcs of parabolas containing Korteweg-de Vries solitons at the junctures. Results of numerical computations of the dynamics of the approximate solutions obtained from the nonstationary Ostrovsky equation are presented. It is found that, in the presence of negative dispersion, the shape of a stationary wave is well predicted by the approximate theory, whereas the calculated wave velocity differs slightly from the theoretical value. The stationary solutions in media with positive dispersion are evidently unstable (at least for sufficiently strong rotation), and numerical computations demonstrate a complicated picture of nonstationary destruction.     

Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion

In this paper, we provide a blow-up mechanism to the modified Camassa–Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result with an initial data having a region of mild oscillation. A key feature of the analysis is the development of the Burgers-type inequalities with focusing property on characteristics, which can be deduced from tracing the ratio between solution and its gradient. Using the continuity and monotonicity of the solutions, we then extend this blow-up criterion to the case of negative linear dispersion, and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that in the case of non-negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile. In contrast to the Camassa–Holm equation with linear dispersion, the effect of linear dispersion of the modified Camassa–Holm equation on the blow-up phenomena is rather delicate.     

New stochastic model for dispersion in heterogeneous porous media: 1. Application to unbounded domains

A new model of solute dispersion in porous media that avoids Fickian assumptions and that can be applied to variable drift velocities as in non-homogeneous or geometrically constricted aquifers, is presented. A key feature is the recognition that because drift velocity acts as a driving coefficient in the kinematical equation that describes random fluid displacements at the pore scale, the use of Ito calculus and related tools from stochastic differential equation theory (SPDE) is required to properly model interaction between pore scale randomness and macroscopic change of the drift velocity. Solute transport is described by formulating an integral version of the solute mass conservation equations, using a probability density. By appropriate linking of this to the related but distinct probability density arising from the kinematical SPDE, it is shown that the evolution of a Gaussian solute plume can be calculated, and in particular its time-dependent variance and hence dispersivity. Exact analytical solutions of the differential and integral equations that this procedure involves, are presented for the case of a constant drift velocity, as well as for a constant velocity gradient. In the former case, diffusive dispersion as familiar from the advection–dispersion equation is recovered. However, in the latter case, it is shown that there are not only reversible kinematical dispersion effects, but also irreversible, intrinsically stochastic contributions in excess of that predicted by diffusive dispersion. Moreover, this intrinsic contribution has a non-linear time dependence and hence opens up the way for an explanation of the strong observed scale dependence of dispersivity.     

对流扩散方程的QC3无网格法

对流扩散方程作为偏微分运动方程的分支,在流体力学、气体动力学等领域有着重要应用.为解决对流扩散方程难以通过解析法得到解析解的难题,采用二阶一致3点积分(Quadratically Consistent 3-Point Integration,简称QC3)提高无网格法的计算效率,通过对积分点上形函数导数的修正,改善无网格法的精度和收敛性.本文将QC3无网格法拓展到对流扩散方程问题中,时域离散采用广义特征线Galerkin法,空间离散采用QC3法.数值结果表明,应用QC3无网格法得到的对流扩散问题数值解与解析解十分接近,验证了QC3无网格法解决对流扩散问题的可行性.     

Extensional wave motion in homogenous isotropic thermoelastic plate by using asymptotic method

The present investigation is concerned with the study of extensional wave motion in an infinite homogenous isotropic, thermoelastic plate by using asymptotic method. The governing equation for the extensional wave motions have been derived from the system of three-dimensional dynamical equations of linear coupled theory of thermoelasticity. All coefficients of the differential operator are expressed as explicit functions of the material parameters. The velocity dispersion equation for the extensional wave motion is deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity are also derived. The thermoelastic Rayleigh–Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established to that of asymptotic method. The dispersion curves for phase velocity and attenuation coefficient are shown graphically for extensional wave motion of the plates.     

The Zakai Equation of Nonlinear Filtering for Jump-Diffusion Observations: Existence and Uniqueness

In this paper we study a nonlinear filtering problem for a general Markovian partially observed system (X,Y), whose dynamics is modeled by correlated jump-diffusions having common jump times. At any time t∈[0,T], the σ-algebra $\mathcal{F}^{Y}_{t}:= \sigma\{ Y_{s}: s\leq t\}$ provides all the available information about the signal X _t . The central goal of stochastic filtering is to characterize the filter, π _t , which is the conditional distribution of X _t , given the observed data $\mathcal{F}^{Y}_{t}$ . In Ceci and Colaneri (Adv. Appl. Probab. 44(3):678–701, 2012) it is proved that π is the unique probability measure-valued process satisfying a nonlinear stochastic equation, the so-called Kushner-Stratonovich equation (in short KS equation). In this paper the aim is to improve the hypotheses to obtain the KS equation and describe the filter π in terms of the unnormalized filter ?, which is solution of a linear stochastic differential equation, the so-called Zakai equation. We prove the equivalence between strong uniqueness of the solution of the KS equation and strong uniqueness of the solution of the Zakai one and, as a consequence, we deduce pathwise uniqueness of the solution of the Zakai equation by applying the Filtered Martingale Problem approach (Kurtz and Ocone in Ann. Probab. 16:80–107, 1988; Ceci and Colaneri in Adv. Appl. Probab. 44(3):678–701, 2012). To conclude, we discuss some particular models.     

The backward group preserving scheme for 1D backward in time advection‐dispersion equation

In this article, we propose a backward group preserving scheme (BGPS) on the advection‐dispersion equation (ADE) for tackling of the contamination problems. The BGPS has been successfully applied on the backward heat conduction problems as well as the backward in time Burgers equation, but it has never been applied to solve the ADE. The BGPS is able to recover the spatial distribution of groundwater contaminant concentration in this work. Several numerical examples are worked out, and we show, based on those numerical examples, that the BGPS is applicable to the ADE and the method can also handle the ADE with piecewise constant dispersion coefficients. When a steep gradient is appeared in the solution, several steps of the BGPS can be used to retrieve the desired initial data and its result is better than the marching‐jury backward beam equation (MJBBE) method as far as our examples are concerned. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010