The aim of this paper is to provide explicitly a sequence of m-dimensional approximate inertial manifolds M(m,j,)j = 1,2,, for each positive integer m, for the Kuramoto-Sivashinsky equations. A very thin neighborhood ...The aim of this paper is to provide explicitly a sequence of m-dimensional approximate inertial manifolds M(m,j,)j = 1,2,, for each positive integer m, for the Kuramoto-Sivashinsky equations. A very thin neighborhood into which the orbits enter with an exponential speed and in a finite time is associated with each manifold. The thickness of these neighborhoods decreases with increasing m for a fixed order j. Besides, the neighborhoods localize the global attractor and aid in the approximate computation of large-time solutions of the Kuramoto-Sivashinsky equations.展开更多
In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a two dimensional generalized anisotropy Kuramoto-Sivashinsky Equation. Then we prove the existence of the gl...In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a two dimensional generalized anisotropy Kuramoto-Sivashinsky Equation. Then we prove the existence of the global attractor. Finally, we get the upper bound estimation of the Haus-dorff and fractal dimension of attractor.展开更多
This paper is concentrated on a nonlinear Galerkin method with sm small- scale components for Kuramoto-Sivashmsky equation, in which convergence results and the analysis of error estimates are given, The conclusion sh...This paper is concentrated on a nonlinear Galerkin method with sm small- scale components for Kuramoto-Sivashmsky equation, in which convergence results and the analysis of error estimates are given, The conclusion shows that this choce of modes is efficient .for The method modifred.展开更多
Nonlinear Galerkin methods are new schemes for integrating dissipative systems:In the present paper, we obtain the estimates to the rate of convergence of such methods for Kuramoto-Sivashinsky equations. In particular...Nonlinear Galerkin methods are new schemes for integrating dissipative systems:In the present paper, we obtain the estimates to the rate of convergence of such methods for Kuramoto-Sivashinsky equations. In particular, by an illustrative example, we show that nonlinear Galerkin methods converge faster than the usual Galerkin method.展开更多
Under the periodic boundary condition, dynamic bifurcation and stability in the modified Kuramoto-Sivashinsky equation with a higher-order nonlinearity i.t（uχ）Puxz are investigated by using the centre manifold redu...Under the periodic boundary condition, dynamic bifurcation and stability in the modified Kuramoto-Sivashinsky equation with a higher-order nonlinearity i.t（uχ）Puxz are investigated by using the centre manifold reduction procedure. The result shows that as the control parameter crosses a critical value, the system undergoes a bifurcation from the trivial solution to produce a cycle consisting of locally asymptotically stable equilibrium points. Furthermore, for cases in which the distances to the bifurcation points are small enough, one-order approximations to the bifurcation solutions are obtained.展开更多
In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is use...In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is used to study the asymptotic behaviour of Kuramoto-Sivashinsky equation and to construct the bifurcation diagrams.展开更多
In this paper, the random Kuramoto-Sivashinsky equation with additive noise is studied numerically, using the finite difference method to simulate the effect of different amplitude of noise on the solitary wave. And n...In this paper, the random Kuramoto-Sivashinsky equation with additive noise is studied numerically, using the finite difference method to simulate the effect of different amplitude of noise on the solitary wave. And numerical experiments show that the white noise does not affect the propagation of the solitary wave, but can increase the amplitude of the solitary wave.展开更多
In this paper, it is considered that the global existence, uniqueness and regularity results for the Cauchy problem of the well-known Kuramoto-Sivashinsky equation [GRAPHICS] only under the condition u(0)(x) is an ele...In this paper, it is considered that the global existence, uniqueness and regularity results for the Cauchy problem of the well-known Kuramoto-Sivashinsky equation [GRAPHICS] only under the condition u(0)(x) is an element of L-2(R-N, R-n). Where u(t, x) = (u(1)(t, x), ..., u(n)(t, x))(T) is the unknown vector-valued function. Results show that for N < 6,.u(0)(x) is an element of L-2(R-N, R-n), the above Cauchy problem admits a unique global solution u(t, x) which belongs to C-infinity,C-infinity(R-N x (0, infinity)).展开更多
The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multidimensions(n ≥ 3) is considered. Based on Green's function method, some ingenious energy estimates are given. Then the global existence ...The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multidimensions(n ≥ 3) is considered. Based on Green's function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the L^p convergence rate of the solution is obtained.展开更多
This paper focuses on the application of Exp-function method to obtain generalized solutions of the KdV-Burgers-Kuramoto equation and the Kuramoto-Sivashinsky equation.It is demonstrated that the Exp-function method p...This paper focuses on the application of Exp-function method to obtain generalized solutions of the KdV-Burgers-Kuramoto equation and the Kuramoto-Sivashinsky equation.It is demonstrated that the Exp-function method provides a mathematical tool for solving the nonlinear evolution equation in mathematical physics.展开更多
In this paper we use the spectral method to analyse the generalized Kuramoto-Sivashinsky equations. We prove the existence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estima...In this paper we use the spectral method to analyse the generalized Kuramoto-Sivashinsky equations. We prove the existence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estimation between spectral approximate solution and exact solution on large time.展开更多
Global well-posedness of the initial-boundary value problem for the stochastic generalized KuramotoSivashinsky equation in a bounded domain D with a multiplicative noise is studied. It is shown that under suitable suf...Global well-posedness of the initial-boundary value problem for the stochastic generalized KuramotoSivashinsky equation in a bounded domain D with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data u0 ∈L2（D×Ω）, this problem has a unique global solution u in the space L2（Ω, C（[0, T ], L2（D））） for any T 〉 0, and the solution map u0 →u is Lipschitz continuous.展开更多
We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points.We aim to identify the unknown number of sources ...We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points.We aim to identify the unknown number of sources and their locations along with their strengths.In our previous work,we proved that minimum measurement points needed under the noise-free setting.In this paper,we extend the proof to cover the noisy cases over a border class of source functions.We show that if the regularization parameter is chosen properly,the problem can be transformed into a poles identification problem.A reconstruction scheme is proposed on the basis of the developed theoretical results.Numerical demonstrations in 2D and 3D conclude the paper.展开更多
文摘The aim of this paper is to provide explicitly a sequence of m-dimensional approximate inertial manifolds M(m,j,)j = 1,2,, for each positive integer m, for the Kuramoto-Sivashinsky equations. A very thin neighborhood into which the orbits enter with an exponential speed and in a finite time is associated with each manifold. The thickness of these neighborhoods decreases with increasing m for a fixed order j. Besides, the neighborhoods localize the global attractor and aid in the approximate computation of large-time solutions of the Kuramoto-Sivashinsky equations.
文摘In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a two dimensional generalized anisotropy Kuramoto-Sivashinsky Equation. Then we prove the existence of the global attractor. Finally, we get the upper bound estimation of the Haus-dorff and fractal dimension of attractor.
文摘This paper is concentrated on a nonlinear Galerkin method with sm small- scale components for Kuramoto-Sivashmsky equation, in which convergence results and the analysis of error estimates are given, The conclusion shows that this choce of modes is efficient .for The method modifred.
文摘Nonlinear Galerkin methods are new schemes for integrating dissipative systems:In the present paper, we obtain the estimates to the rate of convergence of such methods for Kuramoto-Sivashinsky equations. In particular, by an illustrative example, we show that nonlinear Galerkin methods converge faster than the usual Galerkin method.
文摘Under the periodic boundary condition, dynamic bifurcation and stability in the modified Kuramoto-Sivashinsky equation with a higher-order nonlinearity i.t（uχ）Puxz are investigated by using the centre manifold reduction procedure. The result shows that as the control parameter crosses a critical value, the system undergoes a bifurcation from the trivial solution to produce a cycle consisting of locally asymptotically stable equilibrium points. Furthermore, for cases in which the distances to the bifurcation points are small enough, one-order approximations to the bifurcation solutions are obtained.
文摘In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is used to study the asymptotic behaviour of Kuramoto-Sivashinsky equation and to construct the bifurcation diagrams.
文摘In this paper, the random Kuramoto-Sivashinsky equation with additive noise is studied numerically, using the finite difference method to simulate the effect of different amplitude of noise on the solitary wave. And numerical experiments show that the white noise does not affect the propagation of the solitary wave, but can increase the amplitude of the solitary wave.
文摘In this paper, it is considered that the global existence, uniqueness and regularity results for the Cauchy problem of the well-known Kuramoto-Sivashinsky equation [GRAPHICS] only under the condition u(0)(x) is an element of L-2(R-N, R-n). Where u(t, x) = (u(1)(t, x), ..., u(n)(t, x))(T) is the unknown vector-valued function. Results show that for N < 6,.u(0)(x) is an element of L-2(R-N, R-n), the above Cauchy problem admits a unique global solution u(t, x) which belongs to C-infinity,C-infinity(R-N x (0, infinity)).
基金supported by the National Natural Science Foundation of China(11271141)Chongqing Science&Technology Commission(cstc2018jcyjAX0787)
文摘The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multidimensions(n ≥ 3) is considered. Based on Green's function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the L^p convergence rate of the solution is obtained.
基金Supported by the National Natural Science Foundation of China(91024026,10975126)Supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China(200934021100 32)
文摘This paper focuses on the application of Exp-function method to obtain generalized solutions of the KdV-Burgers-Kuramoto equation and the Kuramoto-Sivashinsky equation.It is demonstrated that the Exp-function method provides a mathematical tool for solving the nonlinear evolution equation in mathematical physics.
文摘In this paper we use the spectral method to analyse the generalized Kuramoto-Sivashinsky equations. We prove the existence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estimation between spectral approximate solution and exact solution on large time.
基金Supported by the National Natural Science Foundation of China(No.11571381,11531006,11771449)the National Science Foundation(No.1620449)the High Level Talent Foundation of Qingdao Agricultural University(No.663-1113305)
文摘Global well-posedness of the initial-boundary value problem for the stochastic generalized KuramotoSivashinsky equation in a bounded domain D with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data u0 ∈L2（D×Ω）, this problem has a unique global solution u in the space L2（Ω, C（[0, T ], L2（D））） for any T 〉 0, and the solution map u0 →u is Lipschitz continuous.
文摘We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points.We aim to identify the unknown number of sources and their locations along with their strengths.In our previous work,we proved that minimum measurement points needed under the noise-free setting.In this paper,we extend the proof to cover the noisy cases over a border class of source functions.We show that if the regularization parameter is chosen properly,the problem can be transformed into a poles identification problem.A reconstruction scheme is proposed on the basis of the developed theoretical results.Numerical demonstrations in 2D and 3D conclude the paper.
