In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustr...In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustrate our point of view.展开更多
This paper is concerned with uniform stability and asymptotic behavior for solutions of 2-dimensional Magnetohydrodynamics equations. The author establishes the corresponding temporal decay estimates when the initial ...This paper is concerned with uniform stability and asymptotic behavior for solutions of 2-dimensional Magnetohydrodynamics equations. The author establishes the corresponding temporal decay estimates when the initial data is in the following Sobolev spaces H 2, L 1∩H 2 with ∫(u 0, A 0)dx≠0, or L 1∩H 2 with ∫(u 0, A 0)dx=0, respectively. Most of the decay rates in these estimates are optimal. Moreover, the author proves various uniform stability results, like sup t>0 ‖(w, E, r)(t))‖ YC‖(w 0, E 0)‖ X, where X and Y are Sobolev spaces. It should be pointed out that the decay estimates of the solutions for the case (u 0, A 0)∈L 1∩H 2 follow from the uniform stability estimates. The author utilizes the Fourier splitting method invented by Professor Schonbek and the new elaborate global energy estimates.展开更多
Solidified colloidal material has been widespreadly applied to various fields. Because material is possessed of two special performances: nonhomogeneoiis and changing physical parameter, it is difficult for mechanics ...Solidified colloidal material has been widespreadly applied to various fields. Because material is possessed of two special performances: nonhomogeneoiis and changing physical parameter, it is difficult for mechanics behavior to be described and accordingly the materials excellent function cannot be taken into full play ,which will result in the waste of material. On the premise of conforming to each basic principle, this article gives the basic relation pattern that describes mechanics behavior of solidified colloidal material containing nonhomogeneoiis and changing physical parameter.展开更多
This paper deals with the asymptotic behavior of solutions for the nonlinear Sobolev-Galpernequations.We first show the existence of the global weak attractor in H^2(Ω)∩H_0~1(Ω) for the equations.Andthen by an ener...This paper deals with the asymptotic behavior of solutions for the nonlinear Sobolev-Galpernequations.We first show the existence of the global weak attractor in H^2(Ω)∩H_0~1(Ω) for the equations.Andthen by an energy equation we prove that the global weak attractor is actually the global strong attractor.Thefinite-dimensionality of the global attractor is also established.展开更多
Abstract This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R＋{utt-txx＋ut＋f（u）x=0,t〉0,x∈R＋,u（0,x）=u0（x）→u＋,asx→...Abstract This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R＋{utt-txx＋ut＋f（u）x=0,t〉0,x∈R＋,u（0,x）=u0（x）→u＋,asx→＋∞,ut（0,x）=u1（x）,u（t,0）=ub.For the non-degenerate case f]（u＋） 〈 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u（t,x） which converges to the stationary wave φ（x） uniformly in x ∈ R＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates （including the algebraic convergence rate and the exponential convergence rate） of u（t, x） toward φ（x） are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f＇（ub） 〈 0. The main purpose of this paper is devoted to discussing the case of f＇（ub）= 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.展开更多
基金the Postdoctoral Fund of Huazhong University of Science and Technology under Grant No.0128011006
文摘In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustrate our point of view.
文摘This paper is concerned with uniform stability and asymptotic behavior for solutions of 2-dimensional Magnetohydrodynamics equations. The author establishes the corresponding temporal decay estimates when the initial data is in the following Sobolev spaces H 2, L 1∩H 2 with ∫(u 0, A 0)dx≠0, or L 1∩H 2 with ∫(u 0, A 0)dx=0, respectively. Most of the decay rates in these estimates are optimal. Moreover, the author proves various uniform stability results, like sup t>0 ‖(w, E, r)(t))‖ YC‖(w 0, E 0)‖ X, where X and Y are Sobolev spaces. It should be pointed out that the decay estimates of the solutions for the case (u 0, A 0)∈L 1∩H 2 follow from the uniform stability estimates. The author utilizes the Fourier splitting method invented by Professor Schonbek and the new elaborate global energy estimates.
文摘Solidified colloidal material has been widespreadly applied to various fields. Because material is possessed of two special performances: nonhomogeneoiis and changing physical parameter, it is difficult for mechanics behavior to be described and accordingly the materials excellent function cannot be taken into full play ,which will result in the waste of material. On the premise of conforming to each basic principle, this article gives the basic relation pattern that describes mechanics behavior of solidified colloidal material containing nonhomogeneoiis and changing physical parameter.
基金Supported by the National Natural Science Foundation of China (No.10271034)
文摘This paper deals with the asymptotic behavior of solutions for the nonlinear Sobolev-Galpernequations.We first show the existence of the global weak attractor in H^2(Ω)∩H_0~1(Ω) for the equations.Andthen by an energy equation we prove that the global weak attractor is actually the global strong attractor.Thefinite-dimensionality of the global attractor is also established.
基金This work was supported by two grants from the National Natural Science Foundation of China under contracts 10431060 and 10329101 respectively.
文摘Abstract This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R＋{utt-txx＋ut＋f（u）x=0,t〉0,x∈R＋,u（0,x）=u0（x）→u＋,asx→＋∞,ut（0,x）=u1（x）,u（t,0）=ub.For the non-degenerate case f]（u＋） 〈 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u（t,x） which converges to the stationary wave φ（x） uniformly in x ∈ R＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates （including the algebraic convergence rate and the exponential convergence rate） of u（t, x） toward φ（x） are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f＇（ub） 〈 0. The main purpose of this paper is devoted to discussing the case of f＇（ub）= 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.
