臺灣博碩士論文加值系統

In this paper, we study the existence, uniqueness and regularity dependence results which are established for the initial value problem to the KdVB equation,
$$u_t+uu_x-\alpha u_{xx}+u_{xxx}=0.$$
The methods used in this paper is to regularize the KdVB equation by the addition of the linear term $-\epsilon u_{xxt},$ hence the existence and uniqueness theory for the initial value problem will be developed by the BBMB equation. Finally we consider the limit $\epsilon \to 0$ to show the strong convergence to solutions of the initial value problem for the KdVB equation.

In this paper, we study the existence, uniqueness and regularity dependence results which are established for the initial value problem to the KdVB equation,
$$u_t+uu_x-\alpha u_{xx}+u_{xxx}=0.$$
The methods used in this paper is to regularize the KdVB equation by the addition of the linear term $-\epsilon u_{xxt},$ hence the existence and uniqueness theory for the initial value problem will be developed by the BBMB equation. Finally we consider the limit $\epsilon \to 0$ to show the strong convergence to solutions of the initial value problem for the KdVB equation.

1.Introduction 1
2.The relevant function spaces 4
3.The main result 6
4.Convergence 12
5.Appendix 22
Reference 25

\bibitem{pb1} Amick, J. C., Bona, J. L. {\s AND} Schonbek, M. E. {\it Decay of soluions of some nonlinear wave equation}, J. of Diff. Equation, {\bf 81} (1989) 1-49.
\bibitem{pb2} Bao-ping, L. {\s AND} Pao, C. V. {\it Green's function method for periodic traveling wave solution of the KdV equation}, Appl. Anal., {\bf 14} (1989) 293-301.
\bibitem{pb3} Benjamin, T. B., Bona, J. L. {\s AND} Mahony, J. J. {\it Model equations for long waves in nolinear dispersive systems}, Phil. Trans. R. Soc. Lond., {\bf 272} (1972) 47-78.
\bibitem{pb4} Bona, J. L. {\s AND} Dougalis, V. A. {\it An initial- and boundary-value problem for a model equation for propagation of long waves} J. Math. Anal. Appl., {\bf 75} (1980) 503-522.
\bibitem{pb5} Bona, J. L. {\s AND} Zhang B.-Y. {\it The initial-value problem for forced Korteweg-de Vries equation}, P. R. Soc. Edinburgh, {\bf 126A} (1996) 571-598.
\bibitem{pb6} Bona, J. L. {\s AND} Smith, R. {\it The initial-value problem for the Korteweg-de Vries equation}, Philos. Trans. Roy. Soc. London Ser., {\bf 278} (1978) 555-601.
\bibitem{pb7} Cohen, A. {\it Solution of the Korteweg-de Vries equation from irregular data}, Duke Math. J., {\bf 45} (1978) 149-181.
\bibitem{pb8} Canosa, J. {\s AND} Gazdag, J. {\it The Korteweg-de Vries-Burgers equation}, J. of Comp. phy., {\bf 23} (1977) 3936-403.
\bibitem{pb9} Ching, G. I., {\it The existence theory of the initial-boundary value problem for the GBBM equation}, M.Sc., (1998) National Kaohsiung Normal University.
\bibitem{pb10} Gardner, C. S., Greene, J. M., Kruskal, M. D. {\s AND} Miura, R. M. {\it A method for solving the Korteweg-de Vries equation}, Phys. Rev. Letters. {\bf 19} (1967) 1095-1097.
\bibitem{pb11} Gardner, C. S., Greene, J. M., Kruskal, M. D. {\s AND} Miura, R. M. {\it Korteweg-de Vries equation and generalization. VI. methods for exact solution}, Comm. Pure. Appl. Math., {\bf 27} (1974) 97-133.
\bibitem{pb12} Hsieh, C. F., {\it Existence theory of the modefied BBM equation}, M.Sc., (1995) National Kaohsiung Normal University.
\bibitem{pb13} Korteweg, D. J. {\s AND} de Vries, G. {\it On the change in form of long waves advancing in a rectangular channel, and on a new type of long stationary waves}, Phil. Mag., {\bf 39} (1895) 422-443.
\bibitem{pb14} Mei, M. {\it Large-time behavior of solution for generalized BBMB equation}, Nonlinear Anal., {\bf 33} (1998) 699-714.
\bibitem{pb15} Printems, J. {\it The stochastic Korteweg-de Vries equation in $L^2(R)$}, J. of Diff. Equation, {\bf 153} (1999) 338-373.
\bibitem{pb16} Sjoberg, A. {\it On the Korteweg-de Vries equation : existence and uniqueness}, J. Math. Anal. Appl., {\bf 29} (1970) 569-579.
\bibitem{pb17} Tsutsumi, M. {\s AND} Mukasa, T. {\it Parabolic regularization for the generalized Korteweg-de Vries equation}, Funkcialaj Ekvacioj, {\bf 14} (1991) 89-110.