一类广义Zakharov方程的适定性
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摘要
本文讨论一类广义Zakharov方程组.分别在一维情形和二维情形下,研究该方程组的周期初值问题整体古典解的存在性与唯一性.并且在一维情形下,研究该方程组局部解的blow-up问题.存在唯一性的证明方法是先作Gale¨rkin近似解,得到局部古典解,再由积分先验估计得到整体解.证明了:在一维情形下,当0 < p < 4时,方程的周期初值问题的整体古典解是存在且唯一的;在二维情形下当0 < p < 2及ε0(x)充分小时,方程的初值问题的整体古典解是存在且唯一的.并且在一维情形下得到了:当p≥2(1+πa2E1?0)π+a22√E10+ 3πa2E0及初值满足一定条件时,方程的解在有限时间内出现blow-up.
In this paper, we study a class of generalized Zakharov equations. We considerexistence and uniqueness of the solutions of the periodic initial value problem of thisequations in one dimension space and in two dimension space. The blow-up of the lo-cal solutions of the equations is also discussed. By means of the Gale¨rkin method andthe integral estimates, we obtain the following conclusions: In one dimension space,if 0 < p < 4, the existence and uniqueness of the global classical solution is obtained;In two dimension space, if 0 < p < 2 andε0(x) is enough small, the existenceand uniqueness of the global classical solution is obtained. Then, in one dimensionspace, if p≥2(1+πa2E1?0)π+a22√E10+ 3πa2E0and initial data satisfy some conditions, the localsolution will blow up in finite time.
In this paper, we study a class of generalized Zakharov equations. We considerexistence and uniqueness of the solutions of the periodic initial value problem of thisequations in one dimension space and in two dimension space. The blow-up of the lo-cal solutions of the equations is also discussed. By means of the Gale¨rkin method andthe integral estimates, we obtain the following conclusions: In one dimension space,if 0 < p < 4, the existence and uniqueness of the global classical solution is obtained;In two dimension space, if 0 < p < 2 andε0(x) is enough small, the existenceand uniqueness of the global classical solution is obtained. Then, in one dimensionspace, if p≥2(1+πa2E1?0)π+a22√E10+ 3πa2E0and initial data satisfy some conditions, the localsolution will blow up in finite time.
引文
[1] 郭柏灵, 沈隆钧. Zakharov 方程周期初值问题整体古典解的存在性唯一性[J]. 应用数学学报, 1982, 5(3): 310-324.
[2] 郭柏灵. 广义 Zakharov 方程组的初边值问题 [J]. 高校应用数学学报, 1994,9A(1): 1-12.
[3] 景海荣, 卢圣治. 非均匀等离子体中的 Zakharov 方程 [J]. 北京师范大学学报 (自然科学版), 1996, 32(4): 503-507.
[4] A.V.Babin, M.I.Vishik. Attractors of partial differential equations in an un-bounded domain[J]. Proc. Roy. Soc. Edinburgh, 1990, 116A: 221-243..
[5] A.V.Babin, M.I.Vishik. Attractors of Evolution Equations[J]. Amsterdam, Lon-don, Tokyo, North-Holland, 1992.
[6] C.E.Kenig, G.Ponce, L.Vega. On the Zakharov and Zakhrov-Schulman sys-tems[J]. J.Funct. Anal., 1995, 127: 204-234.
[7] DAI Zheng-de, GUO Bo-ling. Inertial fractal sets for dissipative Zakharov sys-tem[J]. Acta mathematicae applicatae sinica, 1997, 13(3): 279-288.
[8] E.Feireisl. Attractors for semilinear damped wave equations on R3[J]. NonlinearAnal., 1994, 23: 187-195.
[9] G.C.Papanicolaou, C.Sulem, P.L.Sulem, X.P.Wang. Singular solutions of theZakharov equations for Langmuir turbulence[J]. Phys. Fluids, 1991, 3B(4): 969-980.
[10] Glangtas L., Merle F.. Existence of silf-similar blow-up solutions for Zakharovequation in dimension two. I.[J]. Comm. Math. Phys.. 1994, 160(1): 173-215.
[11] Glangtas L., Merle F.. Concentration properties of blow-up solutions and in-stability results for Zakharov equation in dimension two. II.[J]. Comm. Math.Phys.. 1994, 160(2): 349-389.
[12] Guo Boling, Yuan Guangwei. The cauchy problem for the system of Zakharovequations arising from ion-acoustic modes [J]. Proceedings of the Royal Societyof Edinburgh. 1996, 126A: 811-820.
[13] H.Added, S.Added. Existence globale de solutions fortes pour les e′quations dela turbulence de Langmuir en dimension 2 [J]. C.R. Acad. Sci. Paris, 1984, 229:551-554.
[14] I.Flahaut. Attractors for the disssipative Zakhrov system[J]. Nonlinear Anal.,1991, 16: 599-633.
[15] LI Yong-sheng. On the initial boundary value problems for two dimensional sys-tems of Zakharov equations and of complex-Scho¨dinger-real-Booussinesq equa-tions[J]. J.Partial diff. eq., 1992, 5(2): 81-93.
[16] LI Yong-sheng, GUO Bo-ling. Attractor for dissipative Zakharov equations inan unbounded domain[J]. Rev. Math. Phys.,to appear.
[17] Li Yongsheng, Guo Boling. Attractor of dissipative radially symmetric Zakharovequations outside a ball[J]. Math. Methods Apple. Sci. 2004, 27(7): 803-818.
[18] LIN Zheng-guo. Singular limuts of the Zakharov equations[J]. Journal of engi-neering mathematics, 1989, 6(3): 81-87.
[19] M.Landman, G.C.Papanicolaou, C.Sulem, P.L.Sulem, X.P.Wang. Stability ofisotropic self-similar dynamics for scalar collapse[J]. Phys. Rev., 1992, 46A:7869-7876.
[20] Ma Shuqing, Chang Qianshun. Strange attractors on pseudospectral solutionsfor dissipative Zakharov equations[J]. Acta Math. Sci. Ser. B Engl. Ed. 2004,24(3): 321-336.
[21] Masselin Vincent. A result on the blow-up rate for the Zakharov system indimension 3[J]. SIAM J. Math. Anal. 2001, 33(2): 440-447.
[22] Merle Frank. Blow-up results of virial type for Zakharov equations[J]. Comm.Math. Phys.. 1996, 175(2): 433-455.
[23] Merle Frank. Lower bounds for the blow-up rate of solutons of the Zakharovequation in dimension two[J]. Comm. Pure Appl. Math.. 1996, 29A(10): 2493-2498.
[24] P.Constantin, C.Foias, R.Temam. Attractors representing turbulent ?ows[J].Mem. Amer. Math. Sci., 1985, 53(314).
[25] Pecher Hartmut. Global well-posedness below energy space for the 1-dimensional Zakharov system[J]. Internat. Math. Res. Notices, 2001, 19: 1027-1056.
[26] S.H.Schochet, M.I.Weinstein. The nonlinear Schro¨dinger limit of the Zakhrovequation governing Langmuir turbulence[J]. Comm. Math. Phys., 1986, 106:569-580.
[27] T.Ozawa, Y.Tsutsumi. Existence and smoothing effct of solutions for the Za-kharov equation[J]. Diff. Integr. Eqs., 1992, 5: 721-745.
[28] T.Ozawa, Y.Tsutsumi. The nonlnear Schro¨dinger limit and the initial layer of theZakharov equations[J]. Publ.Res. Inst. Math. Sci., 1992, 28: 329-361.
[29] V.E.Zakharov. Collapse of Langmuir waves[J]. Sov. phys. JEPT, 1972, 35: 908-914.
[30] ZHANG Fa-yong, XIANG Xin-min. The global error estimate of the pseu-dospectral method for a class of generalized Zakharov equations(I)[J]. Journalof natural science of Heilongjiang university, 1996, 13(2): 1-6.
[2] 郭柏灵. 广义 Zakharov 方程组的初边值问题 [J]. 高校应用数学学报, 1994,9A(1): 1-12.
[3] 景海荣, 卢圣治. 非均匀等离子体中的 Zakharov 方程 [J]. 北京师范大学学报 (自然科学版), 1996, 32(4): 503-507.
[4] A.V.Babin, M.I.Vishik. Attractors of partial differential equations in an un-bounded domain[J]. Proc. Roy. Soc. Edinburgh, 1990, 116A: 221-243..
[5] A.V.Babin, M.I.Vishik. Attractors of Evolution Equations[J]. Amsterdam, Lon-don, Tokyo, North-Holland, 1992.
[6] C.E.Kenig, G.Ponce, L.Vega. On the Zakharov and Zakhrov-Schulman sys-tems[J]. J.Funct. Anal., 1995, 127: 204-234.
[7] DAI Zheng-de, GUO Bo-ling. Inertial fractal sets for dissipative Zakharov sys-tem[J]. Acta mathematicae applicatae sinica, 1997, 13(3): 279-288.
[8] E.Feireisl. Attractors for semilinear damped wave equations on R3[J]. NonlinearAnal., 1994, 23: 187-195.
[9] G.C.Papanicolaou, C.Sulem, P.L.Sulem, X.P.Wang. Singular solutions of theZakharov equations for Langmuir turbulence[J]. Phys. Fluids, 1991, 3B(4): 969-980.
[10] Glangtas L., Merle F.. Existence of silf-similar blow-up solutions for Zakharovequation in dimension two. I.[J]. Comm. Math. Phys.. 1994, 160(1): 173-215.
[11] Glangtas L., Merle F.. Concentration properties of blow-up solutions and in-stability results for Zakharov equation in dimension two. II.[J]. Comm. Math.Phys.. 1994, 160(2): 349-389.
[12] Guo Boling, Yuan Guangwei. The cauchy problem for the system of Zakharovequations arising from ion-acoustic modes [J]. Proceedings of the Royal Societyof Edinburgh. 1996, 126A: 811-820.
[13] H.Added, S.Added. Existence globale de solutions fortes pour les e′quations dela turbulence de Langmuir en dimension 2 [J]. C.R. Acad. Sci. Paris, 1984, 229:551-554.
[14] I.Flahaut. Attractors for the disssipative Zakhrov system[J]. Nonlinear Anal.,1991, 16: 599-633.
[15] LI Yong-sheng. On the initial boundary value problems for two dimensional sys-tems of Zakharov equations and of complex-Scho¨dinger-real-Booussinesq equa-tions[J]. J.Partial diff. eq., 1992, 5(2): 81-93.
[16] LI Yong-sheng, GUO Bo-ling. Attractor for dissipative Zakharov equations inan unbounded domain[J]. Rev. Math. Phys.,to appear.
[17] Li Yongsheng, Guo Boling. Attractor of dissipative radially symmetric Zakharovequations outside a ball[J]. Math. Methods Apple. Sci. 2004, 27(7): 803-818.
[18] LIN Zheng-guo. Singular limuts of the Zakharov equations[J]. Journal of engi-neering mathematics, 1989, 6(3): 81-87.
[19] M.Landman, G.C.Papanicolaou, C.Sulem, P.L.Sulem, X.P.Wang. Stability ofisotropic self-similar dynamics for scalar collapse[J]. Phys. Rev., 1992, 46A:7869-7876.
[20] Ma Shuqing, Chang Qianshun. Strange attractors on pseudospectral solutionsfor dissipative Zakharov equations[J]. Acta Math. Sci. Ser. B Engl. Ed. 2004,24(3): 321-336.
[21] Masselin Vincent. A result on the blow-up rate for the Zakharov system indimension 3[J]. SIAM J. Math. Anal. 2001, 33(2): 440-447.
[22] Merle Frank. Blow-up results of virial type for Zakharov equations[J]. Comm.Math. Phys.. 1996, 175(2): 433-455.
[23] Merle Frank. Lower bounds for the blow-up rate of solutons of the Zakharovequation in dimension two[J]. Comm. Pure Appl. Math.. 1996, 29A(10): 2493-2498.
[24] P.Constantin, C.Foias, R.Temam. Attractors representing turbulent ?ows[J].Mem. Amer. Math. Sci., 1985, 53(314).
[25] Pecher Hartmut. Global well-posedness below energy space for the 1-dimensional Zakharov system[J]. Internat. Math. Res. Notices, 2001, 19: 1027-1056.
[26] S.H.Schochet, M.I.Weinstein. The nonlinear Schro¨dinger limit of the Zakhrovequation governing Langmuir turbulence[J]. Comm. Math. Phys., 1986, 106:569-580.
[27] T.Ozawa, Y.Tsutsumi. Existence and smoothing effct of solutions for the Za-kharov equation[J]. Diff. Integr. Eqs., 1992, 5: 721-745.
[28] T.Ozawa, Y.Tsutsumi. The nonlnear Schro¨dinger limit and the initial layer of theZakharov equations[J]. Publ.Res. Inst. Math. Sci., 1992, 28: 329-361.
[29] V.E.Zakharov. Collapse of Langmuir waves[J]. Sov. phys. JEPT, 1972, 35: 908-914.
[30] ZHANG Fa-yong, XIANG Xin-min. The global error estimate of the pseu-dospectral method for a class of generalized Zakharov equations(I)[J]. Journalof natural science of Heilongjiang university, 1996, 13(2): 1-6.