某些色散波方程的适定性问题
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摘要
本论文主要研究某些色散波方程Cauchy问题的适定性。所用方法是Fourier限制范数方法，此方法是由J．Bourgain引进的。需要强调的是，所讨论的方程的生成半群的相函数及其一阶，二阶导数有非零奇异点，这就带来一些新的困难。但是，可以利用Fourier限制算子来分离这些奇异点，这样就可以改进已知的一些结果。本论文主要分四章。
在第二章，主要讨论Korteweg-de Vries-Benjamin-Ono方程的适定性问题。证明了当初值属于H~s(R)(s≥-1/8)，其Cauchy问题是局部适定的。当初值属于L~2(R)，其Cauchy问题是整体适定的。
在第三章，主要讨论Hirota方程的Cauchy问题。当方程带有具有导数的非线性项时，证明了若初值属于H~s(R)(s≥1/4)，其Cauchy问题是局部适定的；若初值属于H~s(R)(s≥1)，其Cauchy问题是整体适定的。当方程不带有具有导数的非线性项时，证明了若初值属于H~s(R)(s≥-1/4)，其Cauchy问题是局部适定的；若初值属于H~s(R)(s≥0)，其Cauchy问题是整体适定性的。关于整体适定性的证明，主要思想是解在空间H~s(s≥1)(H~s(s≥0))中的存在区间只依赖于初值的H~1(L~2)模。
在第四章，主要讨论关于涡旋丝的四阶非线性Schrdinger方程的Cauchy问题。证明了当方程具有某些系数限制时，若初值属于H~s(R)(s≥1/2)，其Cauchy问题是局部适定的。
In the dissertation, the Cauchy problems of some dispersive equations are considered by the Fourier restriction norm method, which was first introduced by J.Bourgain. It is important to point out that the phase functions, their first-order derivatives and second-order derivatives have non-zero singular points, which makes the problem much more difficult. However, we can overcome the difficulties by the Fourier restriction operators to separate these singular points. Therefore, some known results can be improved. The dissertation consists of four chapters.
In Chapter 2, the Cauchy Problem to the generalized Korteweg-de Vries-Benjamin-Ono equation is considered. Local well-posedness for data in Hs(R)(s > -1/8) and global well-posedness for data in L2(R) are obtained.
In Chapter 3, the Cauchy problem of Hirota equation is studied. For the equation with derivative in nonlinear terms, the Cauchy problem is locally well-posed for data in Hs(R){s > 1/4) and globally well-posed for data in Hs(R)(s > 1). For the equation without derivative, the Cauchy problem is locally well-posed for data in Hs(R)(s > -1/4) and globally well-posed for data in Hs(R)(s > 0). The main idea for the global well-posedness, based on the generalized trilinear estimates, is that the existence time of the solution in Hs(s > 1)( Hs(R)(s > 0)) only depends on the norm of initial data in H1 (L2).
In Chapter 4, the Cauchy problem for the Fourth-order nonlinear Schrodinger equation related to the vortex filament space is considered. Local result for data in Hs(R)(s > 1/2) is obtained under certain coefficient condition.
在第二章，主要讨论Korteweg-de Vries-Benjamin-Ono方程的适定性问题。证明了当初值属于H~s(R)(s≥-1/8)，其Cauchy问题是局部适定的。当初值属于L~2(R)，其Cauchy问题是整体适定的。
在第三章，主要讨论Hirota方程的Cauchy问题。当方程带有具有导数的非线性项时，证明了若初值属于H~s(R)(s≥1/4)，其Cauchy问题是局部适定的；若初值属于H~s(R)(s≥1)，其Cauchy问题是整体适定的。当方程不带有具有导数的非线性项时，证明了若初值属于H~s(R)(s≥-1/4)，其Cauchy问题是局部适定的；若初值属于H~s(R)(s≥0)，其Cauchy问题是整体适定性的。关于整体适定性的证明，主要思想是解在空间H~s(s≥1)(H~s(s≥0))中的存在区间只依赖于初值的H~1(L~2)模。
在第四章，主要讨论关于涡旋丝的四阶非线性Schrdinger方程的Cauchy问题。证明了当方程具有某些系数限制时，若初值属于H~s(R)(s≥1/2)，其Cauchy问题是局部适定的。
In the dissertation, the Cauchy problems of some dispersive equations are considered by the Fourier restriction norm method, which was first introduced by J.Bourgain. It is important to point out that the phase functions, their first-order derivatives and second-order derivatives have non-zero singular points, which makes the problem much more difficult. However, we can overcome the difficulties by the Fourier restriction operators to separate these singular points. Therefore, some known results can be improved. The dissertation consists of four chapters.
In Chapter 2, the Cauchy Problem to the generalized Korteweg-de Vries-Benjamin-Ono equation is considered. Local well-posedness for data in Hs(R)(s > -1/8) and global well-posedness for data in L2(R) are obtained.
In Chapter 3, the Cauchy problem of Hirota equation is studied. For the equation with derivative in nonlinear terms, the Cauchy problem is locally well-posed for data in Hs(R){s > 1/4) and globally well-posed for data in Hs(R)(s > 1). For the equation without derivative, the Cauchy problem is locally well-posed for data in Hs(R)(s > -1/4) and globally well-posed for data in Hs(R)(s > 0). The main idea for the global well-posedness, based on the generalized trilinear estimates, is that the existence time of the solution in Hs(s > 1)( Hs(R)(s > 0)) only depends on the norm of initial data in H1 (L2).
In Chapter 4, the Cauchy problem for the Fourth-order nonlinear Schrodinger equation related to the vortex filament space is considered. Local result for data in Hs(R)(s > 1/2) is obtained under certain coefficient condition.
引文
[1] J. angulo, Existence and stability of solitary wave solutions of the Benjamin eqyation, J. Diff. Eq., 152(1999), no. 1, 136-159.
[2] D. Bekiranov, T. Ogawa, G. Ponce, Weak solvability and well-posedness of the coupled Schrodinger Korteweg-de Vries equation in the capillary-gravity interaction waves, Proc. Amer. Math. Soc. 125(1997), 2907-2919.
[3] D. Bekiranov, T. Ogawa, G.. Ponce,Interaction equations for short and long dispersive waves, J. Funct. Anal. 158(1998), 357-388.
[4] T. B. Benjamin, A new kind of solitary waves, J. Fluid Meeh., 245(1992), 401-411.
[5] J. Bergh and J. Lfstrm, Interpolation Spaces, Springer-Verlag, 1976.
[6] B. Birnir, G. Ponce, L. Vega, On the ill-posedness for the initial value problem for the modified Korteweg-de Vries equation, preprint.
[7] B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega, On the ilI-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrdinger equations, J. London Math. Soc.(2)53(1996), 551-559.
[8] J. Bona, R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Royal Soc. London Series A 278(1975), 555-601.
[9] J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part Ⅰ: Schrdinger equation, part Ⅱ: the KdV equation, Geom. Funct. Anal., 2(1993), 107-156, 209-262.
[10] J. Bourgain,Refinements of Strichartz' inequality and applications to 2D-NLS with critical non-linearity, Int. Math. Research Notices, 5(1998), 253-283.
[11] J. Bourgain, New global well-posedness results for non-linear Schrdinger equations, AMS 1999.
[12] J. Bourgain, Global well-posedness of defocussing 3D critical NLS in the radial case, J. Amer. Math. Soc. 12(1999), 145-171.
[13] J. Bourgain,Periodic Korteweg de Vries equation with measures as initial data, Selecta Math.(N. S. ) 3(1997),115-159
[14] J. Bourgain, J. Colliander, On the well-posedness of the Zakharov system, IMRN 11(1996), 515-546.
[15] B. Bubnov, On the Cauchy problem for the Korteweg de Vries equation, Soviet Math. Dokl. 21(1980), 502-505.
[16] Xavier Caxvajal, Well-posedness for a higher order nonlinear Schrdinger quation in sobolev spaces of negative indices, preprint.
[17] T. Cazenave, An introduction to nonlinear SchrSdinger equations, Textos de
Metodes Matematicos 22(Rio de Janeiro), 1989.
[18] H. Chen and J. L. BonaExistence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Diff. Eqns. 3(1998)51-84.
[19] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrdinger equations in Hs, Nonlinear Anal. TMA, 14(1990), 807-836.
[20] H. Chihara, The initial value problem for cubic semilinear Schrdinger equations, Publ. Res. Inst. Math. Sci. 32(1996), 445-471.
[21] M. Colin, On the local well-posedness of quasilinear Schrdinger equations in arbitrary space dimension, CPDE 27(2002), 325-354.
[22] J. Colliander, J. Delort, C. Kenig, G. Staffilani,Bilinear Estimates and applications to 2D NLS, preprint.
[23] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T, JAMS 16(2003), 705-749.
[24] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness for the Schrodinger equations with derivative, SIAM J. Math. Anal. 33(2001), 649-669.
[25] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao,Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation, Math Res. Lett. 9(2002), 5-6, 659-682.
[26] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao,A refined global well-posedness for the Schrodinger equations with derivative, SIAM J. Math. Anal. 34(2002), 64-86.
[27] P. Constantin and J. C. Saute, Local smoothing properties of dispersive equations equation, J. Amer. Math. Soc., 1(1988), 413-446.
[28] Y. Fukumoto and H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part Ⅰ. A higher-order asymptotic formula for the velocity, J. Fluid. Mechh, 417(2000), 1-45.
[29] J. Ginibre, Y. Tsutsumi, G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151,(1997)384-436.
[30] A. Gruenrock,A bilinear Airy- estimate with application to gKdV-3, preprint
[31] Boling Guo and Shaobin Tan, Global smooth solution for nonlinear evolution equation of Hirota type, Science In China, 35A(1992), 377-393.
[32] Hasegawa A. & Kodama, Y. Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23(1987), 510-524.
[33] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51(1972),
477-485.
[34] N. Hayashi, T. Ozawa, On the derivative nonlinear Schrdinger equation, Phys D 55(1992), 14-36.
[35] N. Hayashi, T. Ozawa,Finite energy solutions of nonlinear Schrdinger equations of derivative type, SIAM J. Math. Anal. 25(1994), 1488-1503.
[36] N. Hayashi, The initial value problem for the derivative nonlinear Schrdinger equation in the energy space, Nonlinear Anal. 20(1993), 823-833.
[37] R. J. Iorio, On the Cauchy problem for the Benjamin-Ono equation, CPDE 11(1986), 1031-1181.
[38] Rafael Iorio, Valeria de Magalhaes Fourier analysis and Partial Differential Equations, Cambridge university press, 2001.
[39] T. Kato, On the Cauchy problem for the(generalized)Korteweg-de vries equation, Advances in Mathematics supplementary studies in applied Math., 8(1983), 93-128.
[40] C. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the KdV equation, J. Amer. Math. Soc. 4(1991), 323-347
[41] C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46(1993), 527-620.
[42] C. E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Kortewegde Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71(1993), 1-21.
[43] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the Kdv equation, J. Amer. Math. Soc., 9(1996), 573-603.
[44] C. E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40(1991), 33-69.
[45] C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46(1993), 527-620.
[46] C. Kenig, G. Ponce, L. Vega, Quadratic forms for the 1-D semilinear Schrdinger equation, Trans. Amer. Math. Soc. 346(1996), 3323-3353.
[47] Kodama Y., Optical solitons in a monomode fiber, Journal of Statistical Phys. 39(1985), 597-614.
[48] Corinne Laurey, The Cauchy problem for a third order nonlinear Schrdinger equation, Nonlinear Anal. TMA, 29(1997), 121-158
[49] F. Linares, L~2 Global well-posedness of the initial value problem associated
to the Benjamin equation, J. Diff, Eq., 152(1999), 1425-1433.
[50] L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in sobolev spaces of Negative order, Indiana Univ. Math. J., 50(4)(2001), 1745-1776.
[51] L. Molinet, J. C. Sault and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33(2001), 982-988.
[52] E. Ott and N. Sudan, Damping of solitary waves, Phys. Fluids, 13(6)(1970), 1432-1434.
[53] T. Ozawa, On the nonlinear Schrdinger equations of derivative type, Indiana Univ. Math. J., 45(1996), 137-163.
[54] H. Pecher, Global well-posedness below energy space for the 1D Zakharov system, preprint.
[55] H. Pecher, Global solutions of the Klein-Gordon-Schrdinger system with rough data, preprint.
[56] G. Ponce, On the global well-posedness of the Benjamin-Ono equation, DIE 4(1991), 527-542.
[57] L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, [in Italian] Rend. Circ. Mat. Palermo., 22(1906), 117-135.
[58] J. Rauch, Partial Differential Equations springer-Verlag 1991.
[59] I. E. Segal, Space-time decay for solutions of wave equations, Adv. in Math., 22(1976), 302-311.
[60] J. Segata, Well-posedness for the fourth-order nonlinear Schrdinger equation related to the vortex filament, Differential and Integral Equations, 16(2003), 841-864.
[61] Tan Shaobin & Han Yongqian, Long time behavior for solution of generalized nonlinear evolution equations, Chinese Annals of Mathematics, 16A:2(1995), 127-141.
[62] E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality and Oscillatroy Integrals, Princeton University Press, 1993.
[63] G. Staffilani, Quadratic forms for a 2D semilinear Schrdinger equation, Duke Math J. 86(1997), 79-107.
[64] G. Staffilani, On the generalized Korteweg-de Vries-type equations, Diff. and Inte. Eq., 10(1997), 777-796.
[65] R. Strichartz, Restrictions of Fourier transforms to quadratic surface and decay of solutions of wave equations, Duke Math. J., 44(1977), 705-714.
[66] H. Takaoka, Well-posedness for the SchrSdinger equation with the derivative non-linearity, Adv. Diff. Eq. 4(1999), 561-680.
[67] H. Takaoka, Well-posedness for the one dimensional Schrdinger equation with the derivative nonlinearity, Adv. Diff. Eq., 4(1999), 561-680.
[68] H. Takaoka, Global well-posedness for Schrdinger equation with derivative in a nonlinear term and data in low-order Sobolev Spaces, Elec. J. Diff. Eq., 42(2001), 1-23.
[69] T. Tao, Multilinear weighted convolution of L~2 functions, and applications to nonlinear dispersive equation, Amer. J. Math., 123(2001), 839-908.
[2] D. Bekiranov, T. Ogawa, G. Ponce, Weak solvability and well-posedness of the coupled Schrodinger Korteweg-de Vries equation in the capillary-gravity interaction waves, Proc. Amer. Math. Soc. 125(1997), 2907-2919.
[3] D. Bekiranov, T. Ogawa, G.. Ponce,Interaction equations for short and long dispersive waves, J. Funct. Anal. 158(1998), 357-388.
[4] T. B. Benjamin, A new kind of solitary waves, J. Fluid Meeh., 245(1992), 401-411.
[5] J. Bergh and J. Lfstrm, Interpolation Spaces, Springer-Verlag, 1976.
[6] B. Birnir, G. Ponce, L. Vega, On the ill-posedness for the initial value problem for the modified Korteweg-de Vries equation, preprint.
[7] B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega, On the ilI-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrdinger equations, J. London Math. Soc.(2)53(1996), 551-559.
[8] J. Bona, R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Royal Soc. London Series A 278(1975), 555-601.
[9] J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part Ⅰ: Schrdinger equation, part Ⅱ: the KdV equation, Geom. Funct. Anal., 2(1993), 107-156, 209-262.
[10] J. Bourgain,Refinements of Strichartz' inequality and applications to 2D-NLS with critical non-linearity, Int. Math. Research Notices, 5(1998), 253-283.
[11] J. Bourgain, New global well-posedness results for non-linear Schrdinger equations, AMS 1999.
[12] J. Bourgain, Global well-posedness of defocussing 3D critical NLS in the radial case, J. Amer. Math. Soc. 12(1999), 145-171.
[13] J. Bourgain,Periodic Korteweg de Vries equation with measures as initial data, Selecta Math.(N. S. ) 3(1997),115-159
[14] J. Bourgain, J. Colliander, On the well-posedness of the Zakharov system, IMRN 11(1996), 515-546.
[15] B. Bubnov, On the Cauchy problem for the Korteweg de Vries equation, Soviet Math. Dokl. 21(1980), 502-505.
[16] Xavier Caxvajal, Well-posedness for a higher order nonlinear Schrdinger quation in sobolev spaces of negative indices, preprint.
[17] T. Cazenave, An introduction to nonlinear SchrSdinger equations, Textos de
Metodes Matematicos 22(Rio de Janeiro), 1989.
[18] H. Chen and J. L. BonaExistence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Diff. Eqns. 3(1998)51-84.
[19] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrdinger equations in Hs, Nonlinear Anal. TMA, 14(1990), 807-836.
[20] H. Chihara, The initial value problem for cubic semilinear Schrdinger equations, Publ. Res. Inst. Math. Sci. 32(1996), 445-471.
[21] M. Colin, On the local well-posedness of quasilinear Schrdinger equations in arbitrary space dimension, CPDE 27(2002), 325-354.
[22] J. Colliander, J. Delort, C. Kenig, G. Staffilani,Bilinear Estimates and applications to 2D NLS, preprint.
[23] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T, JAMS 16(2003), 705-749.
[24] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness for the Schrodinger equations with derivative, SIAM J. Math. Anal. 33(2001), 649-669.
[25] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao,Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation, Math Res. Lett. 9(2002), 5-6, 659-682.
[26] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao,A refined global well-posedness for the Schrodinger equations with derivative, SIAM J. Math. Anal. 34(2002), 64-86.
[27] P. Constantin and J. C. Saute, Local smoothing properties of dispersive equations equation, J. Amer. Math. Soc., 1(1988), 413-446.
[28] Y. Fukumoto and H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part Ⅰ. A higher-order asymptotic formula for the velocity, J. Fluid. Mechh, 417(2000), 1-45.
[29] J. Ginibre, Y. Tsutsumi, G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151,(1997)384-436.
[30] A. Gruenrock,A bilinear Airy- estimate with application to gKdV-3, preprint
[31] Boling Guo and Shaobin Tan, Global smooth solution for nonlinear evolution equation of Hirota type, Science In China, 35A(1992), 377-393.
[32] Hasegawa A. & Kodama, Y. Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23(1987), 510-524.
[33] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51(1972),
477-485.
[34] N. Hayashi, T. Ozawa, On the derivative nonlinear Schrdinger equation, Phys D 55(1992), 14-36.
[35] N. Hayashi, T. Ozawa,Finite energy solutions of nonlinear Schrdinger equations of derivative type, SIAM J. Math. Anal. 25(1994), 1488-1503.
[36] N. Hayashi, The initial value problem for the derivative nonlinear Schrdinger equation in the energy space, Nonlinear Anal. 20(1993), 823-833.
[37] R. J. Iorio, On the Cauchy problem for the Benjamin-Ono equation, CPDE 11(1986), 1031-1181.
[38] Rafael Iorio, Valeria de Magalhaes Fourier analysis and Partial Differential Equations, Cambridge university press, 2001.
[39] T. Kato, On the Cauchy problem for the(generalized)Korteweg-de vries equation, Advances in Mathematics supplementary studies in applied Math., 8(1983), 93-128.
[40] C. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the KdV equation, J. Amer. Math. Soc. 4(1991), 323-347
[41] C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46(1993), 527-620.
[42] C. E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Kortewegde Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71(1993), 1-21.
[43] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the Kdv equation, J. Amer. Math. Soc., 9(1996), 573-603.
[44] C. E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40(1991), 33-69.
[45] C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46(1993), 527-620.
[46] C. Kenig, G. Ponce, L. Vega, Quadratic forms for the 1-D semilinear Schrdinger equation, Trans. Amer. Math. Soc. 346(1996), 3323-3353.
[47] Kodama Y., Optical solitons in a monomode fiber, Journal of Statistical Phys. 39(1985), 597-614.
[48] Corinne Laurey, The Cauchy problem for a third order nonlinear Schrdinger equation, Nonlinear Anal. TMA, 29(1997), 121-158
[49] F. Linares, L~2 Global well-posedness of the initial value problem associated
to the Benjamin equation, J. Diff, Eq., 152(1999), 1425-1433.
[50] L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in sobolev spaces of Negative order, Indiana Univ. Math. J., 50(4)(2001), 1745-1776.
[51] L. Molinet, J. C. Sault and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33(2001), 982-988.
[52] E. Ott and N. Sudan, Damping of solitary waves, Phys. Fluids, 13(6)(1970), 1432-1434.
[53] T. Ozawa, On the nonlinear Schrdinger equations of derivative type, Indiana Univ. Math. J., 45(1996), 137-163.
[54] H. Pecher, Global well-posedness below energy space for the 1D Zakharov system, preprint.
[55] H. Pecher, Global solutions of the Klein-Gordon-Schrdinger system with rough data, preprint.
[56] G. Ponce, On the global well-posedness of the Benjamin-Ono equation, DIE 4(1991), 527-542.
[57] L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, [in Italian] Rend. Circ. Mat. Palermo., 22(1906), 117-135.
[58] J. Rauch, Partial Differential Equations springer-Verlag 1991.
[59] I. E. Segal, Space-time decay for solutions of wave equations, Adv. in Math., 22(1976), 302-311.
[60] J. Segata, Well-posedness for the fourth-order nonlinear Schrdinger equation related to the vortex filament, Differential and Integral Equations, 16(2003), 841-864.
[61] Tan Shaobin & Han Yongqian, Long time behavior for solution of generalized nonlinear evolution equations, Chinese Annals of Mathematics, 16A:2(1995), 127-141.
[62] E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality and Oscillatroy Integrals, Princeton University Press, 1993.
[63] G. Staffilani, Quadratic forms for a 2D semilinear Schrdinger equation, Duke Math J. 86(1997), 79-107.
[64] G. Staffilani, On the generalized Korteweg-de Vries-type equations, Diff. and Inte. Eq., 10(1997), 777-796.
[65] R. Strichartz, Restrictions of Fourier transforms to quadratic surface and decay of solutions of wave equations, Duke Math. J., 44(1977), 705-714.
[66] H. Takaoka, Well-posedness for the SchrSdinger equation with the derivative non-linearity, Adv. Diff. Eq. 4(1999), 561-680.
[67] H. Takaoka, Well-posedness for the one dimensional Schrdinger equation with the derivative nonlinearity, Adv. Diff. Eq., 4(1999), 561-680.
[68] H. Takaoka, Global well-posedness for Schrdinger equation with derivative in a nonlinear term and data in low-order Sobolev Spaces, Elec. J. Diff. Eq., 42(2001), 1-23.
[69] T. Tao, Multilinear weighted convolution of L~2 functions, and applications to nonlinear dispersive equation, Amer. J. Math., 123(2001), 839-908.