Modified KdV方程和D-S方程的适定性和不适定性问题
|
摘要
本文研究了modified Korteweg-de Vries方程初边值问题并用调和分析作为工具研究了Davey-Stewartson方程柯西问题。
在第二章中，我们得到线性估计和三线性估计，然后证明了四分之一平面上的modified Korteweg-de Vries方程的局部适定性。
在第三章中我们证明了“几乎守恒律”，从而证明了非线性Davey-Stewartson方程在空间H~s(R~2)(这里s＞4/7)整体适定性。
在第四章中我们证明R~d上的一般幂非线性项的聚焦和非聚焦非线性Davey-Stewartson方程，在索伯列夫空间H~s是不适定的，这里指标s小于由scaling或Galilean不变所确定的值，或者当正则性很低不能支持分布函数时。
In this dissertation, we study the initial-boundary-value problem of modified Korteweg-de Vries equation and use the tool of harmonic analysis to study the Cauchy problem of Davey-Stewarson equation.
In the second chapter, We obtain some linear estimates, trilinear estimates .And through these estimates , we prove the local well-posedness of modified Korteweg-de Vries equation in a quarter plane.
In the third chapter, We prove an " almost conservation law " to obtain global-in-time well-posedness for the nonlinear Davey-Stewartson equation in H~S(R~2), and S>4/7.
In the fourth chapter, The nonlinear Davey-Stewartson equations on R~d, with general power nonlinearity and with both the focusing and defocuing signs, are proved to be ill-posed in the Sobolev space H~s whenever the exponent s is lower than that predicated by scaling or Galilean invariance, or when the regularity is too low to support distributional solutions .
在第二章中，我们得到线性估计和三线性估计，然后证明了四分之一平面上的modified Korteweg-de Vries方程的局部适定性。
在第三章中我们证明了“几乎守恒律”，从而证明了非线性Davey-Stewartson方程在空间H~s(R~2)(这里s＞4/7)整体适定性。
在第四章中我们证明R~d上的一般幂非线性项的聚焦和非聚焦非线性Davey-Stewartson方程，在索伯列夫空间H~s是不适定的，这里指标s小于由scaling或Galilean不变所确定的值，或者当正则性很低不能支持分布函数时。
In this dissertation, we study the initial-boundary-value problem of modified Korteweg-de Vries equation and use the tool of harmonic analysis to study the Cauchy problem of Davey-Stewarson equation.
In the second chapter, We obtain some linear estimates, trilinear estimates .And through these estimates , we prove the local well-posedness of modified Korteweg-de Vries equation in a quarter plane.
In the third chapter, We prove an " almost conservation law " to obtain global-in-time well-posedness for the nonlinear Davey-Stewartson equation in H~S(R~2), and S>4/7.
In the fourth chapter, The nonlinear Davey-Stewartson equations on R~d, with general power nonlinearity and with both the focusing and defocuing signs, are proved to be ill-posed in the Sobolev space H~s whenever the exponent s is lower than that predicated by scaling or Galilean invariance, or when the regularity is too low to support distributional solutions .
引文
[1] J. Bona, Shuming Sun, Bing-Yu Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vies equation in a quarter plane. Trans. Amer. Math. Soc., 2 354(2002), 427-490.
[2] C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. on Pure and Appl. Math., 46(1993), 527-620.
[3] E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Eucliden Spaces. Princeton University Press: Princeton, 1971.
[4] C. E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math J., 71(1993), 1-21.
[5] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with application to the Korteweg-De Vries equation. J. of the American Math. Society., 9(1996), 573-603.
[6] C. E. Kenig, G. Ponce, L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106(2001), 617-633.
[7] J. Bona, R. Winther, The Korteweg-de Vries equation posed in a quarter-plane. Siam Y. Math. Anal., 14(1983), 1056-1106.
[8] J. Bona, L. R. Scott, Solutions of the Korteweg-de Vries equations in fractional order Sobolev spaces. Duke Math. J., 43(1976), 87-99.
[9] ) J. Bona, R. Smith, The initial-value problem for the KdV equation. Philos. Trans. Roy. Soc. London Ser. A. 278(1287)(1975), 555-601.
[10] A. Cohen, Solutions of the Korteweg-de Vreis equation from irregular data. Duke Math. J., 45(1978),149-181.
[11] J. Ginibre, Y. Tsutsumi, Uniqueness for the generalized Korteweg-de Vries equa- tions. Siam J. Math. Anal., 20(1989), 1388-1425.
[12] Colliander J., Keel M., Staffilani G., Takaoka H, Tao T., Sharp global wellposedness results for periodic and non-periodic KdV and modified KdV on R and T. JIAMS, 16(2003), 705-749.
[13] Guo Boling, Wang Baoxiang, The Cauchy problem for Davey-Stewantson systems. Comm. on Pure Appl. Math., LII(1999), 1477-1490.
[14] Guo Boling, Wu yaping, Orbital stbility of solitary waves for the nonlinear derivative Schrodinger equation. J. Differential Equation 123(1995), 35-55.
[15] Wang Baoxing, Guo Boling, The Cauchy problem and the existence of scatting for the generalized Davey-Stewartson equations. Chinese Science(A), 44(8)(2001), 994-1002.
[16] Wang Baoxing, Bessel(Riesz) potentials on Banach function spaces and their applications I theory, Acta. Mathematica Sinica, New Series 14(3)(1998), 327-340.
[17] Wang Baoxiang, On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in Hs, Nonlinear Analysis, TMA, 31(1998), 173-187.
[18] Wang Baoxiang, Large time behavior of solutions for critical and subcritical complex Ginzburg-Landau eqations in H~1, Prepint.
[19] Bergh J., Lofstrom J., Interpolation Spaces. Springer-Verlag, 1976.
[20] Shen Caixia, The global wellposedness and scattering of the generalized DaveyStewartson equation. J. Partial Diff. Eqs., 17(2004), 89-96.
[21] Cazenave T., Weissler F. B., The Cauchy problem for critical nonlinear Schrodinger equations in H~s. Nonlinear Anal., TMA, 14(1990), 807-836.
[22] Cazenave T., Weissler F. B., The Cauchy problem for the nonlinear Schrodinger equation in H~1. Manuscipta Math., 61(1988), 477-494.
[23] Keel M., Tao T., Endpoint Strichartz estimates. Amer. J. Math., 120(1998), 955-980.
[24] Davey A., Stewartson K., On three dimensional packets of surface waves. Proc. Roy. Soc. Ser A, 338(1974), 101-110.
[25] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KDV equation. Journal of the A. M. S., 9(1996), 573-603.
[26] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Almost conservation laws and global rough solution to a Nonlinear Schrodinger Equation. Math. Res. Letters, 9(2002), 659-682.
[27] Christ M., Colliander J., Tao T., Ill-posedness for nonlinear Schrodinger and wave equations. To appear, Annales IHP
[28] B. Birnir, G. Ponce, N. Svanstedt, The local ill-posed of the modified KdV equation. Ann. Inst. Henri Poincare 13 4(1996), 529-535.
[29] B. Birnir, F. Linares, Ill-posed for the derivative Schrodinger and generalized Benjamin-Ono equations. Trans. Arner. Math. Soc. 353(2001)9, 3649-3659.
[30] B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega, On the illposedness of the generalized KdV and nonlinear Schrodinger equations. J. London Math. Soc. (2)53(1996), 551-559.
[31] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations. J. Math. Phys., 18(1977), 1794-1797.
[32] L. Molinet, J. C. Saut, N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations. SIAM J. Math. Anal. 33(2001), 982-988.
[33] C. Sulem, P. L. Sulem, The nonlinear Schrodinger equation: Self-focusing and wave collapse. Springer-Verlag, New York 1999.
[34] Cazenave T., An introduction to nonlinear Schrodinger equations. Textos de Metodos Matematicos 26 Instituto de Matematica, Rio de Janeiro, 1996.
[35] Bourgain J., Fourier transform restriction phenomena for certain lattice subsets and applictions to nonlinear evolution equations. Ⅰ. Geom.Funct. Anal, 3(1993), 107-156.
[36] Bourgain J., Fourier transform restriction phenomena for certain lattice subsets and applictions to nonlinear evolution equations. Ⅱ. Geom.Funct. Anal, 3(1993), 209-262.
[37] Bourgain J., Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity. Intern. Mat. Res. Notices, 5(1998), 253-283.
[38] Bourgain J., On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE. Intern. Mat. Res. Notices, 6(1996), 277-304.
[39] )Bourgain J., Approximation of solutions of the cubic nonlinear Schrodinger equations by finite-dimensional eqations and nonsqueezing properties. Internat. Math. Res. Notice, 2(1994), 79-88.
[40] Fonseca G., Linares F., J. Ponce, Global well-posedness for the modified Kortewegde Vries equation. Comm. Partial Differential Equations, 24(3-4)(1999), 683-705.
[41] Colliander J., Staffilani G., Takaoka H., Global wellposedness of KdV below l~2. Math. Res. Letters, 6(1999), 755-778.
[42] 郭柏灵等译，非线性边值问题的一些解法．中山大学出版社．
[43] Folland G. B., Introduction to the partial differential equations. Princeton University Press, 1976.
[44] Miao C．X．，调和分析极其在偏微分方程中的应用．科学出版社，1999．
[45] Pazy A., Semigroups of linear opeartor and application to partial differential equations. Springer-Verlag, 1983.
[46] Triebel H., Theory of function space. Birkhauser Verlag, 1983.
[47] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Global well-posedness for Schrodinger equation with derivative. Siam Y.Math.Anal. 33No.3(2001), 649-669.
[48] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Global well-posedness and scattering for rough solution of a nonlinear Schrodinger equation in R~3. CPAM 57(2004), 987-1014
[49] Colliander J., Keel M., Staffilani G., Takaoka H.. Tao T., Global well-posedness and scattering in the energy space for the critical nonlinear Scrodinger equation in R~3. To apper, Annals of Math.
[50] Tao T., Multilinear weighted convolution of L~2 functions, and application to nonlinear dispersive equations. Amer. J. Math. 123 (2001), 839-908.
[51] Colliander J., Keel M., Staffilani G., Takaoka EL Tao T.. Global well-posedness result for KdV in Sobolev spaces of negative index. EJDE 2001 (2001) No 26, 1-7.
[52] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.. Sharp global well-posedness results for periodic KdV and nonperiodic KdV and modified KdV on R and T. J. Amer. Math. Soc. 16 (2003), 705-749.
[53] Colliander J., Keel M., Staffilani G., Takaoka H.. Tao T.. Global well-posedness of 2D NLS.(in preparation), 2001.
[54] Christ M., Colliander J., Tao T., Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125(2003), to appear.
[55] Staffilani G., On the growth of high Sobolev norms of solutions for KdV and Schrodinger equations. Duke Math. J. 86(1)(1997) 109-142.
[56] R. M. Miura, Korteweg-de Vries equation and generalizations. I.A remarkable explicit nonlinear transformation, J. Mathematical Phy., 9(1968)1202-1204.
[57] R. M. Miura, The Korteweg-de Vries equation:A Survey Results, SIAM Review. 18(3)(1976)412-459.
[58] K. Nakanishi, H. Takaoka, Y. Tsutsumi, Counterexamples to bilinear estimates related to The KdV equation and the nonlinear Schrodinger equation, preprint. 2000.
[2] C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. on Pure and Appl. Math., 46(1993), 527-620.
[3] E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Eucliden Spaces. Princeton University Press: Princeton, 1971.
[4] C. E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math J., 71(1993), 1-21.
[5] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with application to the Korteweg-De Vries equation. J. of the American Math. Society., 9(1996), 573-603.
[6] C. E. Kenig, G. Ponce, L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106(2001), 617-633.
[7] J. Bona, R. Winther, The Korteweg-de Vries equation posed in a quarter-plane. Siam Y. Math. Anal., 14(1983), 1056-1106.
[8] J. Bona, L. R. Scott, Solutions of the Korteweg-de Vries equations in fractional order Sobolev spaces. Duke Math. J., 43(1976), 87-99.
[9] ) J. Bona, R. Smith, The initial-value problem for the KdV equation. Philos. Trans. Roy. Soc. London Ser. A. 278(1287)(1975), 555-601.
[10] A. Cohen, Solutions of the Korteweg-de Vreis equation from irregular data. Duke Math. J., 45(1978),149-181.
[11] J. Ginibre, Y. Tsutsumi, Uniqueness for the generalized Korteweg-de Vries equa- tions. Siam J. Math. Anal., 20(1989), 1388-1425.
[12] Colliander J., Keel M., Staffilani G., Takaoka H, Tao T., Sharp global wellposedness results for periodic and non-periodic KdV and modified KdV on R and T. JIAMS, 16(2003), 705-749.
[13] Guo Boling, Wang Baoxiang, The Cauchy problem for Davey-Stewantson systems. Comm. on Pure Appl. Math., LII(1999), 1477-1490.
[14] Guo Boling, Wu yaping, Orbital stbility of solitary waves for the nonlinear derivative Schrodinger equation. J. Differential Equation 123(1995), 35-55.
[15] Wang Baoxing, Guo Boling, The Cauchy problem and the existence of scatting for the generalized Davey-Stewartson equations. Chinese Science(A), 44(8)(2001), 994-1002.
[16] Wang Baoxing, Bessel(Riesz) potentials on Banach function spaces and their applications I theory, Acta. Mathematica Sinica, New Series 14(3)(1998), 327-340.
[17] Wang Baoxiang, On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in Hs, Nonlinear Analysis, TMA, 31(1998), 173-187.
[18] Wang Baoxiang, Large time behavior of solutions for critical and subcritical complex Ginzburg-Landau eqations in H~1, Prepint.
[19] Bergh J., Lofstrom J., Interpolation Spaces. Springer-Verlag, 1976.
[20] Shen Caixia, The global wellposedness and scattering of the generalized DaveyStewartson equation. J. Partial Diff. Eqs., 17(2004), 89-96.
[21] Cazenave T., Weissler F. B., The Cauchy problem for critical nonlinear Schrodinger equations in H~s. Nonlinear Anal., TMA, 14(1990), 807-836.
[22] Cazenave T., Weissler F. B., The Cauchy problem for the nonlinear Schrodinger equation in H~1. Manuscipta Math., 61(1988), 477-494.
[23] Keel M., Tao T., Endpoint Strichartz estimates. Amer. J. Math., 120(1998), 955-980.
[24] Davey A., Stewartson K., On three dimensional packets of surface waves. Proc. Roy. Soc. Ser A, 338(1974), 101-110.
[25] C. E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KDV equation. Journal of the A. M. S., 9(1996), 573-603.
[26] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Almost conservation laws and global rough solution to a Nonlinear Schrodinger Equation. Math. Res. Letters, 9(2002), 659-682.
[27] Christ M., Colliander J., Tao T., Ill-posedness for nonlinear Schrodinger and wave equations. To appear, Annales IHP
[28] B. Birnir, G. Ponce, N. Svanstedt, The local ill-posed of the modified KdV equation. Ann. Inst. Henri Poincare 13 4(1996), 529-535.
[29] B. Birnir, F. Linares, Ill-posed for the derivative Schrodinger and generalized Benjamin-Ono equations. Trans. Arner. Math. Soc. 353(2001)9, 3649-3659.
[30] B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega, On the illposedness of the generalized KdV and nonlinear Schrodinger equations. J. London Math. Soc. (2)53(1996), 551-559.
[31] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations. J. Math. Phys., 18(1977), 1794-1797.
[32] L. Molinet, J. C. Saut, N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations. SIAM J. Math. Anal. 33(2001), 982-988.
[33] C. Sulem, P. L. Sulem, The nonlinear Schrodinger equation: Self-focusing and wave collapse. Springer-Verlag, New York 1999.
[34] Cazenave T., An introduction to nonlinear Schrodinger equations. Textos de Metodos Matematicos 26 Instituto de Matematica, Rio de Janeiro, 1996.
[35] Bourgain J., Fourier transform restriction phenomena for certain lattice subsets and applictions to nonlinear evolution equations. Ⅰ. Geom.Funct. Anal, 3(1993), 107-156.
[36] Bourgain J., Fourier transform restriction phenomena for certain lattice subsets and applictions to nonlinear evolution equations. Ⅱ. Geom.Funct. Anal, 3(1993), 209-262.
[37] Bourgain J., Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity. Intern. Mat. Res. Notices, 5(1998), 253-283.
[38] Bourgain J., On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE. Intern. Mat. Res. Notices, 6(1996), 277-304.
[39] )Bourgain J., Approximation of solutions of the cubic nonlinear Schrodinger equations by finite-dimensional eqations and nonsqueezing properties. Internat. Math. Res. Notice, 2(1994), 79-88.
[40] Fonseca G., Linares F., J. Ponce, Global well-posedness for the modified Kortewegde Vries equation. Comm. Partial Differential Equations, 24(3-4)(1999), 683-705.
[41] Colliander J., Staffilani G., Takaoka H., Global wellposedness of KdV below l~2. Math. Res. Letters, 6(1999), 755-778.
[42] 郭柏灵等译，非线性边值问题的一些解法．中山大学出版社．
[43] Folland G. B., Introduction to the partial differential equations. Princeton University Press, 1976.
[44] Miao C．X．，调和分析极其在偏微分方程中的应用．科学出版社，1999．
[45] Pazy A., Semigroups of linear opeartor and application to partial differential equations. Springer-Verlag, 1983.
[46] Triebel H., Theory of function space. Birkhauser Verlag, 1983.
[47] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Global well-posedness for Schrodinger equation with derivative. Siam Y.Math.Anal. 33No.3(2001), 649-669.
[48] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Global well-posedness and scattering for rough solution of a nonlinear Schrodinger equation in R~3. CPAM 57(2004), 987-1014
[49] Colliander J., Keel M., Staffilani G., Takaoka H.. Tao T., Global well-posedness and scattering in the energy space for the critical nonlinear Scrodinger equation in R~3. To apper, Annals of Math.
[50] Tao T., Multilinear weighted convolution of L~2 functions, and application to nonlinear dispersive equations. Amer. J. Math. 123 (2001), 839-908.
[51] Colliander J., Keel M., Staffilani G., Takaoka EL Tao T.. Global well-posedness result for KdV in Sobolev spaces of negative index. EJDE 2001 (2001) No 26, 1-7.
[52] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.. Sharp global well-posedness results for periodic KdV and nonperiodic KdV and modified KdV on R and T. J. Amer. Math. Soc. 16 (2003), 705-749.
[53] Colliander J., Keel M., Staffilani G., Takaoka H.. Tao T.. Global well-posedness of 2D NLS.(in preparation), 2001.
[54] Christ M., Colliander J., Tao T., Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125(2003), to appear.
[55] Staffilani G., On the growth of high Sobolev norms of solutions for KdV and Schrodinger equations. Duke Math. J. 86(1)(1997) 109-142.
[56] R. M. Miura, Korteweg-de Vries equation and generalizations. I.A remarkable explicit nonlinear transformation, J. Mathematical Phy., 9(1968)1202-1204.
[57] R. M. Miura, The Korteweg-de Vries equation:A Survey Results, SIAM Review. 18(3)(1976)412-459.
[58] K. Nakanishi, H. Takaoka, Y. Tsutsumi, Counterexamples to bilinear estimates related to The KdV equation and the nonlinear Schrodinger equation, preprint. 2000.