リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「Long range scattering and blow-up problem for nonlinear dispersive equations」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。
リケラボ

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

Long range scattering and blow-up problem for nonlinear dispersive equations

Komada Koichi 東北大学

2021.03.25

概要

This thesis is concerned with the dynamics of nonlinear dispersive equations. A nonlinear dispersive equation is a partial differential equation with linear dispersion and nonlinear interaction, which describes the propagation of waves in several physics. Due to the com- petition among dispersion and nonlinear interaction, there are several types of behavior of solutions for nonlinear dispersive equations. In particular, there are three typical types of behavior as follows:

・(Scattering) When dispersion is predominant over nonlinearity, the amplitude of solutions decays as time evolves and nonlinearity can be neglected for sufficiently large time. In particular, we say that a solution scatters if it behaves like a solution of the linearized equation for large time.

・(Blow-up) When the contribution of nonlinearity is much larger than that of dis- persion, the amplitude of solutions rapidly increases and some norm of solutions divergent in finite time.

・(Soliton) When dispersion and nonlinearity balance each other, there are solutions of which amplitude does not decay or increase. Typical examples in this case are standing wave solutions and traveling wave solutions. In particular, spatially local- ized standing wave solutions and traveling wave solutions are called soliton.

In this thesis, we consider three problems about dynamics for nonlinear dispersive equations. The first one is the scattering problem for nonlinear dispersive equations with a nonlocal dispersive term and a critical nonlinearity. The second one is the blow- up problem for nonlinear Schr¨odinger equations with anisotropic fourth-order dispersion. The third one is the blow-up problem for the quantum Zakharov system which is the system of a fourth-order Schr¨odinger equation and a fourth-order wave equation. These contents are based on the three papers [37, 38, 39].

論文の公開元へ

関連論文

関連論文をもっと見る

参考文献

[1] Abdelouhab, L., Bona, J. L., Felland, M. and Saut, J.-C., Nonlocal models for non- linear dispersive waves, Phys. D 40 (1989), 360–392.

[2] Aceves, A. B., de Angelis, C., Rubenchik, A. M. and Turitsyn, S. K., Multidimen- sional solitons in fiber arrays, Opt. Lett. 19 (1995), 329–331.

[3] Albert, J. P., Bona, J. L. and Henry, D. B., Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, Phys. D 24 (1987), 343– 366.

[4] Bahouri, H., Chemin, Y. J. and Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer-Verlag, Berlin-Heidelberg-New York, 2011.

[5] Bergh, J. and L¨ofstr¨om, J., Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.

[6] Bona, J. L., Lannes, D. and Saut, J.-C., Asymptotic models for internal waves, J. Math. Pures Appl. 89 (2008), 538–566.

[7] Bonheure, D., Cast´eras, J-B., Gou, T. and Jeanjean, L., Strong instability of ground states to a fourth order Schr¨odinger equation, Int. Math. Res. Not. IMRN (2019), 5299–5315.

[8] Bouchel, O., Remarks on NLS with higher order anisotropic dispersion, Adv. Differ- ential Equations 13 (2008), 169–198.

[9] Boulenger, T. and Lenzmann, E., Blowup for biharmonic NLS, Ann. Sci. E´c. Norm. Sup´er. (4), 50 (2017), 503–544.

[10] Br´ezis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.

[11] Chen, H. H. and Lee, Y. C., Internal-wave solutions of fluids with finite depth, Phys. Rev. Lett. 43 (1979), 264–266.

[12] Chen, T. J., Fang, Y. F. and Wang, K. H., Low regularity global well-posedness for the quantum Zakharov system in 1D, Taiwanese J. Math. 21 (2017), 341–361.

[13] Fang, Y. F., Lin, C. K. and Segata, J., The fourth order nonlinear Schr¨odinger limit for quantum Zakharov system, Z. Angew. Math. Phys. (2016) 67:145.

[14] Fang, Y. F. and Nakanishi, K., Global well-posedness and scattering for quantum Zakharov system in L2, Proc. Amer. Math. Soc. Series B 6 (2019), 21–32.

[15] Fang, Y. F., Segata, J. and Wu, T. F., On the standing waves of quantum Zakharov system, J. Math. Anal. Appl. 458 (2018), 1427–1448.

[16] Fang, Y. F., Shin, H. W. and Wang, K. H., Local well-posedness for the quantum Zakharov system in one spatial dimension, J. Hyperbolic Differ. Equ. 14 (2017), 157–192.

[17] Fibich, G., Ilan, B. and Papanicolaou, G., Self-focusing with fourth-order dispersion, SIAM J. Appl. Math. 62 (2002), 1437–1462.

[18] Fibich, G., Ilan, B. and Schochet, S., Critical exponents and collapse of nonlin- ear Schr¨odinger equations with anisotropic fourth-order dispersion, Nonlinearity 16 (2003), 1809–1821.

[19] Garcia, L. G., Hass, F., Oliveira, L. P. L. and Goedert, J., Modified Zakharov equa- tions for plasmas with a quantum correction, Phys. Plasmas 12 (2005), 012302.

[20] Glassey, R. T., On the blowing-up of solutions to the Cauchy problem for the nonlinear Schr¨odinger equations, J. Math. Phys. 18 (1977), 1794–1797.

[21] Glangetas, L. and Merle, F., Existence of self-similar blow-up solutions for the Za- kharov equation in dimension two. Part I, Commun. Math. Phys. 160 (1994), 173– 215.

[22] Glangetas, L. and Merle, F., Concentration properties of blow-up solutions and in- stability results for Zakharov equation in dimension two. Part II, Commun. Math. Phys. 160 (1994), 349–389.

[23] Guo, C. and Cui, S., Solvability of the Cauchy problem of non-isotropic Schr¨odinger equations in Sobolev spaces, Nonlinear Anal. 68 (2008), 768–780.

[24] Guo, C. and Cui, S., Well-posedness of the Cauchy problem of high dimension non- isotropic fourth-order Schr¨odinger equations in Sobolev spaces, Nonlinear Anal. 70 (2009), 3761–3772.

[25] Guo, C., Zhao, X. and Wei, X., Cauchy problem for higher-order Schr¨odinger equa- tions in anisotropic Sobolev space, Appl. Anal. 88 (2009), 1329–1338.

[26] Guo, Z. and Nakanishi, K., The Zakharov system in 4D radial energy space below the ground state, e-print arXiv:1810.05794

[27] Guo, Z., Nakanishi, K. and Wang, S., Global dynamics below the ground state energy for the Zakharov system in the 3D radial case, Adv. Math. 238 (2013), 412–441.

[28] Gustafson, S., Nakanishi, K. and Tsai, T. P., Scattering for the Gross-Pitaevskii equation, Math. Res. Lett. 13 (2006), 273–285.

[29] Hass, F., Garcia, L. G. and Goedert, J., Quantum Zakharov equations, Fourth In- ternational Winter Conference on Mathematical Methods in Physics, CBPF, Rio de Janeiro (SISSA, WC2004), J. High Energy Phys. (2005) 015.

[30] Hass, F. and Shukla, P. K., Quantum and classical dynamics of Langmuir wave packets, Phys. Rev. E 79 (2009), 066402.

[31] Hayashi, N. and Naumkin, P. I., Final state problem for Korteweg-de Vries type equations, J. Math. Phys. 47 (2006), 123501.

[32] Hayashi, N., Naumkin, P. I. and S´anchez-Su´arez, I., Asymptotic for the modified Whitham equation, Commun. Pure Appl. Anal. 17 (2018), 1407–1448.

[33] Joseph, R. I., Solitary waves in a finite depth fluid, J. Phys. A 10 (1977), 225–227.

[34] Keel, M. and Tao, T., Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955–980.

[35] Kenig, C. E., Ponce, G. and Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33–69.

[36] Kodama, Y., Ablowitz, M. J. and Satsuma, J., Direct and inverse scattering problems of the nonlinear intermediate long wave equation, J. Math. Phys. 23 (1982), 564–576.

[37] Komada, K., Final state problem for class of nonlinear nonlocal dispersive equation, J. Math. Anal. Appl. 480 (2019), 123434.

[38] Komada, K., Existence of blow-up solutions to nonlinear Schr¨odinger equations with anisotropic fourth-order dispersion, preprint.

[39] Komada, K., Existence of radially symmetric blow-up solutions for quantum Zakharov system, preprint.

[40] Lebedev, D. R. and Radul, A. O., Generalized internal long wave equations: con- struction, Hamiltonian structure, and conservation laws, Comm. Math. Phys. 91 (1983), 543–555.

[41] Lieb, E., On the lowest eigenvalue of the Laplacian for the intersection of two do- mains, Invent. Math. 74 (1983), 441–448.

[42] Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case. I and II, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 1 (1984), 109–145 and 223–283.

[43] Martel, Y., Blow-up for the nonlinear Schr¨odinger equation in nonisotropic spaces, Nonlinear Anal. 28 (1997), 1903–1908.

[44] Merle, F., Blow-up results of virial type for Zakharov equations, Comm. Math. Phys. 175 (1996), 433–455.

[45] Naumkin, P. I. and S´anchez-Su´arez, I., On the modified intermediate long-wave equa- tion, Nonlinearity 31 (2018), 980–1008.

[46] Ogawa, T. and Tsutsumi, Y., Blow-up of H1 solutions for the nonlinear Schr¨odinger equation, J. Differential Equations 92 (1991), 317–330.

[47] Pausader, B., Global well-posedness for energy critical fourth-order Schr¨odinger equa- tions in the radial case, Dyn. Partial Differ. Equ. 4 (2007), 197–225.

[48] Ponce, G., On the global well-posedness of the Benjamin-Ono equation, Differential integral Equations 4 (1991), 527–542.

[49] Santini, P. M., Ablowitz, M. J. and Fokas, A. S., On the limit from the intermediate long wave equation to the Benjamin-Ono equation, J. Math. Phys. 25 (1984), 892– 899.

[50] Satsuma, J., Ablowitz, M. J. and Kodama, Y., On an internal wave equation de- scribing a stratified fluid with finite depth, Phys. Lett. A 73 (1979), 283–286.

[51] Satsuma, J., Ablowitz, M. J. and Kodama, Y., Nonlinear intermediate long-wave equation: analysis and method of solution, Phys. Rev. Lett. 46 (1981), 687–690.

[52] Satsuma, J., Taha, T. R. and Ablowitz, M. J., On a B¨acklund transformation and scattering problem for the modified intermediate long wave equation, J. Math. Phys. 25 (1984), 900–904.

[53] Saut, J-C. and Segata, J., Asymptotic behavior in time of solution to the nonlinear Schr¨odinger equation with higher order anisotropic dispersion, Discrete. Contin. Dyn. Syst. 39 (2019), 219–239.

[54] Saut, J-C. and Segata, J., Long range scattering for the nonlinear Schr¨odinger equa- tion with higher order anisotropic dispersion in two dimensions, J. Math. Anal. Appl. 483 (2020), 123638, 17 pp.

[55] Scoufis, G. and Cosgrove, C. M., An application of the inverse scattering transform to the modified intermediate long wave equation, J. Math. Phys. 46 (2005), 103501.

[56] Smith, R., Nonlinear Kelvin and continental-shelf waves, J. Fluid. Mech. 57 (1972), 379–391.

[57] Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.

[58] Wen, S. and Fan, D., Spatiotemporal instabilities in nonlinear kerr media in the presence of arbitrary higher order dispersion, J. Opt. Soc. Am. B 19 (2002) 1653– 1659.

[59] Whitham, G. B., Linear and Nonlinear Waves, Pure and Applied Mathematics, Wiley-Interscience, New York, 1974.

[60] Zhang, X. Y., A note on the illposedness for anisotropic nonlinear Schr¨odinger equa- tion, Acta Math. Sin. (English series) 24 (2008), 891–900.

[61] Zhao, X., Guo, C., Sheng, W. and Wei, X., Well-posedness of the fourth-order per- turbed Schr¨odinger type equation in non-isotropic Sobolev spaces, J. Math. Anal. Appl. 382 (2011), 97–109.

参考文献をもっと見る