Local well-posedness of quasi-linear systems generalizing KdV

Communications on Pure and Applied Analysis

Volumn 12, Issue 2, 2013, Pages 899-921

  Akhunov, Timur ^a  

^a UNIVERSITY OF CALGARY   (Canada)

Author keywords

Dispersive; Energy method; KdV; Partial differential equations; Quasilinear

Indexed keywords

EID: 84873320837     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2013.12.899     Document Type: Article

References (23)

1. T. Akhunov, "Local Well Posedness of Dispersive Systems in One Dimension, " Ph.D. thesis, University of Chicago, 2011.
2. D. M. Ambrose, G. Simpson, J. D. Wright, and D. G. Yang, Ill-posedness of degenerate dispersive equations, Nonlinearity 25 (2012), no. 9.
3. D. M. Ambrose and J. D. Wright, Dispersion vs. anti-diffusion: well-posedness in variable coefficient and quasilinear equations of KdV-type, ArXiv e-prints (2012).
4. J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.
5. A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
6. P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.
7. W. Craig, J. Goodman, Linear dispersive equations of Airy type, J. Differential Equations 87 (1990), no. 1, 38-61.
8. W. Craig, T. Kappeler and W. Strauss, Gain of regularity for equations of KdV type, Ann. Inst. H. Poincare Anal. Non Lineaire, 9 (1992), 147-186.
9. W. Craig, T. Kappeler and W. Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math., 48 (1995), 769-860.
10. S. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions, J. Math. Kyoto Univ., 34 (1994), 319-328.
11. N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension, Differential Integral Equations, 7 (1994), 453-461.
12. L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations, " Mathematiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997.
13. T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93-128.
14. C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 10 (1993), 255-288.
15. C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.
16. C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., 158 (2004), 343-388.
17. C. E. Kenig and G. Stalani, Local well-posedness for higher order nonlinear dispersive systems, J. Fourier Anal. Appl., 3 (1997), 417-433.
18. S. N. Kruzhkov and A. V. Faminski, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Mat. Sb. (N.S.) 120 (1983), 396-425.
19. W. K. Lim and G. Ponce, On the initial value problem for the one dimensional quasi-linear Schrödinger equations, SIAM J. Math. Anal., 34 (2002), 435-459 (electronic).
20. J. L. Marzuola, J. Metcalfe and D. Tataru, Quasilinear Schrödinger equations i: Small data and quadratic interactions, Advances in Mathematics 231 (2012), 1151-1172.
21. S. Mizohata, "On the Cauchy Problem, " Notes and Reports in Mathematics in Science and Engineering, vol. 3, Academic Press Inc., Orlando, FL, 1985.
22. P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J., 55 (1987), 699-715.
23. L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878.