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Local well-posedness of quasi-linear systems generalizing KdV
1. | Department of Mathematics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada |
References:
[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.
doi: 10.1088/0951-7715/25/9/2655. |
[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.
doi: 10.1098/rsta.1975.0035. |
[5] |
A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[6] |
P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.
doi: S0894-0347-1988-0928265-0. |
[7] |
W. Craig, J. Goodman, Linear dispersive equations of Airy type, J. Differential Equations, 87 (1990), 38-61.
doi: 10.1016/0022-0396(90)90014-G. |
[8] |
W. Craig, T. Kappeler and W. Strauss, Gain of regularity for equations of KdV type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 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.
doi: 10.1002/cpa.3160480802. |
[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," Mathématiques & 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. Poincaré Anal. Non Linéaire, 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.
doi: 10.1090/S0894-0347-96-00200-7. |
[16] |
C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., 158 (2004), 343-388.
doi: 10.1007/s00222-004-0373-4. |
[17] |
C. E. Kenig and G. Staffilani, Local well-posedness for higher order nonlinear dispersive systems, J. Fourier Anal. Appl., 3 (1997), 417-433.
doi: 10.1007/BF02649104. |
[18] |
S. N. Kruzhkov and A. V. Faminskiĭ, 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).
doi: 10.1137/S0036141001399520. |
[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.
doi: 10.1016/j.aim.2012.06.010. |
[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.
doi: 10.1215/S0012-7094-87-05535-9. |
[23] |
L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878.
doi: 10.1090/S0002-9939-1988-0934859-0. |
show all references
References:
[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.
doi: 10.1088/0951-7715/25/9/2655. |
[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.
doi: 10.1098/rsta.1975.0035. |
[5] |
A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[6] |
P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.
doi: S0894-0347-1988-0928265-0. |
[7] |
W. Craig, J. Goodman, Linear dispersive equations of Airy type, J. Differential Equations, 87 (1990), 38-61.
doi: 10.1016/0022-0396(90)90014-G. |
[8] |
W. Craig, T. Kappeler and W. Strauss, Gain of regularity for equations of KdV type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 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.
doi: 10.1002/cpa.3160480802. |
[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," Mathématiques & 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. Poincaré Anal. Non Linéaire, 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.
doi: 10.1090/S0894-0347-96-00200-7. |
[16] |
C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., 158 (2004), 343-388.
doi: 10.1007/s00222-004-0373-4. |
[17] |
C. E. Kenig and G. Staffilani, Local well-posedness for higher order nonlinear dispersive systems, J. Fourier Anal. Appl., 3 (1997), 417-433.
doi: 10.1007/BF02649104. |
[18] |
S. N. Kruzhkov and A. V. Faminskiĭ, 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).
doi: 10.1137/S0036141001399520. |
[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.
doi: 10.1016/j.aim.2012.06.010. |
[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.
doi: 10.1215/S0012-7094-87-05535-9. |
[23] |
L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878.
doi: 10.1090/S0002-9939-1988-0934859-0. |
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