American Institute of Mathematical Sciences

March  2013, 12(2): 899-921. doi: 10.3934/cpaa.2013.12.899

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

Received  September 2011 Revised  August 2012 Published  September 2012

In this article we prove local well-posedness of quasilinear dispersive systems of PDE generalizing KdV. These results adapt the ideas of Kenig-Ponce-Vega from the Quasi-Linear Schrödinger equations to the third order dispersive problems. The main ingredient of the proof is a local smoothing estimate for a general linear problem that allows us to proceed via the artificial viscosity method.
Citation: Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure and Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899
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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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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