Local well-posedness of quasi-linear systems generalizing KdV

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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

Abstract
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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 Schrödinger 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.

Keywords: energy method., quasilinear, KdV, dispersive, partial differential equations.

Mathematics Subject Classification: Primary: 35Q53, 35G2.

Citation: Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & 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, (2011).
[2]	D. M. Ambrose, G. Simpson, J. D. Wright, and D. G. Yang, Ill-posedness of degenerate dispersive equations,, Nonlinearity \textbf{25} (2012), 25 (2012). 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)., (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. 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. doi: 10.1007/s002200050801.
[6]	P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations,, J. Amer. Math. Soc., 1 (1988), 413. doi: S0894-0347-1988-0928265-0.
[7]	W. Craig, J. Goodman, Linear dispersive equations of Airy type,, J. Differential Equations \textbf{87} (1990), 87 (1990), 38. 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. Poincar\'e Anal. Non Lin\'eaire, 9 (1992), 147.
[9]	W. Craig, T. Kappeler and W. Strauss, Microlocal dispersive smoothing for the Schrödinger equation,, Comm. Pure Appl. Math., 48 (1995), 769. 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.
[11]	N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension,, Differential Integral Equations, 7 (1994), 453.
[12]	L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations,", Math\'ematiques & Applications (Berlin) [Mathematics & Applications], (1997).
[13]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, Studies in applied mathematics, (1983), 93.
[14]	C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 10 (1993), 255.
[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. 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. 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. 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.) \textbf{120} (1983), 120 (1983), 396.
[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. 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 \textbf{231} (2012), 231 (2012). doi: 10.1016/j.aim.2012.06.010.
[21]	S. Mizohata, "On the Cauchy Problem,", Notes and Reports in Mathematics in Science and Engineering, (1985).
[22]	P. Sjölin, Regularity of solutions to the Schrödinger equation,, Duke Math. J., 55 (1987), 699. 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. 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, (2011).
[2]	D. M. Ambrose, G. Simpson, J. D. Wright, and D. G. Yang, Ill-posedness of degenerate dispersive equations,, Nonlinearity \textbf{25} (2012), 25 (2012). 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)., (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. 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. doi: 10.1007/s002200050801.
[6]	P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations,, J. Amer. Math. Soc., 1 (1988), 413. doi: S0894-0347-1988-0928265-0.
[7]	W. Craig, J. Goodman, Linear dispersive equations of Airy type,, J. Differential Equations \textbf{87} (1990), 87 (1990), 38. 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. Poincar\'e Anal. Non Lin\'eaire, 9 (1992), 147.
[9]	W. Craig, T. Kappeler and W. Strauss, Microlocal dispersive smoothing for the Schrödinger equation,, Comm. Pure Appl. Math., 48 (1995), 769. 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.
[11]	N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension,, Differential Integral Equations, 7 (1994), 453.
[12]	L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations,", Math\'ematiques & Applications (Berlin) [Mathematics & Applications], (1997).
[13]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, Studies in applied mathematics, (1983), 93.
[14]	C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 10 (1993), 255.
[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. 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. 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. 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.) \textbf{120} (1983), 120 (1983), 396.
[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. 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 \textbf{231} (2012), 231 (2012). doi: 10.1016/j.aim.2012.06.010.
[21]	S. Mizohata, "On the Cauchy Problem,", Notes and Reports in Mathematics in Science and Engineering, (1985).
[22]	P. Sjölin, Regularity of solutions to the Schrödinger equation,, Duke Math. J., 55 (1987), 699. 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. doi: 10.1090/S0002-9939-1988-0934859-0.

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