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

Timur Akhunov ^1,

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 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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