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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

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).   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[5]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45.  doi: 10.1007/s002200050801.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

S. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions,, J. Math. Kyoto Univ., 34 (1994), 319.   Google Scholar

[11]

N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension,, Differential Integral Equations, 7 (1994), 453.   Google Scholar

[12]

L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations,", Math\'ematiques & Applications (Berlin) [Mathematics & Applications], (1997).   Google Scholar

[13]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, Studies in applied mathematics, (1983), 93.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Mizohata, "On the Cauchy Problem,", Notes and Reports in Mathematics in Science and Engineering, (1985).   Google Scholar

[22]

P. Sjölin, Regularity of solutions to the Schrödinger equation,, Duke Math. J., 55 (1987), 699.  doi: 10.1215/S0012-7094-87-05535-9.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

T. Akhunov, "Local Well Posedness of Dispersive Systems in One Dimension,", Ph.D. thesis, (2011).   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[5]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45.  doi: 10.1007/s002200050801.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

S. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions,, J. Math. Kyoto Univ., 34 (1994), 319.   Google Scholar

[11]

N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension,, Differential Integral Equations, 7 (1994), 453.   Google Scholar

[12]

L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations,", Math\'ematiques & Applications (Berlin) [Mathematics & Applications], (1997).   Google Scholar

[13]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, Studies in applied mathematics, (1983), 93.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Mizohata, "On the Cauchy Problem,", Notes and Reports in Mathematics in Science and Engineering, (1985).   Google Scholar

[22]

P. Sjölin, Regularity of solutions to the Schrödinger equation,, Duke Math. J., 55 (1987), 699.  doi: 10.1215/S0012-7094-87-05535-9.  Google Scholar

[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.  Google Scholar

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