American Institute of Mathematical Sciences

Advanced
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

[1]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503
[2]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024
[3]

Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
[4]

Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
[5]

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1275-1326. doi: 10.3934/cpaa.2015.14.1275
[6]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[7]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[8]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[9]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[10]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[11]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032
[12]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019045
[13]

Carlos E. Kenig. The method of energy channels for nonlinear wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6979-6993. doi: 10.3934/dcds.2019240
[14]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345
[15]

Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747
[16]

Juan-Ming Yuan, Jiahong Wu. A dual-Petrov-Galerkin method for two integrable fifth-order KdV type equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1525-1536. doi: 10.3934/dcds.2010.26.1525
[17]

Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929
[18]

Figen Özpinar, Fethi Bin Muhammad Belgacem. The discrete homotopy perturbation Sumudu transform method for solving partial difference equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 615-624. doi: 10.3934/dcdss.2019039
[19]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231
[20]

Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences