June  2022, 12(2): 447-473. doi: 10.3934/mcrf.2021030

Local well-posedness for a class of 1D Boussinesq systems

1. 

Mathematics Department, Universidad del Cauca, Popayán, Cauca, Colombia

* Corresponding author: Alex M. Montes

Received  October 2020 Revised  March 2021 Published  June 2022 Early access  May 2021

Fund Project: A. Montes and R. Córdoba were supported by Universidad del Cauca

In this paper we study the local well-posedness for the Cauchy problem associated with a special class of one-dimensional Boussinesq systems that model the evolution of long water waves with small amplitude in the presence of surface tension.

Citation: Alex M. Montes, Ricardo Córdoba. Local well-posedness for a class of 1D Boussinesq systems. Mathematical Control and Related Fields, 2022, 12 (2) : 447-473. doi: 10.3934/mcrf.2021030
References:
[1]

B. Alvarez-Samaniego and X. Carvajal, On the local well-posedness for some systems of coupled KdV equations, Nonlinear Analysis, 69 (2008), 692-715.  doi: 10.1016/j.na.2007.06.009.

[2]

D. BekiranovT. Ogawa and G. Ponce, Interaction equations for short and long dispersive waves, J. Funct. Anal., 158 (1998), 357-388.  doi: 10.1006/jfan.1998.3257.

[3]

J. BonaM. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media Ⅰ: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.

[4]

J. BonaM. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media Ⅱ: The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[5]

J. BonaZ. Grujić and H. Kalisch, A KdV-type Boussinesq system: From the energy level to analytic spaces, Discret. Contin. Dyn. Syst., 26 (2010), 1121-1139.  doi: 10.3934/dcds.2010.26.1121.

[6]

J. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equations, Discret. Contin. Dyn. Syst., 23 (2009), 1241-1252.  doi: 10.3934/dcds.2009.23.1241.

[7]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅰ, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[8]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅱ, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[9]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.

[10]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.

[11]

E. Compaan and N. Tzirakis, Well-posedness and nonlinear smoothing for the "good" Boussinesq equation on the half-line, J. Differential Equations, 262 (2017), 5824-5859.  doi: 10.1016/j.jde.2017.02.016.

[12]

A. Esfahani and L. Farah, Local well-posedness for the sixth-order Boussinesq equation, J. Math. Anal. Appl., 385 (2012), 230-242.  doi: 10.1016/j.jmaa.2011.06.038.

[13]

L. Farah, Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation, Comm. Partial Differential Equations, 34 (2009), 52-73.  doi: 10.1080/03605300802682283.

[14]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.

[15]

S. Li, M. Chen and B. Zhang, Low regularity solutions of non-homogeneous boundary value problems of a higher order Boussinesq equation in a quarter plane, J. Math. Anal. App., 492 (2020), 124406, 1–35. doi: 10.1016/j.jmaa.2020.124406.

[16]

C. KenigG. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.

[17]

C. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[18]

C. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[19]

C. KenigG. 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.

[20]

C. KenigG. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation, Trans. Amer. Math. Soc., 348 (1996), 3323-3353.  doi: 10.1090/S0002-9947-96-01645-5.

[21]

F. Linares, $L^{2}$ global well-posedness of the initial value problem associated to the Benjamin equation, J. Differential Equations, 152 (1999), 377-393.  doi: 10.1006/jdeq.1998.3530.

[22]

J. Quintero, Solitary water waves for a 2D Boussinesq type system, J. Partial Differ. Equ., 23 (2010), 251-280. 

[23]

J. Quintero and A. Montes, Existence, physical sense and analyticity of solitons for a 2D Boussinesq-Benney-Luke system, Dyn. Partial Differ. Equ., 10 (2013), 313-342.  doi: 10.4310/DPDE.2013.v10.n4.a1.

[24]

J. Quintero and A. Montes, On the Cauchy and solitons for a class of 1D Boussinesq systems, Differ. Equ. Dyn. Syst., 24 (2016), 367-389.  doi: 10.1007/s12591-015-0264-8.

[25]

J. Quintero and A. Montes, Periodic solutions for a class of one-dimensional Boussinesq systems, Dyn. Partial Differ. Equ., 13 (2016), 241-261.  doi: 10.4310/DPDE.2016.v13.n3.a3.

[26]

J. Quintero, A. Montes and R. Córdoba, On the stability of a Boussinesq system, work in progress.

[27]

X. Yang and B. Zhang, Local well-posedness of the coupled KdV-KdV systems on $\mathbb R$, preprint, 2020, arXiv: 1812.08261v2.

show all references

References:
[1]

B. Alvarez-Samaniego and X. Carvajal, On the local well-posedness for some systems of coupled KdV equations, Nonlinear Analysis, 69 (2008), 692-715.  doi: 10.1016/j.na.2007.06.009.

[2]

D. BekiranovT. Ogawa and G. Ponce, Interaction equations for short and long dispersive waves, J. Funct. Anal., 158 (1998), 357-388.  doi: 10.1006/jfan.1998.3257.

[3]

J. BonaM. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media Ⅰ: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.

[4]

J. BonaM. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media Ⅱ: The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[5]

J. BonaZ. Grujić and H. Kalisch, A KdV-type Boussinesq system: From the energy level to analytic spaces, Discret. Contin. Dyn. Syst., 26 (2010), 1121-1139.  doi: 10.3934/dcds.2010.26.1121.

[6]

J. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equations, Discret. Contin. Dyn. Syst., 23 (2009), 1241-1252.  doi: 10.3934/dcds.2009.23.1241.

[7]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅰ, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[8]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅱ, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[9]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.

[10]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.

[11]

E. Compaan and N. Tzirakis, Well-posedness and nonlinear smoothing for the "good" Boussinesq equation on the half-line, J. Differential Equations, 262 (2017), 5824-5859.  doi: 10.1016/j.jde.2017.02.016.

[12]

A. Esfahani and L. Farah, Local well-posedness for the sixth-order Boussinesq equation, J. Math. Anal. Appl., 385 (2012), 230-242.  doi: 10.1016/j.jmaa.2011.06.038.

[13]

L. Farah, Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation, Comm. Partial Differential Equations, 34 (2009), 52-73.  doi: 10.1080/03605300802682283.

[14]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.

[15]

S. Li, M. Chen and B. Zhang, Low regularity solutions of non-homogeneous boundary value problems of a higher order Boussinesq equation in a quarter plane, J. Math. Anal. App., 492 (2020), 124406, 1–35. doi: 10.1016/j.jmaa.2020.124406.

[16]

C. KenigG. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.

[17]

C. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[18]

C. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[19]

C. KenigG. 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.

[20]

C. KenigG. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation, Trans. Amer. Math. Soc., 348 (1996), 3323-3353.  doi: 10.1090/S0002-9947-96-01645-5.

[21]

F. Linares, $L^{2}$ global well-posedness of the initial value problem associated to the Benjamin equation, J. Differential Equations, 152 (1999), 377-393.  doi: 10.1006/jdeq.1998.3530.

[22]

J. Quintero, Solitary water waves for a 2D Boussinesq type system, J. Partial Differ. Equ., 23 (2010), 251-280. 

[23]

J. Quintero and A. Montes, Existence, physical sense and analyticity of solitons for a 2D Boussinesq-Benney-Luke system, Dyn. Partial Differ. Equ., 10 (2013), 313-342.  doi: 10.4310/DPDE.2013.v10.n4.a1.

[24]

J. Quintero and A. Montes, On the Cauchy and solitons for a class of 1D Boussinesq systems, Differ. Equ. Dyn. Syst., 24 (2016), 367-389.  doi: 10.1007/s12591-015-0264-8.

[25]

J. Quintero and A. Montes, Periodic solutions for a class of one-dimensional Boussinesq systems, Dyn. Partial Differ. Equ., 13 (2016), 241-261.  doi: 10.4310/DPDE.2016.v13.n3.a3.

[26]

J. Quintero, A. Montes and R. Córdoba, On the stability of a Boussinesq system, work in progress.

[27]

X. Yang and B. Zhang, Local well-posedness of the coupled KdV-KdV systems on $\mathbb R$, preprint, 2020, arXiv: 1812.08261v2.

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