July  2015, 14(4): 1259-1274. doi: 10.3934/cpaa.2015.14.1259

A remark on the well-posedness of a degenerated Zakharov system

1. 

Universidade Federal da Bahia, Instituto de Matemática, Av. Adhemar de Barros , Ondina, 40170-110, Salvador, Bahia, Brazil

2. 

IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320

Received  September 2013 Revised  March 2014 Published  April 2015

We extend the local well-posedness theory for the Cauchy problem associated to a degenerated Zakharov system. The new main ingredients are the derivation of Strichartz and maximal function norm estimates for the linear solution of a Schrödinger type equation with missing dispersion in one direction. The result here improves the one in [10].
Citation: Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259
References:
[1]

I. Bejenaru, S. Herr, J. Holmer, and D. Tataru, On the 2D Zakharov system with $L^2$ Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007.  Google Scholar

[2]

M. Colin and M. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations, 17 (2004), 297-330.  Google Scholar

[3]

T. Colin and G. Métivier, Instabilities in Zakharov equations for lazer propagation in a plasma, Phase space analysis of partial differential equations, Progr. Nonlinear Differential Equations Appl., 69 (2006), 297-330, Birkhäuser Boston, MA. doi: 10.1007/978-0-8176-4521-2_6.  Google Scholar

[4]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.  Google Scholar

[5]

J. Ginibre, Y. 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.  Google Scholar

[6]

C. E. Kenig and S. N. Ziesler, Maximal function estimate with aplications to a modified Kadomtsev-Petviashvili equation, Comm. Pure Appl. Anal., 4 (2005), 45-91. doi: 10.3934/cpaa.2005.4.45.  Google Scholar

[7]

C. E. Kenig, G. 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.  Google Scholar

[8]

C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, 10 (1993), 255-288.  Google Scholar

[9]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equation, Springer, New York, 2009, 256 pp.  Google Scholar

[10]

F. Linares, G. Ponce and J. C. Saut, On a degenerate Zakharov System, Bull. Braz. Math. Soc., 36 (2005), 1-23. doi: 10.1007/s00574-005-0025-3.  Google Scholar

[11]

T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., 28 (1992), 329-361. doi: 10.2977/prims/1195168430.  Google Scholar

[12]

G. Riazuelo, Etude théorique et numérique de l'influence du lissage optique sur la filamentation des faisceaux lasers dans les plasmas sous-critiques de fusion inertielle,, Th\`ese de l'Universit\'e Paris XI., ().   Google Scholar

[13]

V. E. Zakharov, Collapse of Langmuir Waves, Sov. Phys. JETP, 35 (1972), 908-914. Google Scholar

show all references

References:
[1]

I. Bejenaru, S. Herr, J. Holmer, and D. Tataru, On the 2D Zakharov system with $L^2$ Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007.  Google Scholar

[2]

M. Colin and M. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations, 17 (2004), 297-330.  Google Scholar

[3]

T. Colin and G. Métivier, Instabilities in Zakharov equations for lazer propagation in a plasma, Phase space analysis of partial differential equations, Progr. Nonlinear Differential Equations Appl., 69 (2006), 297-330, Birkhäuser Boston, MA. doi: 10.1007/978-0-8176-4521-2_6.  Google Scholar

[4]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.  Google Scholar

[5]

J. Ginibre, Y. 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.  Google Scholar

[6]

C. E. Kenig and S. N. Ziesler, Maximal function estimate with aplications to a modified Kadomtsev-Petviashvili equation, Comm. Pure Appl. Anal., 4 (2005), 45-91. doi: 10.3934/cpaa.2005.4.45.  Google Scholar

[7]

C. E. Kenig, G. 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.  Google Scholar

[8]

C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, 10 (1993), 255-288.  Google Scholar

[9]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equation, Springer, New York, 2009, 256 pp.  Google Scholar

[10]

F. Linares, G. Ponce and J. C. Saut, On a degenerate Zakharov System, Bull. Braz. Math. Soc., 36 (2005), 1-23. doi: 10.1007/s00574-005-0025-3.  Google Scholar

[11]

T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., 28 (1992), 329-361. doi: 10.2977/prims/1195168430.  Google Scholar

[12]

G. Riazuelo, Etude théorique et numérique de l'influence du lissage optique sur la filamentation des faisceaux lasers dans les plasmas sous-critiques de fusion inertielle,, Th\`ese de l'Universit\'e Paris XI., ().   Google Scholar

[13]

V. E. Zakharov, Collapse of Langmuir Waves, Sov. Phys. JETP, 35 (1972), 908-914. Google Scholar

[1]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126

[2]

Akansha Sanwal. Local well-posedness for the Zakharov system in dimension d ≤ 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021147

[3]

Hartmut Pecher. Local well-posedness for the Klein-Gordon-Zakharov system in 3D. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1707-1736. doi: 10.3934/dcds.2020338

[4]

Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019

[5]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087

[6]

Zijun Chen, Shengkun Wu. Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021161

[7]

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

[8]

Alex M. Montes, Ricardo Córdoba. Local well-posedness for a class of 1D Boussinesq systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021030

[9]

Gustavo Ponce, Jean-Claude Saut. Well-posedness for the Benney-Roskes/Zakharov- Rubenchik system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 811-825. doi: 10.3934/dcds.2005.13.811

[10]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[11]

Shinya Kinoshita. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1479-1504. doi: 10.3934/dcds.2018061

[12]

Isao Kato. Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2247-2280. doi: 10.3934/cpaa.2016036

[13]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075

[14]

Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic & Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029

[15]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813

[16]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

[17]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

[18]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

[19]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195

[20]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]