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,, \emph{Nonlinearity}, 22 (2009), 1063. doi: 10.1088/0951-7715/22/5/007.

[2]

M. Colin and M. Colin, On a quasilinear Zakharov system describing laser-plasma interactions,, \emph{Differential Integral Equations}, 17 (2004), 297.

[3]

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

[4]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation,, \emph{Differential Equations}, 31 (1995), 1002.

[5]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, \emph{J. Funct. Anal.}, 151 (1997), 384. doi: 10.1006/jfan.1997.3148.

[6]

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

[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,, \emph{Comm. Pure Appl. Math.}, 46 (1993), 527. doi: 10.1002/cpa.3160460405.

[8]

C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations,, \emph{Ann. Inst. Henri Poincar\'e}, 10 (1993), 255.

[9]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equation,, Springer, (2009).

[10]

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

[11]

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

[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., ().

[13]

V. E. Zakharov, Collapse of Langmuir Waves,, \emph{Sov. Phys. JETP}, 35 (1972), 908.

show all references

References:
[1]

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

[2]

M. Colin and M. Colin, On a quasilinear Zakharov system describing laser-plasma interactions,, \emph{Differential Integral Equations}, 17 (2004), 297.

[3]

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

[4]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation,, \emph{Differential Equations}, 31 (1995), 1002.

[5]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, \emph{J. Funct. Anal.}, 151 (1997), 384. doi: 10.1006/jfan.1997.3148.

[6]

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

[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,, \emph{Comm. Pure Appl. Math.}, 46 (1993), 527. doi: 10.1002/cpa.3160460405.

[8]

C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations,, \emph{Ann. Inst. Henri Poincar\'e}, 10 (1993), 255.

[9]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equation,, Springer, (2009).

[10]

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

[11]

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

[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., ().

[13]

V. E. Zakharov, Collapse of Langmuir Waves,, \emph{Sov. Phys. JETP}, 35 (1972), 908.

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