• DOI:10.1006/JFAN.1997.3148
• Corpus ID: 120543995

On the Cauchy Problem for the Zakharov System

@article{Ginibre1997OnTC,
  title={On the Cauchy Problem for the Zakharov System},
  author={Jean Ginibre and Yoshio Tsutsumi and Giorgio Velo},
  journal={Journal of Functional Analysis},
  year={1997},
  volume={151},
  pages={384-436}
}

• J. Ginibre, Y. Tsutsumi, G. Velo
• Published 15 December 1997
• Mathematics
• Journal of Functional Analysis

Abstract We study the local Cauchy problem in time for the Zakharov system, (1.1) and (1.2), governing Langmuir turbulence, with initial data ( u (0), n (0), ∂ t n (0))∈ H k ⊕ H lscr; ⊕ H l−1 , in arbitrary space dimension ν . We define a natural notion of criticality according to which the critical values of ( k , l) are ( ν /2−3/2, ν /2−2). Using a method recently developed by Bourgain, we prove that the Zakharov system is locally well posed for a variety of values of ( k , l). The results cover the whole subcritical range for ν ⩾4. For ν ⩽3, they cover only part of it and the lowest admissible values are ( k , l)=(1/2, 0) for ν =2, 3 and ( k , l)=(0, −1/2) for ν =1. As a by product of the one dimensional result, we prove well-posedness of the Benney system, (1.14) and (1.15), governing the interaction of short and long waves for the same values of ( k , l).