On the Cauchy Problem for the Zakharov System

✍ Scribed by J. Ginibre; Y. Tsutsumi; G. Velo

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 672 KB
• Volume: 151
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3148

✦ Synopsis

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 Ä H l ÄH l&1 , in arbitrary space dimension &. We define a natural notion of criticality according to which the critical values of (k, l) are (&Â2&3Â2, &Â2&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 & 4. For & 3, they cover only part of it and the lowest admissible values are (k, l)=(1Â2, 0) for &=2, 3 and (k, l)=(0, &1Â2) for &=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).