Electron. J. Diff. Eqns., Vol. 2002(2002), No. 75, pp. 1-11.

Regularity bounds on Zakharov system evolutions

James Colliander & Gigliola Staffilani

Spatial regularity properties of certain global-in-time solutions of the Zakharov system are established. In particular, the evolving solution $u(t)$ is shown to satisfy an estimate $\|u(t)\|_{H^s} \leq C |t|^{(s-1)+}$, where $H^s$ is the standard spatial Sobolev norm. The proof is an adaptation of earlier work on the nonlinear Schrodinger equation which reduces matters to bilinear estimates.

Submitted March 15, 2002. Published August 20, 2002.
Math Subject Classifications: 35Q55.
Key Words: initial value problems, bilinear estimates, Zakharov system, weak turbulence.

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James Colliander
Department of Mathematics
University of Toronto
Toronto, ON M5R 1W8 Canada
e-mail: colliand@math.toronto.edu
Gigliola Staffilani
Department of Mathematics
Massachusets Institute of Technology
77 Massachusets Avenue
Cambridge, MA 02139-4307 USA
e-mail: gigliola@math.mit.edu

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