Global solutions of the  wave-Schrödinger system with rough data

Takafumi Akahori

doi:10.3934/cpaa.2005.4.209

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2005, Volume 4, Issue 2: 209-240. Doi: 10.3934/cpaa.2005.4.209

This issue Previous Article Next Article Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients

Global solutions of the wave-Schrödinger system with rough data

Takafumi Akahori^1,

1.
Mathematical Institute, Tohoku University, Sendai 980-8578

Received: December 31, 2003

Revised: November 30, 2004

Published: May 2005

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Abstract

We prove that the wave-Schröodinger system is globally well-posed for data in $(H^{s_{1}} \times \dot{H}^{s_{2}} \times \dot{H}^{s_{2}-1})(\mathbb^{d})$, where $d=3,4$ and $s_{1},s_{2}> q_{d} \ (q_{3}=(\sqrt{57}-5)/4, \ q_{4}=\sqrt{3}-1)$. Our proof is based on the I-method. We introduce the space $\Omega^{s,b}$ which controls the low frequency part and the modified multiplier for I-method to work in the space $\Omega^{s,b}$.

Keywords:

Globalwell-posedness,
wave-Schrödingersystem.

Mathematics Subject Classification: 35L05,35Q55.

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