On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data

Yang Han

doi:10.3934/dcds.2005.12.233

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2005, Volume 12, Issue 2: 233-242. Doi: 10.3934/dcds.2005.12.233

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On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data

Yang Han^1,

1.
Department of Mathematics, South-west Jiaotong University, Chengdu 610031

Received: April 30, 2003

Revised: May 31, 2004

Published: January 2005

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Abstract

An interaction equations of the complex scalar nucleon field and real scalar meson field is considered. we show that the Cauchy problem of the Klein-Gordon-Schrödinger system

$ iu_{t}+u_{x x}=-uv, $

$ v_{t t}-v_{x x}+v=|u|^2,$

$u(0, x)= u_0(x), v(0, x)= v_0(x), v_t(0, x)= v_1(x)$

is locally well-posed for weak initial data $(u_0, v_0, v_1)\in H^s\times H^{s-1/2}\times H^{s-3/2}$ with $s\geq 0$. We use the analogous method for estimate the nonlinear couple terms developed by Bourgain and refined by Kenig, Ponce and Vega.

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Mathematics Subject Classification: Primary 35Q99, 35B65.

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