A completely KdV-type Boussinesq system in low regularity spaces
Keywords:Cauchy problem, Korteweg-de Vries equations, Global well posedness, Bourgain spaces, almost conservation laws
In this paper we study the well-posedness of Cauchy problem for a Boussinesq system formed by two Kortewegde Vries equations coupled through the linear part and the non-linear terms. First we proof its local well-posedness
in the Sobolev spaces Hs (R) x Hs (R), s > -3/4, using the bilinear estimate established by Kenig, Ponce and Vega in the Fourier transform restriction spaces [4, 12]. After, we prove the global well-posedness in Hs (R) x Hs (R) for s > -3/10, our proof proceeds by the method of almost conservation laws, sometimes called the “I-method”[5, 6].
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