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Well-posedness for KdV-type equations with quadratic nonlinearity (1812.10002v1)

Published 25 Dec 2018 in math.AP

Abstract: We consider the Cauchy problem of the KdV-type equation [ \partial_t u + \frac{1}{3} \partial_x3 u = c_1 u \partial_x2u + c_2 (\partial_x u)2, \quad u(0)=u_0. ] Pilod (2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space $Hs(\mathbb{R})$ for any $s \in \mathbb{R}$ if $c_1 \neq 0$. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in $H2(\mathbb{R})$ with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in $H1(\mathbb{R})$ with bounded primitives.

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