The well-posedness of Cauchy problem for dissipative modified Korteweg de Vries equations

Details The well-posedness of Cauchy problem for dissipative modified Korteweg de Vries equations The well-posedness of Cauchy problem for dissipative modified Korteweg de Vries equations

2007-01-22

arxiv.org/abs/math/0701602v1

Differential and Integral Equations Volume 20, Number 11(2007)1285-1301

Mathematics

Analysis of PDEs

20 pages

Scientific paper

In this paper we consider some dissipative versions of the modified Korteweg
de Vries equation $u_t+u_{xxx}+|D_x|^{\alpha}u+u^2u_x=0$ with $0<\alpha\leq 3$.
We prove some well-posedness results on the associated Cauchy problem in the
Sobolev spaces $H^s({\Bbb R})$ for $s>1/4-\alpha/4$ on the basis of the $[k;
Z]-$multiplier norm estimate obtained by Tao in \cite{Tao} for KdV equation.

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