Well-posedness and ill-posedness of the fifth-order modified KdV equation

Kwon, Soonsik

Electronic journal of differential equations, Tome 2008 (2008)

Résumé

We consider the initial value problem of the fifth-order modified KdV equation on the Sobolev spaces.

$\displaylines{ \partial_t u - \partial_x^5u + c_1\partial_x^3(u^3) + c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0\cr u(x,0)= u_0(x) }$

where $ u:\mathbb{R}\times\mathbb{R} \to \mathbb{R} $ and $c_j$'s are real. We show the local well-posedness in $H^s(\mathbb{R})$ for $s\geq 3/4$ via the contraction principle on $X^{s,b}$ space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below $H^{3/4}(\mathbb{R})$. The counter example is obtained by approximating the fifth order mKdV equation by the cubic NLS equation.

Détail

Classification : 35Q53
Keywords: local well-posedness, ill-posedness, mkdv hierarchy