A PRIORI ESTIMATES FOR THE FIFTH-ORDER MODIFIED KDV EQUATIONS IN BESOV SPACES WITH LOW REGULARITY

Abstract

We get a priori estimates for the fifth-order modified KdV equations in Besov spaces with low regularity which cover the full subcritical range. These estimates are obtained from the power series expansion of the perturbation determinant associated to the Lax pair. More precisely, we get the global in time bounds of the $B^s_{2,r}$ norm of the solution for $-1/2< s < 1$, $1\leq r\leq \infty$. Then we can obtain the sharp global well-posedness in $H^s$ for $s\geq 3/4$, which is the minimal regularity threshold for which the well-posedness problem can be solved via the contraction principle.