Low regularity problems of the fifth-order KdV and the modified KdV equations.

摘要

In this dissertation we consider the fifth-order equations arising from the KdV and modified KdV hierarchies, the so-called fifth-order KdV equation and the fifth-order mKdV equation. We study the initial value problem of these equations with initial data in the Sobolev spaces Hs( R ). For both equations we have low regularity local well-posedness and ill-posedness results in some sense.;In the first part we prove that the fifth-order KdV equation 6tu+65x u+c16xu62 xu+c2u63xu+0 u0,x=u0 x is locally well-posed in Hs( R ) for s ≥ 52 . Also, we prove the solution map of the equation is not uniformly continuous on a bounded set. The local well-posedness proof is based on the modified energy method and uses some linear estimates coming from dispersion effect. For the negative result, we use a similar example introduced for the Benjamin-Ono equation by Koch-Tzvetkov [20]. In the counter example, strong low-high interaction plays a crucial role.;In the second part, we show the initial value problem of the fifth-order modified KdV equation 6tu-65x u+c163x+c2 u6xu62xu+c 3uu63xu=0 ux,0=u0 x is locally well-posed in Hs( R ) for s ≥ ¾ via the contraction principle on the Xs,b space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below H¾ ( R ). The counter example is obtained by approximating the fifth-order mKdV equation by the cubic NLS equation.