Globalwell-posedness and I method for the fifth order Korteweg-de Vries equation

Abstract

The Kawahara equation has fewer symmetries than the KdV equation; in particular, it has no invariant scaling transform and is not completely integrable. Thus its analysis requires different methods. We prove that the Kawahara equation is locally well posed in H −7/4 , using the ideas of anF s -type space [8]. Then we show that the equation is globally well posed in H s for s ≥ −7/4, using the ideas of the "I-method" [7].

Well-posedness issues on the periodic modified Kawahara equation

“…The Cauchy problems for the Kawahara and modified Kawahara equations posed on R have been extensively studied. For the Kawahara equation ((1.2) with p = 2), we refer to [18,17,74,13,32,12,37,39,64] for the well-and ill-posedness results. As the best result in the sense of the low regularity Cauchy problem, Kato [37,39] proved the local well-posedness for s ≥ −2 by modifying X s,b space, the global well-posedness for s > − 38 21 and the ill-posedness for s < −2 in the sense that the flow map is discontinuous at zero.…”

Norm inflation for the generalized Boussinesq and Kawahara equations

“…This equation arises in modeling capillary-gravity waves on a shallow layer and magneto-sound propagation in plasmas ( [15]). Many authors have studied well-posedness of (1.3) (see [4,25,3,5,14] and references therein). Kato [14] proved that (1.3) is well-posed in H s (R) with s ≥ −2 and ill-posed in H s (R) with s < −2.…”

Decay properties for solutions of fifth order nonlinear dispersive equations

“…These results heavily depend on the structure of the nonlinearity P(·) considered. For the precise statements of these kind of results see [18], [2], [13], [3], [8] and references therein. In particular, Kenig and Pilod [16] established local and global results in the energy space for the IVP associated to the equation (1.1) with P(·) as in (1.3) (see also [7]).…”