Global well-posedness for the generalized derivative nonlinear Schrödinger equation

In this article we study the well-posedness of the generalized derivative nonlinear Schr & ouml;dinger equation (gDNLS) iut+uxx=i|u|2 sigma ux, for small powers sigma. We analyze this equation at both low and high regularity, and are able to establish global well-posedness in Hs when s is an element of[1,4 sigma) and sigma is an element of(32,1). Our result when s = 1 is particularly relevant because it corresponds to the regularity of the energy for this problem. Moreover, a theorem of Liu et al (2013 J. Nonlinear Sci. 23 557-83) establishes the orbital stability of the gDNLS solitons, provided that there is a suitable H1 well-posedness theory. To our knowledge, this is the first low regularity well-posedness result for a quasilinear dispersive model where the nonlinearity is both rough and lacks the decay necessary for global smoothing type estimates. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates. While the tools developed in this article are used to study gDNLS, we believe that they should be applicable in the study of local well-posedness for other dispersive equations of a similar character. It should also be noted that the high regularity well-posedness presents a novel issue, as the roughness of the nonlinearity limits the potential regularity of solutions. Our high regularity well-posedness threshold s<4 sigma is twice as high as one might na & iuml;vely expect, given that the function z|z|2 sigma is only C1,2 sigma-1 H & ouml;lder continuous. Moreover, although we cannot prove H1 well-posedness when sigma <= 32, we are able to establish Hs well-posedness in the high regularity regime s is an element of(2-sigma,4 sigma) for the full range of sigma is an element of(12,1). This considerably improves the known local results, which had only been established in either H2 or in weighted Sobolev spaces.