Low Regularity Local Well-Posedness of the Derivative Nonlinear Schrödinger Equation with Periodic Initial Data

Abstract

The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local well-posedness for data u 0 in the space b H s r (T), defined by the normsis shown in the parameter range s ≥ 1 2 , 2 > r > 4 3 . The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method.2000 Mathematics Subject Classification. 35Q55.

Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition inH12

“…It was made clear in [12] that the above frequency cancelations are essential in establishing the estimate that deals with the derivative-cubic nonlinearity in the scale of FourierLebesgue spaces FL s,r (T). However, in Sobolev spaces H s (T), for s ≥ 1 2 , one can handle the cubic-derivative term u 2 ∂ x u without the frequency cancelations (see Lemma 4.2 below due to Herr [17]).…”

“…We also recall an observation of Grünrock and Herr [12,Remark 2], that in the periodic setting, due to the presence of a translation in space operator in the gauge transformation G β (see (2.42)), at any regularity level, the uniform continuity of the solution map of (1.1) fails without fixing the mass on bounded subsets of H s (T) (see also [18, Theorem 3.1.1.(ii)]). Nevertheless, for the gauge equivalent equation (2.43) one does not face the uniform continuity bottleneck due to the translation operator and it was for this equation that the contraction mapping argument was applied in [17].…”

“…In fact, the local uniform continuity of the solution map of the periodic DNLS fails without fixing the mass on bounded subsets of H s (T), at any regularity level (see [18, Theorem 3.1.1.(ii)]). Although counterexamples to the trilinear estimate (1.19) for s < 1 2 were given in [12], to the best of the author's knowledge, a direct argument towards the (mild) ill-posedness of (1.1) in H s (T) can not be found in the literature. Note that for the gauge equivalent equation (1.18), one does not face the local uniform continuity bottleneck due to the translation operator and it was for this equation that the contraction mapping argument was applied in [17].…”

“…Regarding possible improvements to the estimate (1.19), a remarkable property of (1.18) was uncovered by Grünrock and Herr [12] while working in the scale of FourierLebesgue spaces FL s,r (T) (defined by (2.2) below). By appropriately modifying the classical X s,b norms, they established a local well-posedness result in FL 20) owing to the periodic gauge transformation.…”

“…12) one obtains that the energy E[u] controls theḢ 1 -norm of a solution u of DNLS, provided that (1.8) holds. Finally, we mention that in [14,15], uniqueness of H 1 (R)-solutions was also obtained, but conditional to the auxiliary spaces L 12 t (R; H the linear part of the equation at hand (defined by (2.5) on T, their definition on R being analogous).…”