Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

Abstract

In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities.In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity.Finally, we investigate the KdV equation at the endpoint regularity H −3/4 , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [−T, T ] and a locally Lipschitz continuous map taking the initial data in H −3/4 to a distributional solution u ∈ C 0 ([−T, T ]; H −3/4 ) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.

Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

“…In this paper, we prove GWP for k = 0, s = − 1 2 , using a scheme based on mass conservation (1.2) and subcritical slack in certain multilinear estimates at this regularity threshold. In [12], it is shown that the onedimensional LWP theory of [11] is effectively sharp by adapting techniques of [5] and [7]. Thus, we establish GWP in the largest space for which LWP holds.…”

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

“…In his work, of particular importance is the gauge transformation defined by 6) with J (f )(x) :=´x −∞ |f (y)| 2 dy, x ∈ R. Through the transformation v = G 1 2 (u), DNLS reduces to i∂ t v + ∂ 2 x v = i|v| 2 ∂ x v (1.7) and the theory of [45] for smooth initial data with sufficiently small H 1 -norm can be applied. This result was improved in the papers by Hayashi and Ozawa [15,16] by reducing DNLS to a system of two semi-linear Schrödinger equations (with no derivatives in the nonlinearities), where it was obtained 3 the global existence of H 1 -solutions under the assumption while the norms of interest remain essentially unchanged, i.e.…”

On the Propagation of Regularity and Decay of Solutions to thek-Generalized Korteweg-de Vries Equation

“…After the works of [23], [1] and [13] gathering the local and the global well-posedness results, in [15], [2], [16], [3], [4], [25], [11], [12] and [17] one has:…”