A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

Abstract

We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle. We first consider the real-valued setting and show global well-posedness of the (usual) renormalized mKdV equation in the Fourier-Lebesgue spaces. In the complex-valued setting, we observe that the momentum plays an important role in the well-posedness theory. In particular, we prove that the complex-valued mKdV equation is ill-posed in the sense of non-existence of solutions when the momentum is infinite, in the spirit of the work on the nonlinear Schrödinger equation by Guo-Oh (2018). This nonexistence result motivates the introduction of the second renormalized mKdV equation, which we propose as the correct model in the complex-valued setting outside H 1 2 (T). Furthermore, imposing a new notion of finite momentum for the initial data, at low regularity, we show existence of solutions to the complex-valued mKdV equation. In particular, we require an energy estimate, from which conservation of momentum follows.2020 Mathematics Subject Classification. 35Q53.

A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier–Lebesgue Spaces

“…In this paper, we continue the study of the well-posedness of the periodic complex-valued mKdV equation (1.1) initiated in [4], by adapting the method introduced by Deng et al [9] to the mKdV equation (1.1). Before stating our main results, we review the known wellposedness theory for the mKdV equation (1.1).…”

“…In view of the ill-posedness of (1.1), in [4], we proposed an alternative model for the complex-valued mKdV equation (1.1) at low regularity. Similarly to the first gauge transform G 1 which exploited the conservation of mass, we introduced a second renormalization of the equation through the following gauge transformation depending on the momentum G 2 (u)(t, x) = e ∓i P(u)t u(t, x).…”

“…4 11 p .+ ϕ T (DN R C,> + DN R D,> ) w, w, ϕ T • GA,≥ [w, u, u] + ϕ T (DN R C,> + DN R D,> ) w, w, ϕ T • G A,> [w, u, u] + ϕ T (DN R C,> + DN R D,> ) w, w, ϕ T • G B,≥ [w, w, u] + ϕ T (DN R C,> + DN R D,> ) w, w, ϕ T • G B,> [w, w, u] + ϕ T B 2 A,≥ w, ϕ T • G A,≥ [w, u, u], u + B 2 A,≥ w, ϕ T • G A,> [w, u, u], u + ϕ T B 2 A,≥ w, ϕ T • G B,≥ [w, w, u], u + B 2 A,≥ w, ϕ T • G B,> [w, w, u], u + ϕ T B 2 A,> w, ϕ T • G A,≥ [w, u, u], u + B 2 A,> w, ϕ T • G A,> [w, u, u], u + ϕ T B 2 A,> w, ϕ T • G B,≥ [w, w, u], u + B 2 A,> w, ϕ T • G B,> [w, w, u], u + ϕ T B 3 A,≥ w, u, ϕ T • G A,≥ [w, u, u] + B 3 A,≥ w, u, ϕ T • G A,> [w, u, u] + ϕ T B 3 A,≥ w, u, ϕ T • G B,≥ [w, w, u] + B 3 A,≥ w, u, ϕ T • G B,> [w, w, u] + ϕ T B 3 A,> w, u, ϕ T • G A,≥ [w, u, u] + B 3 A,> w, u, ϕ T • G A,> [w, u, u] + ϕ T B 3 A,> w, u, ϕ T • G B,≥ [w, w, u] + B 3 A,> w, u, ϕ T • G B,> [w, w, u] + ϕ T B 3 B,≥ w, w, ϕ T • G A,≥ [w, u, u] + B 3 B,≥ w, w, ϕ T • G A,> [w, u, u] + ϕ T B 3 B,≥ w, w, ϕ T • G B,≥ [w, w, u] + B 3 B,≥ w, w, ϕ T • G B,> [w, w, u] + ϕ T B 3 B,> w, w, ϕ T • G A,≥ [w, u, u] + B 3 B,> w, w, ϕ T • G A,> [w, u, u] + ϕ T B 3 B,> w, w, ϕ T • G B,≥ [w, w, u] + B 3 B,> w, w, ϕ T • G B,> [w, w, u] . (A.1)…”

Long time decay and asymptotics for the complex mKdV equation

“…Local wellposedness has also been show in the weighted Sobolev spaces H s ∩ |x| −m L 2 for s ≥ 2m, m ∈ Z + in [39] and in H s ∩ |x| −m L 2 for s ≥ 1/4, s ≥ 2m in [13]. For the equation set on the torus, wellposedness was shown by Chapouto in a range of Fourier-Lebesgue spaces [3,4]. Relaxing the requirement of uniformly continuous dependence on the initial data, Harrop-Griffiths, Killip, and Visan used the complete integrability of the equation to prove a weaker form of wellposedness in H s for s > −1/2 in [28].…”

“…They also show that for s ≤ −1/2, the equation exhibits instantaneous norm inflation, so no wellposedness result is possible. Outside the scale of H s spaces, Grünrock proved in [21] that the equation is locally wellposed for real-valued initial data in the spaces H s r defined by the norms u H s r := ξ û L r ′ for the parameter range 4 3 < r ≤ 2, s ≥ 1 2 − 1 2r . The parameter range was later improved by Grünrock and Herr in [22] to 1 < r ≤ 2, s ≥ 1 2 − 1 2r , which has the scaling-critical space H 0 1 as the (exclude) endpoint.…”