Abstract
We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2021), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside H^frac{1}{2}({mathbb {T}}). Here, we employ the method introduced by Deng et al. (Commun Math Phys 384(1):1061–1107, 2021) to prove local well-posedness of the second renormalized mKdV equation in the Fourier–Lebesgue spaces {mathcal {F}}L^{s,p}({mathbb {T}}) for sge frac{1}{2} and 1le p <infty . As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier–Lebesgue spaces with infinite momentum.
Highlights
1.1 Modified Korteweg-de Vries EquationWe consider the Cauchy problem for the complex-valued modified Korteweg-de Vries equation on the one-dimensional torus T = R/(2πZ):∂t u + ∂x3u = ±|u|2∂x u, u|t=0 = u0, (t, x) ∈ R × T, (1.1)where u is a complex-valued function
As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) without the second renormalization for initial data in these Fourier–Lebesgue spaces with infinite momentum
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]
Summary
The best known result in the Sobolev scale is due to Kappeler and Topalov [21] who exploited the completely integrable structure of (1.2) They showed that the real-valued defocusing mKdV equation, (1.2) with ‘+’, is globally wellposed in H s(T) for s ≥ 0. We showed that is the limit for the local well-posedness of the complex-valued equation (1.1), as it is ill-posed outside in the sense of non-existence of solutions. This result is closely related to the momentum. Since the main focus of this paper is on improving the previous well-posedness result of mKdV2 (1.8), we will not discuss further how to recover solutions of the complex-valued mKdV equation (1.1) from those of mKdV2 (1.8)
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