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

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 $$H^\frac{1}{2}({\mathbb {T}})$$ H 1 2 ( 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 $${\mathcal {F}}L^{s,p}({\mathbb {T}})$$ F L s , p ( T ) for $$s\ge \frac{1}{2}$$ s ≥ 1 2 and $$1\le p <\infty $$ 1 ≤ p < ∞ . 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.

Probabilistic local Cauchy theory of the cubic nonlinear wave equation in negative Sobolev spaces

“…In the context of nonlinear Schrödinger type equations, such instability results without renormalization in negative Sobolev spaces are known even deterministically; see [30,49]. See also [16,17] for a similar instability result on the complex-valued mKdV equation in the deterministic setting.…”

“…for k ∈ N, where the positive constant c is independent of k and N . Noting that the right-hand side is summable in N ∈ N, we can invoke the Borel-Cantelli lemma to conclude that there exists Ω k of full probability such that (16) for each ω ∈ Ω k , there exists M = M (ω) 1 such that for any N M , we have…”

On Strichartz Estimates from ℓ 2-Decoupling and Applications

“…This shows that the unrenormalized equation is ill-posed in Fourier Lebesgue spaces of negative Sobolev regularity. Further recent local well-posedness results on the complex-valued mKdV equation in Fourier Lebesgue spaces are due to Chapouto [10,11]. We also mention the recent work of Forlano [19], who unified and simplified previous proofs of global well-posedness for the modified Korteweg-de Vries equation in Sobolev spaces H s (K) with 0 ≤ s < ε N ε f L 2 (T) , I have proved in [38] that there is no data-to-solution mapping to the unrenormalized equation in the following sense in H s (T) for s < 0: A map S : H s (T) → C([−T, T ], H s (T)) where T = T ( u 0 H s ) > 0 is referred to as data-to-solution mapping to (81) if it satisfies the following properties:…”

“…We emphasize that the decoupling arguments do not distinguish between the square and irrational tori, which means we can allow for general T 2 γ . We suppose that for i = 1, 2, 3 (11) supp( fi ) ⊆ B(ξ * i , N 3 ) for some ξ * i ∈ B(0, 2N 1 ) and (12) |ξ * 1 − ξ * 2 | N 1 . Consider the expression (13) [0,N −α 1 ]×T 2 γ |e it∆ f 1 e it∆ f 2 e it∆ f 3 | 4/3 dxdt.…”

A remark on the well-posedness of the modified KdV equation in $L^2$

“…To understand why this is the case, we recall that Chapouto [6,7] showed that (1.9) and even its renormalized variant, in the sense of the corresponding gauge transformation to (1.8), are ill-posed below H 1/2 (T). The cause of the instability is the possibility that the momentum P (q) = Im ˆT qq ′ dx may be infinite outside H 1/2 (T).…”