Low regularity a priori bounds for the modified Korteweg-de Vries equation

Abstract

Abstract:We study the local well-posedness in the Sobolev space H s (R) for the modified Korteweg-de Vries (mKdV) equation ∂ t u+∂ 3x u±∂ x u 3 = 0 on R. Kenig-Ponce-Vega [10] and Christ-Colliander-Tao [1] established that the data-to-solution map fails to be uniformly continuous on a fixed ball in H s (R) when s <

On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations

“…This will be done in Section 5. We will use a variant of the function spaces from [13,4] to prove a priori estimates in the first step. Next, L 2 -Lipschitz dependence for initial data of higher regularity is discussed.…”

“…However, in Euclidean space we do not have to use the Fourier transform in time, which allows for a simplification of the construction compared to the periodic case. Short-time L 2 -valued U p -/V p -spaces will be utilized like in [4,29]. Here, we will be very brief and instead refer to these works for a presentation of the basic function space properties.…”

On global well-posedness of the modified KdV equation in modulation spaces

“…In [17,18], Kenig-Ponce-Vega exploited the dispersive nature of the equation and proved local well-posedness of (1.1) in H s (R), s ≥ 1 4 . In [27], Tao gave an alternative proof of the local well-posedness in H 1 4 (R) by using the Fourier restriction norm method. We also mention recent papers [23,22] on unconditional uniqueness of solutions to (1.1) in H s (R), s > 1 4 .…”

“…This in particular implies that one can not use a contraction argument to construct solutions to (1.1) in this regime. One possible approach to study rough solutions outside H 1 4 (R) is to use a more robust energy method. In [4], Christ-Holmer-Tataru employed an energy method in the form of the short-time Fourier restriction norm method and proved global existence of solutions to the real-valued mKdV (1.3) in H s (R) for − 1 8 < s < 1 4 .…”

“…3 The upper bound 1 − 1 p is not essential and we expect that this restriction can be relaxed by a consideration similar to that in Section 3 of [20]. 4 The modulation spaces are based on the unit cube decomposition of the frequency space and thus there is no scaling for the modulation spaces. We, however, say that M 2,∞ 0 (R) is a critical space in view of the embedding (1.5) with s = 0 and p = ∞.…”