Well-posedness issues on the periodic modified Kawahara equation

Abstract

This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on T), which is well-known as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime [26]. We show in this paper some well-posedness results, mainly the global well-posedness in L 2 (T). The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works [69,60], which weakens the non-trivial resonance in the cubic interactions (a kind of smoothing effect) for the local result, and the global well-posedness result immediately follows from L 2 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in H s (T), s > 0, due to the lack of L 4 -Strichartz estimate for arbitrary L 2 data, a slight modification, thus, is needed to attain the local well-posedness in L 2 (T). This is the first low regularity (global) well-posedness result for the periodic modified Kwahara equation, as far as we know. A direct interpolation argument ensures the unconditional uniqueness in H s (T), s > 1 2 , and as a byproduct, we show the weak ill-posedness below H 1 2 (T), in the sense that the flow map fails to be uniformly continuous.2010 Mathematics Subject Classification. Primary 35Q53, 76B15, Secondary 35G25. 1 2 (T)). However, we do not take their argument for the proof of Theorem 1.5 in order to avoid abusing the normal form mechanism.Remark 1.6. Theorems 1.4 and 1.5 will be improved in the forthcoming work by developing the argument inspired by, for instance, Okamoto [64] and Molinet, Pilod and Vento [57], respectively. 3 The function space X s,b T is the time localization of the standard X s,b introduced by Bourgain [6]

On the stability of periodic traveling waves for the modified Kawahara equation

“…The Cauchy problem associated to the evolution equation (1.1) is locally well-posed in the energy space H 2 per ([0, L]) according with the recent development in[10]. The global well-posedness in the same space can be obtained by combining the local theory with the Gagliardo-Nirenberg inequality.…”

Local well‐posedness for a type of periodic fifth‐order Korteweg–de Vries equations without nonlinear dispersive term

“…x , is a contraction map. Then, applying the standard contraction mapping principle, we immediately get that Equation (1.6) admits a unique local solution as u 0 ∈ H s (T) with s ⩾ 5 8 . Lipschitz continuity of the solution S T ∶ H s (T) → B R,s map is implied by Equation (4.3).…”

“…One may refer to other studies 2‐4 for more details on fifth‐order KdV equations with or without dispersive terms. And one may also refer the introduction in Kwak 5 for the differences of shallow‐water, KdV equation, and Kawahara equation.…”

“…Theorem 1.1 (Local well-posedness). Let s ≥ 5 8 . For any u 0 ∈ H s (T), there exists T = T(||u 0 || H s ) > 0 such that Equation (1.6) admits a unique solution u satisfying u ∈ C([−T, T]; H s (T)).…”

The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation

“…Kwak [28] discussed equation (4) (posed on T) under the condition α = −1, µ = − µ 3 , k = 3, i.e., u t −…”