Well‐posedness, blow‐up phenomena and analyticity for a two‐component higher order Camassa–Holm system

Abstract

In this paper, the local well-posedness for the Cauchy problem of a two-component higher-order Camassa-Holm system (2HOCH) is established in Besov spaces(and also in Sobolev spaces × −2 = 2,2 × −2 2,2 with > 5∕2), which improves the corresponding results for higher-order Camassa-Holm in [7,24,25], where the Sobolev index = 3, > 7∕2, ≥ 7∕2 is required, respectively. Then the precise blow-up mechanism and global existence for the strong solutions of 2HOCH are determined in the lowestFinally, the Gevrey regularity and analyticity of the 2HOCH are presented. K E Y W O R D S 2HOCH, analyticity, Gevrey regularity, local well-posedness, wave breaking M S C ( 2 0 1 0 ) 35G25, 35L15, 35Q53, 35Q58This family of so-called -equations possess peakon solution and wave breaking(see [20,27] and references therein). By using Painlevé analysis [15,16], there are only two asymptotically integrable equations within this family: the Camassa-Holm (CH) equation (Eq. (1.2) with = 2, c.f. [4,5]) and the Degasperis-Procesi (DP) equation (Eq. (1.2) with = 3, c.f. [15]). Integrable Mathematische Nachrichten. 2018;291:1595-1619.How to cite this article: Zhou S. Well-posedness, blow-up phenomena and analyticity for a two-component higher order Camassa-Holm system.

Continuity properties of the data-to-solution map for the two-component higher order Camassa–Holm system

“…Applying the transport equation theory combined with the method of the Besov spaces, one may obtain the following local well-posedness result for system (2.1), more details can be seen in [37,38].…”

“…Recently, He and Yin [22], Chen and Zhou [4] established the local well-posedness of (1.1) in Besov spaces. Zhou [38], Zhang and Li [37] investigated the local well-posedness, blow-up criteria and Gevrey regularity of the solutions to (1.1) with σ = 2. When ρ ≡ 0, α = 0 and b = 2, (1.1) reduces to a Camassa-Holm equation with fractional order inertia operator, whose geometrical interpretation and local well-posedness can be seen in [12,13,22], and if we further assume 2 ≤ σ ∈ Z + , (1.1) becomes a higher order Camassa-Holm equation derived as the Euler-Poincaré differential equation on the Bott-Virasoro group with respect to the H σ metric [32].…”

Lipschitz metric for the modified coupled Camassa–Holm system

“…The well‐posedness and blow‐up solutions of periodic case for the system () were discussed by Fu et al [30] Tian et al established the local well‐posedness and blow‐up solutions for the system () in Besov space [31]. Non‐uniform dependence, persistence properties, and more results of the system () also were studied in previous studies [32–34]. Moreover, Tian and Xu researched the attractor of the system () [35].…”