A note on the solution map for the periodic Camassa–Holm equation

Tang, Hao; Zhao, Yongye; Liu, Zhengrong

doi:10.1080/00036811.2013.847923

Cited by 32 publications

(26 citation statements)

References 41 publications

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“…Recently, Tang et al [35] studied the continuous properties of the solution map for the periodic CH equation in Besove spaces B s p,r with 1 ≤ p, r ≤ ∞ and s > max{1 + 1/ p, 3/2}. In this paper, we aim at establishing the similar results in critical Besove space B 3/2 2,r , r = 1, ∞.…”

Section: Introductionmentioning

confidence: 60%

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The dependences on initial data for the b-family equation in critical Besov space

2014

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This paper is concerned with the periodic boundary problem for the b-family equation. At first, we use a different method to prove the local well-posedness in the critical Besov space B 3/2 2,1 . Then we show that if a weaker B q p,r -topology is used, the solution map becomes Hölder continuous. Moreover, we show that the dependence on initial data is optimal in B 3/2 2,1 in the sense that the solution map is continuous but not uniformly continuous. Finally, we obtain the periodic peaked solutions to the b-family equation and apply them to obtain the ill-posedness in B 3/2 2,∞ .

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Section: Introductionmentioning

confidence: 60%

“…with 1 ≤ p, r ≤ ∞ and s > max{ 3 2 , 1 + 1 p } (see [19,35]), can not be obtained for p = 2, r = 1 and s = 3/2. Consequently, we cannot directly obtain the convergence in B 1/2 2,1 .…”

Section: Introductionmentioning

confidence: 97%

The dependences on initial data for the b-family equation in critical Besov space

2014

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“…We note that the issue of nonuniform dependence on initial data has been the subject of many papers (see, e.g., [32,43]). In this paper, it is to be regretted that we can not find a feasible method to study the uniform continuity of the solution map 0 → defined by problem (1)-(2) for general ( ) and ( ).…”

Section: Introduction and Resultsmentioning

confidence: 99%

The Periodic Boundary Value Problem for a Quasilinear Evolution Equation in Besov Spaces

Liu

Xiang

2015

Journal of Function Spaces

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“…Since then different techniques have been successfully applied to various nonlinear dispersive and integrable equations, see [2,45,47] for example. Particularly, for the incompressible Euler equation, we refer to [35,60], and for CH type equations, we refer to [33,34,[61][62][63] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On a Stochastic Camassa–Holm Type Equation with Higher Order Nonlinearities

Rohde

Tang

2020

J Dyn Diff Equat

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The subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces H s with s > 3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.

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A note on the solution map for the periodic Camassa–Holm equation

Cited by 32 publications

References 41 publications

The dependences on initial data for the b-family equation in critical Besov space

The dependences on initial data for the b-family equation in critical Besov space

The Periodic Boundary Value Problem for a Quasilinear Evolution Equation in Besov Spaces

On a Stochastic Camassa–Holm Type Equation with Higher Order Nonlinearities

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