1. Ill-posedness for the Cauchy problem of the Camassa-Holm equation in [formula omitted].
- Author
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Guo, Yingying, Ye, Weikui, and Yin, Zhaoyang
- Subjects
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BESOV spaces , *BANACH algebras , *EQUATIONS , *PROBLEM solving - Abstract
For the famous Camassa-Holm equation, the well-posedness in B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞) and the ill-posedness in B p , r 1 + 1 p (R) with p ∈ [ 1 , ∞ ] , r ∈ (1 , ∞ ] had been studied in [13,14,16,23] , that is to say, it only left an open problem in the critical case B ∞ , 1 1 (R) proposed by Danchin in [13,14]. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in B ∞ , 1 1 (R). Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critical Besov spaces B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞ ] have been completed. Finally, since the norm inflation occurs by choosing an special initial data u 0 ∈ B ∞ , 1 1 (R) but u 0 x 2 ∉ B ∞ , 1 0 (R) (an example implies B ∞ , 1 0 (R) is not a Banach algebra), we then prove that this condition is necessary. That is, if u 0 x 2 ∈ B ∞ , 1 0 (R) holds, then the Camassa-Holm equation has a unique solution u (t , x) ∈ C T (B ∞ , 1 1 (R)) ∩ C T 1 (B ∞ , 1 0 (R)) and the norm inflation will not occur. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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