Ill-posedness for the Cauchy problem of the Camassa-Holm equation in [formula omitted

Abstract

For the famous Camassa-Holm equation, the well-posedness in Bp,11+1p(R) with p∈[1,∞) and the ill-posedness in Bp,r1+1p(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∞,11(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∞,11(R). Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critical Besov spaces Bp,11+1p(R) with p∈[1,∞] have been completed. Finally, since the norm inflation occurs by choosing an special initial data u0∈B∞,11(R) but u0x2∉B∞,10(R) (an example implies B∞,10(R) is not a Banach algebra), we then prove that this condition is necessary. That is, if u0x2∈B∞,10(R) holds, then the Camassa-Holm equation has a unique solution u(t,x)∈CT(B∞,11(R))∩CT1(B∞,10(R)) and the norm inflation will not occur.