The failure of Hölder regularity of solutions for the Camassa--Holm type equation in Besov spaces

2024/01/20 by Li, Jinlu, Yu, Yanghai, Zhu, Weipeng
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.11097

Abstract

It is proved that if u₀∈ B^s_p,r with s>1+\frac1p, (p,r)∈[1,+∞]×[1,+∞) or s=1+\frac1p, (p,r)∈[1,+∞)× \1\, the solution of the Camassa--Holm equation belongs to C([0,T];B^s_p,r). In the paper, we show that the continuity of the solution can not be improved to the Hölder continuity. Precisely speaking, the solution of the Camassa--Holm equation belongs to C([0,T];B^s_p,r) but not to C^α([0,T];B^s_p,r) with any α∈(0,1).

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