Analysis on continuity of the solution map for the Whitham equation in Besov spaces

Zhengyan Liu and Xinglong Wu

Mathematische Nachrichten, 2024, vol. 297, issue 4, 1451-1467

Abstract: This paper is devoted to studying the continuity of the solution map for the Cauchy problem of the Whitham equation. First, the continuity dependence of solution is established in B2,rs$\mathrm{B}^{s}_{2,r}$ in the sense of Hadamard. Next, by constructing approximate solutions, we show that the data‐to‐solution map is not uniformly continuous in Besov spaces B2,rs$\mathrm{B}^{s}_{2,r}$ (s>32,1≤r≤∞$s>\frac{3}{2}, 1\le r\le \infty$) on the periodic case and on the line. The crucial technical tool used to prove this result is Lemma 5.1, which generalizes the result of Lemma 3 in [23].

Date: 2024
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