Well-posedness and Continuity Properties of the Fornberg_Whitham equation in Besov space B¹_∞,1(\R)

2024/02/17 by Guorong Qu, Xing Wu, Qu, Guorong +3
Mathematics · #35L30 #35Q35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2402.11247

openalex publication_date 2024/02/17 · openalex created_date 2024/02/22 · openalex updated_date 2026/07/28

Abstract

For the Fornberg-Whitham equation, the local well-posedness in the critical Besov space B_{p, 1}^1+(1)/(p)(ℝ) with 1≤ p <∞ has been studied in (Guo, Nonlinear Anal. RWA., 2023). However, for the endpoint case p=∞, whether it is locally well-posed or ill-posed in B_{∞, 1}¹(ℝ) is still unknown. In this paper, we prove that the Fornberg-Whitham equation is well-posed in the critical Besov space B_{∞, 1}¹(ℝ) with solutions depending continuously on initial data, which is different from that of the Camassa-Holm equation (Guo et al., J. Differ. Equ., 2022). In addition, we show that this dependence is sharp by showing that the solution map is not uniformly continuous on the initial data.

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