Local well-posedness for a generalized sixth-order Boussinesq equation

2024/03/07 by Long Zhong, Zhong, Long, Shenghao Li +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2403.04295

openalex publication_date 2024/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A formally second order correct Boussinesq-type equation that describes unidirectional shallow water waves is derived, u_tt - u_xx - u_xxxx - u_xxxxxx - (u²)_xx - (u²)_xxxx - (uu_xx)_xx - (u³)_xx = 0. Such equation is analogous to original Boussinesq equation but with higher order approximation which may ensure a more accuracy description on a long time scale. Moreover, through a rigorous derivation from Boussiensq systems, it has redeemed all the non-linear terms neglected in the sixth order Boussinesq equation (SOBE), u_tt - u_xx - u_xxxx - u_xxxxxx - (u²)_xx = 0. The Cauchy problem for this generalized SOBE is then considered under the Bourgain space, X^s,b, framework. The multi-linear estimates for (u²)_xx, (u²)_xxxx, (uu_xx)_xx and (u³)_xx are given, the local wellposedness of the gSOBE is established for s>(1)/(2).

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