Well-posedness for a higher order water wave model on modulation spaces

2024/08/31 by Xavier Carvajal, Carvajal, Xavier, Mahendra Panthee +1
Earth and Planetary Sciences · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.2409.00467

openalex publication_date 2024/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Considered in this work is the initial value problem (IVP) associated to a higher order water wave model \begincases η_t+η_x-γ₁ η_xxt+γ₂η_xxx+δ₁ η_xxxxt+δ₂η_xxxxx+(3)/(2)ηη_x+γ(η²)_xxx-(7)/(48)(η_x²)_x-(1)/(8)(η³)_x=0,
η(x,0) = η₀(x). \endcases The main interest is in addressing the well-posedness issues of the IVP when the given initial data are considered in the modulation space M_s^2,p(ℝ) or the L^p-based Sobolev spaces H^s,p(ℝ), 1≤ p<∞. We derive some multilinear estimates in these spaces and prove that the above IVP is locally well-posed for data in M_s^2,p(ℝ) whenever s>1 and p≥ 1, and in H^s,p(ℝ) whenever p∈ [1,∞) and s≥ max\ \frac1p+\frac12, 1 \. We also use a combination of high-low frequency technique and an \em a priori estimate, and prove that the local solution with data in the modulation spaces M_s^2,p(ℝ) can be extended globally to the time interval [0, T] for any given T≫1 if 1≤ \frac32-\frac1p

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