Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions

2015/12/04 by Isao Kato, Kato, Isao
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1512.01510

openalex publication_date 2015/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension d ≥ 4 with radial or non-radial initial datum (u, ∂_t u, n, ∂_t n)|_t=0∈ H^s+1(ℝ^d) × H^s(ℝ^d) × H^s(ℝ^d) × H^s-1(ℝ^d). The critical value of s is s=s_c=d/2-2. If the initial datum is radial, then we prove the small data global well-posedness and scattering at the critical space in d ≥ 4 by applying the radial Strichartz estimates and U², V² type spaces. On the other hand, if the initial datum is non-radial, then we prove the local well-posedness at s=1/4 when d=4 and s=s_c+1/(d+1) when d ≥ 5 by applying the U², V² type spaces.

Citations

• Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations

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