Improved Local Well-Posedness in Sobolev Spaces for Two-Dimensional Compressible Euler Equations

2025/12/09 by Zhang, Huali
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions

paper · doi:10.48550/arxiv.2512.08581

openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28

Abstract

We establish the local existence and uniqueness of solutions to the two-dimensional compressible Euler equations with initial velocity \bv₀, logarithmic density ρ₀, and specific vorticity \(w₀\), which satisfy (\bv₀, ρ₀, w₀, ∇ w₀)∈ H^\frac74+(ℝ²)× H^\frac74+(ℝ²) × H^\frac32(ℝ²) × L⁸(ℝ²). The proof applies Smith-Tataru method \citeST and the inherent wave-transport structure of the two-dimensional compressible Euler equations. The key observation is that Strichartz estimates hold when the regularity requirement for vorticity is lower than that for velocity and density, even though the gradient of vorticity appears as a source term in the velocity wave equation. Furthermore, our result presents an improvement of (1)/(4)-order regularity compared to previous results \citeZ1 and \citeZ2.

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