[Paper Review] On the rough solutions of 3D compressible Euler equations: an alternative proof

Huali Zhang, Lars Andersson|arXiv (Cornell University)|Apr 26, 2021

Navier-Stokes equation solutions27 references4 citations

TL;DR

This paper presents an alternative, simplified proof of the local well-posedness of the 3D compressible Euler equations using Smith-Tataru's wave-transport approach. It establishes existence, uniqueness, and stability of solutions for initial data with velocity and density in $H^s$ and specific vorticity in $H^{s_0}$, where $2 < s_0 < s$, improving on prior results by Q. Wang with a more streamlined framework based on microlocal analysis and Strichartz-type estimates.

ABSTRACT

The well-posedness of Cauchy problem of 3D compressible Euler equations is studied. By using Smith-Tataru's approach \cite{ST}, we prove the local existence, uniqueness and stability of solutions for Cauchy problem of 3D compressible Euler equations, where the initial data of velocity, density, specific vorticity $v, ρ\in H^s, \varpi \in H^{s_0} (2

Motivation & Objective

• To provide a simplified and alternative proof of the local well-posedness of the 3D compressible Euler equations under low regularity assumptions.
• To extend the applicability of Smith-Tataru's wave-transport framework to the compressible Euler system with non-zero vorticity.
• To establish existence, uniqueness, and stability of solutions when initial velocity and density are in $H^s$ and specific vorticity in $H^{s_0}$ with $2 < s_0 < s$.
• To improve upon Q. Wang's result [42] by offering a more streamlined and accessible proof using microlocal and space-time estimates.

Proposed method

• Adopting Smith-Tataru's approach based on wave packet decompositions and space-time $L^p$ estimates for quasilinear wave equations.
• Utilizing a wave-transport system formulation to decouple the dynamics of vorticity and density from the main fluid evolution.
• Applying microlocal techniques to control the nonlinear interactions in the system via frequency-localized energy estimates.
• Employing Strichartz-type estimates and refined space-time norms to handle low regularity data in Sobolev spaces.
• Matching initial data via superposition of normalized wave packets to ensure compatibility with the required regularity.
• Using the vector-field method and curvature decomposition to control error terms arising from nonlinearity and metric dependence.

Experimental results

Research questions

• RQ1 Can the well-posedness of the 3D compressible Euler equations be established with lower regularity assumptions on initial data using an alternative method?
• RQ2 How can Smith-Tataru's wave-transport framework be adapted to handle the compressible Euler system with non-zero vorticity?
• RQ3 What is the minimal regularity threshold for local existence, uniqueness, and stability of solutions in the compressible case?
• RQ4 Can the proof of Q. Wang [42] be simplified and made more transparent using modern microlocal analysis tools?

Key findings

• The Cauchy problem for 3D compressible Euler equations is locally well-posed for initial data with velocity $v_0$, density $ ho_0$ in $H^s$, and specific vorticity $ar{\omega}_0$ in $H^{s_0}$, where $2 < s_0 < s$.
• The proof establishes existence, uniqueness, and stability of solutions under the same regularity assumptions as in Q. Wang [42], but through a more streamlined and conceptually clearer framework.
• The method relies on wave packet decompositions and space-time estimates, particularly Strichartz-type estimates, to control the nonlinear terms in the system.
• The authors achieve the well-posedness result using only $H^s$ regularity for velocity and density, and slightly lower $H^{s_0}$ regularity for vorticity, avoiding Hölder-type assumptions on curl of vorticity.
• The proof is an alternative to Wang's [42] and simplifies the analysis by focusing on the wave-transport structure and microlocal techniques.