Small scale creation of the Lagrangian flow in 2d perfect fluids

2024/01/12 by Ayman Rimah Said, Said, Ayman Rimah
Engineering · Mathematics · #35Q31 #35S99 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2401.06476

openalex publication_date 2024/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this paper we prove that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time at least like t^(1)/(3). This initial data dependent norm quantifies the exact L² decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which we then show to be the quantitative phenomenon behind the Lyapunov construction by Shnirelman.

Related

• Small scale creation for solutions of the incompressible two dimensional Euler equation
• Lagrangian solutions to the 2D euler system with L¹ vorticity and infinite energy
• Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow
• Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations
• Ill-posedness for the incompressible Euler equations in critical Sobolev spaces