Propagation of logarithmic regularity and inviscid limit for the 2D Euler equations

Gennaro Ciampa; Gianluca Crippa; Stefano Spirito; Gennaro Ciampa; Gianluca Crippa; Stefano Spirito

doi:10.3934/mine.2024020

Mathematics in Engineering

2024, Volume 6, Issue 4: 494-509. doi: 10.3934/mine.2024020

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Propagation of logarithmic regularity and inviscid limit for the 2D Euler equations

1.
DISIM–Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi dell'Aquila, Via Vetoio, L'Aquila 67100, Italy
2.
Departement Mathematik Und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland

Received: 12 February 2024 Revised: 24 June 2024 Accepted: 25 June 2024 Published: 12 July 2024

The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with $ L^p $ initial vorticity, provided that $ p\geq 4 $. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order $ |\log\nu|^{-\alpha/2} $ with $ \alpha > 0 $.
- incompressible Euler equations,
- vorticity,
- propagation of regularity,
- modulus of continuity,
- inviscid limit
Citation: Gennaro Ciampa, Gianluca Crippa, Stefano Spirito. Propagation of logarithmic regularity and inviscid limit for the 2D Euler equations[J]. Mathematics in Engineering, 2024, 6(4): 494-509. doi: 10.3934/mine.2024020

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Abstract

The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with $ L^p $ initial vorticity, provided that $ p\geq 4 $. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order $ |\log\nu|^{-\alpha/2} $ with $ \alpha > 0 $.

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