Intermittent singular solutions of the stationary 2D Navier-Stokes equations in sharp Sobolev spaces

2025/06/01 by Ashkarian, Estepan, Bhargava, Ataleshvara, Gismondi, Nicholas +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.00841

Abstract

In this paper we construct non-trivial solutions to the stationary Navier-Stokes equations on the two dimensional torus which lie in \bigcap_{ε∈ (0,1)} L^2-ε(\mathbbT²) ∩ H^-ε(\mathbbT²). Due to the fact that our solutions are not square integrable, we must redefine the notion of solution. Our result gives a sharp extension of recent work of Lemarié-Rieusset, who proved a similar result in the space H^-1 ∩ BMO^-1. The main new ingredient is the incorporation of intermittency into the construction of the solutions.

Citations

• Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data
• Sharp non-uniqueness for the Navier-Stokes equations in R³
• Highly singular (frequentially sparse) steady solutions for the 2D Navier-Stokes equations on the torus
• Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
• Weak solutions of ideal MHD which do not conserve magnetic helicity
• Convex integration and phenomenologies in turbulence
• Wild solutions of the Navier-Stokes equations whose singular sets in\n time have Hausdorff dimension strictly less than 1
• Nonuniqueness of weak solutions to the Navier-Stokes equation
• Nonuniqueness of weak solutions to the SQG equation
• Nonuniqueness of weak solutions of the nonlinear Schroedinger equation

Cited by

• Intermittent solutions of the stationary 2D surface quasi-geostrophic equation

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