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Admissible solutions of the 2D Onsager's conjecture

Lili Du, Xinliang Li, Weikui Ye

TL;DR

The work addresses Onsager’s conjecture in two dimensions by constructing dissipative weak solutions to the 2D incompressible Euler equations with Hölder regularity $C^{\frac{1}{3}-}$ and with a prescribed energy profile $e(t)$. The authors develop a novel convex integration framework that combines traveling-wave temporal oscillations, a gluing procedure, and a multiple-iteration energy-corrector scheme (Newton–Nash plus Picard) to produce energy-dissipating solutions below the Onsager threshold. They establish the density of initial data that generate such dissipative solutions in Besov spaces $B^{\gamma}_{\infty,r}$ (for $0<\gamma<\frac{1}{3}$ and $r<\infty$), implying abundant non-uniqueness in 2D. The results extend the Onsager program to 2D with strict energy dissipation and provide a robust mechanism potentially applicable to other fluid models with similar dissipative behavior.

Abstract

We show that for any $γ< \frac{1}{3}$ there exist Hölder continuous weak solutions $v \in C^γ([0,T] \times \mathbb{T}^2)$ of the two-dimensional incompressible Euler equations that strictly dissipate the total kinetic energy, improving upon the elegant work of Giri and Radu [Invent. Math., 238 (2), 2024]. Furthermore, we prove that the initial data of these \textit{admissible} solutions are dense in $B^γ_{\infty,r<\infty}$. Our approach introduces a new class of traveling waves, refining the traditional temporal oscillation function first proposed by Cheskidov and Luo [Invent. Math., 229(3), 2022], to effectively modulate energy on any time intervals. Additionally, we propose a novel ``multiple iteration scheme'' combining Newton-Nash iteration with a Picard-type iteration to generate an energy corrector for controlling total kinetic energy during the perturbation step. This framework enables us to construct dissipative weak solutions below the Onsager critical exponent in any dimension $d \geq 2$.

Admissible solutions of the 2D Onsager's conjecture

TL;DR

The work addresses Onsager’s conjecture in two dimensions by constructing dissipative weak solutions to the 2D incompressible Euler equations with Hölder regularity and with a prescribed energy profile . The authors develop a novel convex integration framework that combines traveling-wave temporal oscillations, a gluing procedure, and a multiple-iteration energy-corrector scheme (Newton–Nash plus Picard) to produce energy-dissipating solutions below the Onsager threshold. They establish the density of initial data that generate such dissipative solutions in Besov spaces (for and ), implying abundant non-uniqueness in 2D. The results extend the Onsager program to 2D with strict energy dissipation and provide a robust mechanism potentially applicable to other fluid models with similar dissipative behavior.

Abstract

We show that for any there exist Hölder continuous weak solutions of the two-dimensional incompressible Euler equations that strictly dissipate the total kinetic energy, improving upon the elegant work of Giri and Radu [Invent. Math., 238 (2), 2024]. Furthermore, we prove that the initial data of these \textit{admissible} solutions are dense in . Our approach introduces a new class of traveling waves, refining the traditional temporal oscillation function first proposed by Cheskidov and Luo [Invent. Math., 229(3), 2022], to effectively modulate energy on any time intervals. Additionally, we propose a novel ``multiple iteration scheme'' combining Newton-Nash iteration with a Picard-type iteration to generate an energy corrector for controlling total kinetic energy during the perturbation step. This framework enables us to construct dissipative weak solutions below the Onsager critical exponent in any dimension .

Paper Structure

This paper contains 18 sections, 24 theorems, 153 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Overview of the approach and comments
  3. Overview of the approach
  4. Comments on main results
  5. The construction of dissipative weak solutions
  6. Iteration proposition
  7. Mollification step and gluing procedure
  8. Mollification step
  9. Classical exact flows
  10. Gluing procedure
  11. Traveling waves
  12. Multiple iteration scheme
  13. Iterative model: Newton-Nash iteration
  14. A Picard-type iteration
  15. Proof of Proposition \ref{['p:main-prop']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $e(t): [0,1] \to \mathbb{R}$ be a positive smooth function and $\mathbb{T}^2$ represent the two-dimensional torus. Then there exists a continuous vector field $v \in C^{\frac{1}{3}-}(\mathbb{T}^2 \times [0,1], \mathbb{R}^2)$ that solves the 2D incompressible Euler equations in the sense of distributions and such that

Figures (4)

  • Figure 2: Multiple iteration scheme
  • Figure : (a) The supports of traveling waves $g_{k,i,n}(t,x_1)$. For any two distinct values of $n = 0, 1, 2, \cdots, L-1$, the supports of the corresponding traveling waves are pairwise disjoint sets.
  • Figure : (a) The supports of traveling waves $g_{k,i,n}(t,x_1)$. For any two distinct values of $n = 0, 1, 2, \cdots, L-1$, the supports of the corresponding traveling waves are pairwise disjoint sets.
  • Figure : (b) The supports of the temporal oscillation function $g_k(t)$. As in the construction of VGR23, no energy is added to the solution during the time intervals between two colored regions.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Proposition 3.1: Main iteration proposition
  • Proposition 3.2
  • Proposition 3.3: Estimates for mollified functions
  • Proposition 3.4: Local well-posedness
  • Proposition 3.5: Estimates for $(\bar{v}_q, \mathring{\bar{R}}_q)$
  • ...and 27 more