[Paper Review] The Onsager conjecture in 2D: a Newton-Nash iteration

Vikram Giri, Răzvan-Octavian Radu|arXiv (Cornell University)|May 29, 2023

Navier-Stokes equation solutions4 citations

TL;DR

This paper resolves the flexible part of the Onsager conjecture in two dimensions by constructing nontrivial weak solutions to the 2D incompressible Euler equations with compact temporal support and Hölder regularity $ u \in C^{\gamma} $ for any $ \gamma < 1/3 $. The method combines a Newton-Nash iteration, using time-oscillatory perturbations derived from the linearized Euler equations to overcome non-interaction challenges in 2D, thereby achieving Onsager-critical regularity without relying on Mikado flows or spatial intermittency.

ABSTRACT

For any $γ&lt;1/3$, we construct a nontrivial weak solution $u$ to the two-dimensional, incompressible Euler equations, which has compact support in time and satisfies $u\in C^γ(\mathbb R_t imes \mathbb T^2_x)$. In particular, the constructed solution does not conserve energy and, thus, settles the flexible part of the Onsager conjecture in two dimensions. The proof involves combining the Nash iteration technique with a new linear Newton iteration.

Motivation & Objective

• To close the rigidity/flexibility gap in the Onsager conjecture for two-dimensional incompressible Euler flows.
• To construct non-conservative weak solutions with Hölder regularity $ \gamma < 1/3 $ and compact support in time in 2D.
• To overcome the topological obstruction in 2D that prevents the use of Mikado flows due to line intersections.
• To develop a Newton-Nash iteration framework that uses time-oscillatory correctors to decouple error-erasing mechanisms.
• To provide a new proof of the flexible Onsager conjecture that avoids intermittency and Mikado flow constructions.

Proposed method

• The authors employ a Nash-type iterative scheme to construct weak solutions by successively adding highly oscillatory perturbations that erase errors in the Euler equations.
• A novel Newton iteration is introduced, where perturbations are defined as solutions to the linearized Euler equations with a temporally oscillatory forcing term.
• The time-oscillatory nature of the perturbations ensures non-interaction between different directional error components, circumventing the 2D line intersection problem.
• The method relies on Calderón-Zygmund type operators to handle the pressure and divergence-free constraints in the iterative correction process.
• The iteration is constructed in a Lagrangian framework using flows of the velocity field, ensuring regularity propagation through the sequence of approximations.
• Global existence is established by patching local solutions over time intervals using uniqueness and continuity arguments.

Experimental results

Research questions

• RQ1 Can non-conservative weak solutions to the 2D incompressible Euler equations be constructed with Hölder regularity $ \gamma < 1/3 $?
• RQ2 How can the non-interaction requirement in the Nash iteration be achieved in 2D, where Mikado flows fail due to line intersections?
• RQ3 Can a Newton-type linearization be used to design time-oscillatory perturbations that decouple error correction in 2D?
• RQ4 Does the absence of spatial intermittency or Mikado flows preclude achieving Onsager-critical regularity in 2D?
• RQ5 Is it possible to construct compactly supported, non-conservative solutions in 2D using only time-oscillatory corrections in a Newton-Nash framework?

Key findings

• The paper constructs a nontrivial weak solution $ u \in C^{\gamma}(\mathbb{R}_t \times \mathbb{T}^2_x) $ to the 2D incompressible Euler equations for any $ \gamma < 1/3 $ with compact support in time.
• The solution does not conserve kinetic energy, thereby confirming the flexible part of the Onsager conjecture in two dimensions.
• The method avoids the use of Mikado flows and spatial intermittency, providing a third independent proof of the flexible Onsager conjecture.
• The Newton-Nash iteration framework successfully decouples error correction in 2D by using time-oscillatory perturbations derived from the linearized Euler equations.
• The construction is robust in any dimension $ d \geq 2 $, as the method does not rely on spatial dimension-specific structures.
• The solution sequence converges in $ C^1_t C^{N-1+\alpha}_x $ and $ C_t C^{N+\alpha}_x $, ensuring the existence of a global, regular limit solution.