[Paper Review] Transporting microstructure and dissipative Euler flows

Tristan Buckmaster, Camillo De Lellis+1 more|arXiv (Cornell University)|Feb 12, 2013

Navier-Stokes equation solutions19 references21 citations

TL;DR

This paper presents a streamlined proof of the existence of periodic, dissipative solutions to the 3D incompressible Euler equations in the Hölder space $ C^{1/5 - \varepsilon} $, achieving the improved regularity threshold established by Isett. Using a modified convex integration scheme with superposed perturbed Beltrami flows across multiple scales, the authors resolve the key obstruction from linear transport errors in fast-slow flow interactions, thereby confirming Onsager's conjecture for Hölder exponents below $ 1/3 $.

ABSTRACT

Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces (arXiv:1202.1751 and arXiv:1205.3626 (2012)). The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper P. Isett (arXiv:1211.4065) has improved upon our methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent - albeit below the one conjectured by Onsager. In this paper we give a shorter proof of Isett's final result, adhering more to the original scheme and introducing some new devices. More precisely we show that for any positive $ε$ there exist periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy and belong to the Hölder class $C^{\frac{1}{5}-ε}$.

Motivation & Objective

• To provide a shorter, more streamlined proof of Isett's result on dissipative Euler solutions with Hölder regularity $ C^{1/5 - \varepsilon} $.
• To isolate and clarify the novel techniques introduced by Isett that overcome the transport error obstruction in the convex integration scheme.
• To maintain consistency with the original iterative framework of De Lellis and Székelyhidi while improving regularity bounds.
• To demonstrate that the improved Hölder exponent $ 1/5 - \varepsilon $ is achievable in the context of prescribed energy profiles $ e(t) $.
• To extend the method to produce solutions with compactly supported time dependence, though with less control over energy shape.

Proposed method

• Employing an iterative convex integration scheme to construct solutions to the Euler-Reynolds system, balancing Reynolds stress errors at each step.
• Using superposed perturbations based on fast periodic Beltrami flows to generate microstructure across multiple scales.
• Introducing a refined analysis of the commutator $ [b, \mathcal{R}] $ to control transport errors arising from slow velocity fields acting on fast oscillatory flows.
• Applying a hierarchy of commutator estimates via Propositions E.1 and D.1 to bound error terms in Hölder norms, with careful scaling of amplitude $ \delta_q^{1/2} $ and frequency $ \lambda_q $.
• Utilizing the identity $ \mathscr{S}(bae^{i\lambda k\cdot x}) - b\mathscr{S}(ae^{i\lambda k\cdot x}) = \frac{aA(b)}{\lambda^2}e^{i\lambda k\cdot x} $ to decompose and estimate non-local error terms.
• Applying interpolation and Leibniz rule identities to reorder and bound higher-order terms in the commutator expansion, ensuring decay in $ \lambda^{-N} $.

Experimental results

Research questions

• RQ1 Can the Hölder regularity threshold for dissipative solutions of the 3D incompressible Euler equations be improved beyond $ 1/10 - \varepsilon $ using convex integration?
• RQ2 What specific modifications to the iterative scheme are required to control the transport error in fast-slow flow interactions?
• RQ3 How can Isett’s improved Hölder exponent of $ 1/5 - \varepsilon $ be re-derived with a more direct and transparent proof?
• RQ4 To what extent can the original convex integration framework be preserved while achieving better regularity?
• RQ5 What is the role of commutator estimates in resolving the transport obstruction in microstructure-based constructions?

Key findings

• The paper establishes the existence of a continuous vector field $ v \in C^{1/5 - \varepsilon}(\mathbb{T}^3 \times [0,1], \mathbb{R}^3) $ solving the incompressible Euler equations in the sense of distributions.
• The constructed solution dissipates total kinetic energy, satisfying $ \int_{\mathbb{T}^3} |v(x,t)|^2 \, dx = e(t) $ for any given smooth positive function $ e(t) $.
• The velocity field belongs to the Hölder class $ C^{1/5 - \varepsilon} $, achieving the improved regularity threshold previously obtained by Isett.
• The pressure field $ p $ is shown to be in $ C^{2/5 - 2\varepsilon} $, consistent with the regularity of the velocity field.
• The proof provides a more direct and transparent derivation of Isett’s result, emphasizing the core geometric and analytic mechanisms behind the improved Hölder exponent.
• The method can be adapted to produce nontrivial solutions with compactly supported time dependence, though the energy profile is not precisely controlled in this variant.