An Onsager-type Theorem for General 2D Active Scalar Equations

TL;DR

This paper proves an Onsager-type theorem for 2D active scalar equations with specific Fourier multiplier symbols, establishing the existence of nonconservative weak solutions below a certain regularity threshold, thus advancing the understanding of energy conservation in fluid dynamics.

Contribution

It introduces a sharp threshold for energy conservation in 2D active scalar equations with odd homogeneous symbols, extending Onsager's conjecture to a broader class of equations.

Findings

Existence of nontrivial weak solutions that do not conserve Hamiltonian below a regularity threshold.

Sharp regularity threshold for energy conservation in terms of the symbol's homogeneity.

Extension of results to 2D and 3D even active scalar equations in the appendix.

Abstract

This paper concerns the Onsager-type problem for general 2-dimensional active scalar equations of the form: $\partial_t \theta+u\cdot\nabla \theta= 0$, with $u=T[\theta]$ being a divergence-free velocity field and $T$ being a Fourier multiplier operator with symbol $m$. It is shown that if $m$ is a odd and homogeneous symbol of order $\delta$: $m(\lambda\xi)=\lambda^{\delta} m(\xi)$, where $\lambda>0, -1\le\delta\le0$, then there exists a nontrivial temporally compact-supported weak solution $\theta\in C_t^0 C_x^{\frac{2\delta}{3}-}$, which fails to conserve Hamiltonian. This result is sharp since all weak solutions of class $C_t^0C_x^{\frac{2\delta}{3}+}$ will necessarily conserve the Hamiltonian (which is proved by P. Isett and A. Ma in arXiv:2403.08279, 2024.) and thus resolves the flexible part of the generalized Onsager conjecture for general 2D odd active scalar equations. Also, in the appendix, analogous results have been obtained for general 2D and 3D even active scalar equations. The proof is achieved by using convex integration scheme at the level $v=-\nabla^{\perp}\cdot\theta$ together with a Newton scheme recently introduced by V. Giri and R. O. Radu (2D Onsager conjecture: a Newton-Nash iteration. Invent. math. (2024).). Moreover, a novel algebraic lemma and sharp estimates for some complicated trilinear Fourier multipliers are established to overcome the difficulties caused by the generality of the equations.