An Onsager-type Theorem for General 2D Active Scalar Equations
Xuanxuan Zhao
TL;DR
The paper resolves the flexible part of the Onsager-type conjecture for general 2D odd active scalar equations by constructing temporally compact-supported weak solutions with Hölder regularity $\Lambda^{-1}\theta\in C^{\gamma}$ for $\gamma$ in $\left[\frac{1+\delta}{2},1+\frac{2\delta}{3}\right)$, where the multiplier has homogeneity degree $\delta\in[-1,0]$. It develops a unified Newton–Nash convex integration framework operating at the potential level, aided by a new algebraic lemma and sharp multi-linear Fourier multiplier estimates, to handle the general multiplier structure $m$ and the corresponding anti-divergence operator $\mathcal{R}(m)$. This approach unifies and extends previous 2D Euler and SQG Onsager-type results, providing sharp thresholds for Hamiltonian conservation and establishing an effective method applicable to a broad class of 2D ASEs; the appendix also extends analogous results to even multipliers and to general 2D/3D settings. The work thereby advances the flexible part of the Onsager conjecture for a wide family of active scalar models and offers a robust framework potentially extensible to higher dimensions and other nonlocal PDE systems.
Abstract
This paper concerns the Onsager-type problem for general 2-dimensional active scalar equations of the form: $\partial_t θ+u\cdot\nabla θ= 0$, with $u=T[θ]$ 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 $δ$: $m(λξ)=λ^δ m(ξ)$, where $λ>0, -1\leδ\le0$, then there exists a nontrivial temporally compact-supported weak solution $θ\in C_t^0 C_x^{\frac{2δ}{3}-}$, which fails to conserve Hamiltonian. This result is sharp since all weak solutions of class $C_t^0C_x^{\frac{2δ}{3}+}$ 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 $v=-\nabla^{\perp}\cdotθ$ 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.