Xuanxuan Zhao | Searcharxiv

arXiv2026

Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations

We construct nonzero compactly supported stationary distributional solutions $u\in L^2(\mathbb{R}^3;\mathbb{R}^3)$ of the unforced incompressible Navier--Stokes equations, together with compactly supported pressures in $L^1$, with both norms arbitrarily small. The construction addresses the three-dimensional finite-energy endpoint at which the usual single-scale intermittent Mikado mechanism loses its algebraic gain against the Laplacian. The main new ingredient is a logarithmic Mikado profile, built from a truncated two-dimensional harmonic dipole and spread over logarithmically many transverse scales. The same perturbation yields a localized $h$-principle: nonzero stationary solutions are strongly dense in the localized $L^p$ classes for every $1\leq p<2$ and in $H^{-1}$, and weakly dense at $p=2$, while strong $L^2$ density fails because of an exact isotropic quadratic-moment constraint. We also prescribe arbitrary positive $L^2$ norms, periodize the construction to $\mathbb{T}^3$, and obtain a stationary-versus-Leray nonuniqueness mechanism for the evolutionary equations.

math.AP↗

arXiv2025

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

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.

math.AP↗