arXiv subjects

Lin Lü

Publications and source records attributed to Lin Lü.

4 recordsLinked to original sources

arXiv2026

Derivation of the focusing $\Phi^6_1$ measure in the optimal mass regime from many-body quantum Gibbs states

We derive the focusing $\Phi^6_1$ measure on the torus $\mathbb{T}$ as the high-temperature/mean-field limit of many-body quantum Gibbs states with an attractive three-body interaction. The main difficulty in the focusing setting is to relate the classical mass cutoff to the quantum particle-number cutoff. Our result reaches the optimal mass threshold for the classical field identified by Oh, Sosoe, and Tolomeo (2022), and thereby extends the earlier work of Rout and Sohinger (2025). At the critical threshold, the short-range interaction is allowed to shrink to a Dirac delta function on a logarithmic scale in the temperature parameter. Strictly below the threshold, the same convergence holds with a polynomial dependence on the temperature. Moreover, we establish a quantum-level phase transition at the same mass threshold. The proof develops the variational framework of Lewin, Nam, and Rougerie (2015) in the focusing setting and relies on two new ingredients: a non-factorized trial state construction and a delicate tail estimate for the interacting lower symbol. These allow us to control the localization and relative entropy errors caused by the particle-number cutoff, as well as the contribution of the focusing exponential weight.

math-ph↗

arXiv2025

A proof of Onsager's conjecture for the stochastic 3D Euler equations

This paper investigates the stochastic 3D Euler equations on a periodic domain $\mathbb{T}^3$, driven by a $GG^*$-Wiener process $B$ of trace class: \begin{align*} \mathrm{d} u+\mathrm{div}(u\otimes u)\,\mathrm{d} t+\nabla p\,\mathrm{d}t=\mathrm{d}B, \quad \mathrm{div} u=0. \end{align*} First, for any $\vartheta<1/3$, we construct infinitely many global-in-time probabilistically strong and analytically weak solutions $u\in C([0,\infty),C^{\vartheta}(\mathbb{T}^3,\mathbb{R}^3))$. These solutions dissipate the energy pathwisely up to a stopping time $\mathfrak{t}$, which can be chosen arbitrarily large with high probability, i.e. it holds almost surely \begin{align*} \|u(t\wedge\mathfrak{t})\|_{L^2}^2< \|u(s\wedge\mathfrak{t})\|_{L^2}^2 +2 \int_{s\wedge\mathfrak{t}}^{t\wedge\mathfrak{t}} \big\langle u(r), \mathrm{d} B(r) \big\rangle +\mathrm{Tr}\big(GG^*\big) (t\wedge\mathfrak{t}-s\wedge\mathfrak{t}), \end{align*} for any $0\leq s < t<\infty$. We also provide a brief proof of energy conservation for $\vartheta>1/3$ based on \cite{CET94}, thereby confirming the Onsager theorem for the stochastic 3D Euler equations. Second, let $0<\bar{\vartheta}<\bar{\beta}<1/3$, we construct infinitely many global-in-time probabilistically strong and analytically weak solutions in $C([0,\infty),C^{\bar{\vartheta}}(\mathbb{T}^3,\mathbb{R}^3))$ for arbitrary divergence-free initial data in $C^{\bar{\beta}}(\mathbb{T}^3,\mathbb{R}^3)$. Our construction relies on the convex integration method developed in the deterministic setting by \cite{Ise18}, adapting it to the stochastic context by introducing a novel energy inequality into the convex integration scheme and combining stochastic analysis arguments with a Wong--Zakai type estimate.

math.PR↗

arXiv2024

H\"{o}lder continuous solutions to stochastic 3D Euler equations via stochastic convex integration

In this paper, we are concerned with the three dimensional Euler equations driven by an additive stochastic forcing. First, we construct global H\"{o}lder continuous (stationary) solutions in $C(\mathbb{R};C^{\vartheta})$ space for some $\vartheta>0$ via a different method from \cite{LZ24}. Our approach is based on applying stochastic convex integration to the construction of Euler flows in \cite{DelSze13} to derive uniform moment estimates independent of time. Second, for any divergence-free H\"{o}lder continuous initial condition, we show the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in $L^p_{\rm{loc}}([0,\infty);C^{\vartheta'}) \cap C_{\rm{loc}}([0,\infty);H^{-1})$ for all $p\in [1,\infty)$ and some $\vartheta'>0$.

math.PR↗

arXiv2024

Stationary solutions to stochastic 3D Euler equations in H\"older space

We establish the existence of infinitely many global and stationary solutions in $C(\mathbb{R};C^{\vartheta})$ space for some $\vartheta>0$ to the three dimensional Euler equations driven by an additive noise. The result is based on a new stochastic version of the convex integration method, incorporating the stochastic convex integration method developed in \cite{HZZ22b} and pathwise estimates to derive uniform moment estimates independent of time.

math.PR↗