arXiv subjects

Huaxiang Lü

Publications and source records attributed to Huaxiang Lü.

8 recordsLinked to original sources

arXiv2026

The Wave Kinetic Theory for Quasilinear MMT Equation

We study the one-dimensional quasilinear Majda--McLaughlin--Tabak (MMT) equation on a large torus $[0,L]$: \begin{align*} i \partial_t u +2π|\nabla|^σu +λ^{2}|\nabla|^β\left[ \left||\nabla|^βu\right|^{2} |\nabla|^βu\right]=0. \end{align*} Our focus is on the well-posedness of its dynamics and the emergence of kinetic behavior where the domain size $L$ tends to infinity and the nonlinearity $α=λ^2L^{-1}$ vanishes. In contrast to semilinear dispersive models, the quasilinear structure leads to unavoidable derivative loss, which prevents the construction of solutions via iteration of the Duhamel formula. Our results exhibit a dichotomy depending on the dispersion exponent $σ$. For $σ\in(1,2]$, we prove that, with high probability, solutions exist up to time scales $T_0 \sim α^{-\frac54+} \wedge α^{-\frac1{1-β}+}$, and that only trivial resonances occur, leading to a degenerate wave kinetic equation. For $σ\in(0,1)$, we prove the existence up to time scales $T_0 \sim α^{-1-}$ and show that the second-order statistics are well approximated by the wave kinetic equation. In both cases, the solutions remain smooth while exhibiting smallness in suitable $L^\infty$-based norms despite having large total energy. The proof proceeds in two main steps. First, we establish the propagation of randomness for a suitably truncated equation, which allows us to overcome the derivative loss and recover the kinetic description. Then, we perform deterministic high-order energy estimates and a bootstrap argument to extend the solution up to time $T_0$.

math.AP↗

arXiv2026

Non-Uniqueness for Nonlinear Fokker--Planck Equations and Their Associated Distribution-Dependent SDEs

In this paper, we study distribution-dependent stochastic differential equations on the domain $\mathcal O=\mathbb T^d$ or $\mathbb R^d$, $d\geq 2$, of the form \begin{align*} {\rm d}X_t = v(t,X_t,ρ_t)\,{\rm d}t + \sqrt{2}\, σ(t,X_t,ρ_t)\,{\rm d}W_t, \qquad ρ_t:=\frac{{\rm d}μ_t}{{\rm d}x}, \end{align*} where $μ_t=\operatorname{Law}(X_t)$. Our main construction is carried out at the level of the associated nonlinear Fokker--Planck equations. We first build non-unique probability solutions to these PDEs and then use the superposition principle to obtain non-unique martingale solutions to the corresponding DDSDEs. We establish two main non-uniqueness results concerning stationary states, both on the torus and in the whole space, under the corresponding structural assumptions. First, we construct a divergence-free drift $v\in C_tL^{d-}$ such that the DDSDE admits \emph{infinitely many} distinct solutions starting from the stationary initial density. This result lies at the natural critical regularity threshold: in several models, well-posedness is expected for drifts in $C_tL^{d+}$. Second, for $d\geq 3$ and every prescribed $N\in\mathbb{N}$, we construct a divergence-free drift for which the DDSDE admits at least $N$ distinct stationary martingale solutions. The resulting multiplicity of equilibrium states is reminiscent of multistability and phase-transition phenomena in physical systems.

math.PR↗

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 $0<\bar{\vartheta}<\barβ<1/3$, 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 $C^{\barβ}(\mathbb{T}^3,\mathbb{R}^3)$. 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↗

arXiv2025

Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space

In this paper, we investigate the stochastic differential equation on $\mathbb{R}^d,d\geq2$: \begin{align*} \dif X_t&=v(t,X_t)\dif t+\sqrt{2} \dif W_t. \end{align*} For any finite collection of initial probability measures $\{μ^i_0\}_{1\leq i\leq M}$ on $\mathbb{R}^d$ and $\frac{d}{p}+\frac{1}{r}>1$, we construct a divergence-free drift field $v\in L_t^rL^p\cap C_tL^{d-}$ such that the associated SDE admits at least two distinct weak solutions originating from each initial measure $μ^i_0$. This result is sharp in view of the well-known uniqueness of strong solutions for drifts in $C_tL^{d+}$, as established in \cite{KR05}. As a corollary, there exists a measurable set $A\subset\mathbb{R}^d$ with positive Lebesgue measure such that for any $x\in A$, the SDE with drift $v$ admits at least two weak solutions when with start in $x\in A$. The proof proceeds by constructing two distinct probability solutions to the associated Fokker-Planck equation via a convex integration method adapted to all of $\mathbb{R}^d$ (instead of merely the torus), together with refined heat kernel estimate.

math.PR↗

arXiv2025

Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations

We are concerned with the (stochastic) Lagrangian trajectories associated with Euler or Navier-Stokes equations. First, in the vanishing viscosity limit, we establish sharp non-uniqueness results for positive solutions to transport equations advected by weak solutions of the 3D Euler equations that exhibit kinetic energy dissipation with $C_{t,x}^{1/3-}$ regularity. As a corollary, in conjunction with the superposition principle, this yields the non-uniqueness of associated (deterministic) Lagrangian trajectories. Second, in dimension $d\geq2$, for any $\frac{1}{p}+\frac{1}{r}>1$ or $p\in(1,2),r=\infty$, we construct solutions to the Euler or Navier-Stokes equations in the space $L_t^rL^p\cap L_t^1W^{1,1}$, demonstrating that the associated (stochastic) Lagrangian trajectories are not unique. Our result is sharp in 2D in the sense that: (1) in the stochastic case, for any vector field $v\in C_tL^p$ with $p>2$, the associated stochastic Lagrangian trajectory associated with $v$ is unique (see \cite{KR05}); (2) in the deterministic case, the LPS condition guarantees that for any weak solution $v\in C_tL^p$ with $p>2$ to the Navier-Stokes equations, the associated (deterministic) Lagrangian trajectory is unique. Our result is also sharp in dimension $d\geq2$ in the sense that for any divergence-free vector field $v\in L_t^1W^{1,s}$ with $s>d$, the associated (deterministic) Lagrangian trajectory is unique (see \cite{CC21}).

math.AP↗

arXiv2024

Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise

We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on $\mathbb{T}^2$ \begin{align*} \dif u+÷(u\otimes u)\dif t+\nabla p\dif t&=Δu\dif t + (-Δ)^{\fa/2}\dif B_t,\ \ \ \ ÷u=0,\notag \end{align*} driven by derivative of space-time white noise, where $\fa\in[0,\frac13)$. In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution $N(0,\frac12(-Δ)^{\fa-1})$. The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces.

math.PR↗

arXiv2023

Sharp Non-uniqueness of Solutions to 2D Navier-Stokes Equations with Space-Time White Noise

In this paper we are concerned with the 2D incompressible Navier-Stokes equations driven by space-time white noise. We establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions $u$ for every divergence free initial condition $u_0\in L^p\cup C^{-1+δ},\ p\in(1,2),δ>0$. More precisely, there exist infinitely many solutions such that $u-z\in C([0,\infty);L^p)\cap L^2_{\rm{loc}}([0,\infty);H^ζ)\cap L^1_{\rm{loc}}([0,\infty);W^{\frac13,1})$ for some $ζ\in(0,1)$, where $z$ is the solution to the linear equation. This result in particular implies non-uniqueness in law. Our result is sharp in the sense that the solution satisfying $u-z\in C([0,\infty);L^2)\cap L^2_{\rm{loc}}([0,\infty);H^ζ)$ for some $ζ\in(0,1)$ is unique.

math.PR↗

arXiv2022

Global-in-time probabilistically strong solutions to stochastic power-law equations: existence and non-uniqueness

We are concerned with the power-law fluids driven by an additive stochastic forcing in dimension $d\geq3$. For the power index $r\in(1,\frac{3d+2}{d+2})$, we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in $L^p_{loc}([0,\infty);L^2)\cap C([0,\infty);W^{1,\max\{1,r-1\}}),p\geq1$ for every divergence free initial condition in $L^2\cap W^{1,\max\{1,r-1\}}$. This result in particular implies non-uniqueness in law. Our result is sharp in the three dimensional case in the sense that the solution is unique if $r\geq \frac{3d+2}{d+2}$.

math.PR↗