Nonuniqueness and Turbulence

An example elucidating the mathematical situation in the statistical non-uniqueness problem of turbulence

An instructive example is presented to elucidate the mathematical situation in the non-uniqueness problem of the infinite Friedmann-Keller hierarchy of equations for all multi-point moments within the theory of spatially unbounded Navier-Stokes turbulence. It is shown that the non-uniqueness problem of the Friedmann-Keller hierarchy emerges from the property that the system of equations is defined forward recursively. As a result, this system does not possess a unique general solution, even when the complete infinite system is formally considered. That is, even when imposing a sufficient numbe…

On uniqueness of transfer rates in magnetohydrodynamic turbulence

In hydrodynamic and magnetohydrodynamic turbulence, expressions for the transfer rates rely on integrals over wavenumber triads $(\textbf{k,p,q})$ satisfying $\textbf{k+p+q=0}$. As an example $S_E^{uu}(\textbf{k|p,q})$ denotes the kinetic energy transfer rate to the mode $\textbf{k}$, from the two other modes in the triad, $\textbf{p}$ and $\textbf{q}$. As noted by Kraichnan (1958), in $S_E^{uu}(\textbf{k|p,q})$, what fraction of the energy transferred to the mode $\textbf{k}$ originated from $\textbf{p}$ and which from $\textbf{q}$ is unknown . Such an expression is thus incongruent with the …

神经湍流闭合的适定性与切向耗散

Well-posedness of neural turbulence closures and tangent dissipation

该研究针对神经湍流闭合，证明全局切向耗散保证单调算子的适定性，并提出两种促进方法，在槽道流测试中验证了唯一性并大幅降低逆灵敏度。

Rigorous results in space-periodic two-dimensional turbulence

We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier-Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value problem and the associated Markov process, we formulate results on the existence, uniqueness and mixing of a stationary measure. We next turn to various consequences of these properties: strong law of large numbers, central limit theorem, and random attractors related to a unique stationary measure. We also discuss the Donsker-Varadhan and Freidlin-Wentzell type…

Navier-Stokes方程在几乎相同的初始条件下光滑解的非唯一性

Non-uniqueness of smooth solutions of the Navier-Stokes equations from almost the same initial conditions

该研究通过清洁数值模拟展示Navier-Stokes方程在极小初始条件差异下存在多个光滑解，可能对方程的唯一性和存在性问题提供新见解。

Anomalous Dissipation in Passive Scalar Transport

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible $C^\infty([0,T)\times \mathbb{T}^d)\cap L^1([0,T]; C^{1-}(\mathbb{T}^d))$ velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows non-uniqueness of solutions to the transport equation with an incompressible $L^1([0,T]; C^{1-}(\mathbb{T}^d))$ drift, which is smooth except at one point in time. We also provide three sufficient conditions for anomalous dissipation provided solutions to the inviscid equation become…