低于临界标度的随机平流纳维 - 斯托克斯方程中的涨落动力学
Fluctuation dynamics in randomly advected Navier-Stokes equations below critical scaling
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中文总结 AI 辅助
研究随机平流纳维 - 斯托克斯方程，引入双参数模型族。在完全亚临界情形下证明大数定律，二维中更强假设下确定主导涨落，给出解的收敛情况及涨落特性。
中文摘要 AI 辅助
我们研究随机平流的不可压缩纳维 - 斯托克斯方程，其中平流场是均值为零、无散度、时空平稳且具有光滑一阶相关性的速度场。引入一个双参数模型族，平流在快速时间尺度\(\varepsilon^2\)上加速且具有空间相关长度\(\delta\)，临界情形\(\varepsilon = \delta\)对应纳维 - 斯托克斯方程的自然抛物标度。在完全亚临界情形\(\varepsilon = o (\delta)\)下，在二维和三维证明了大数定律：解收敛到一个由格林 - 久保公式给出增强扩散系数的确定性纳维 - 斯托克斯系统。在二维中，在更强假设\(\varepsilon = o (\delta^{1 + \iota})\)（\(\iota > 0\)）下，确定了主导涨落：减去满足纳维 - 斯托克斯型非线性系统的确定性宏观修正后，重标涨落收敛到由乘性时空白噪声驱动的线性化纳维 - 斯托克斯方程的高斯场。
英文摘要
We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale $\varepsilon^2$ and has spatial correlation length $δ$; the critical regime $\varepsilon = δ$ corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime $\varepsilon = o (δ)$, we prove a law of large numbers in dimensions $d = 2, 3$: the solutions converge to a deterministic Navier--Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption $\varepsilon = o (δ^{1 + ι})$ for some $ι> 0$, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.
