具有粗糙数据的二维Navier-Stokes方程的大向前离散自相似解

中文摘要 AI 辅助

我们构造了二维Navier-Stokes方程在$L^2_{loc}(\mathbb{R}^2\setminus\{0\})$中具有任意大数据的向前离散自相似（DSS）解。对于这种粗糙的初始数据，我们首先在离散缩放下建立了一个尖锐的线性估计，该估计将对数时间中一个基本环上的初始数据的$L^2$-范数与其次热提升在一个周期内的Dirichlet能量等同起来。这连同线性轮廓与非线性余项在对数时间的一个周期内的结构抵消，产生了余项的一致$\dot{H}^1$界，其中初始数据的自然$L^2_{loc}$正则性似乎是尖锐的。由于二维Leray算子没有$L^2$-强制性，我们通过利用$L^p$估计（$1<p<2$），以及关键的bootstrap论证和周期性，恢复了余项的一致能量控制。然后，只要初始数据属于$C^2_{loc}(\mathbb{R}^2\setminus\{0\})$，我们首先利用进一步的加权估计和不动点论证构造光滑的DSS解。这些更强的加权界在粗糙数据的逼近下不是一致的。相反，过渡到$L^2_{\rm loc}$数据仅依赖于非加权能量估计。最后，我们利用极限周期轮廓的强时间连续性来恢复验证初始数据所需的一致空间紧性。

英文摘要

We construct forward discretely self-similar (DSS) solutions of two-dimensional Navier-Stokes equations with arbitrarily large data in $L^2_{loc}(\mathbb{R}^2\setminus\{0\})$. For such rough initial data, we first establish a sharp linear estimate under the discrete scaling, which identifies the \(L^2\)-norm of the initial datum on a fundamental annulus with the Dirichlet energy of its caloric lift over one period in logarithmic time. This, together with a structural cancellation between the linear profile and the nonlinear remainder over one period in logarithmic time, yields a uniform \(\dot{H}^1\)-bound for the remainder, where the natural $L^2_{loc}$-regularity of the initial data seems to be sharp. Since the two-dimensional Leray operator has no \(L^2\)-coercivity, we recover uniform energy control for the remainder by exploiting \(L^p\)-estimates ($1<p<2)$, together with a key bootstrap argument and periodicity. We then first construct smooth DSS solutions using further weighted estimates and a fixed-point argument as long as the initial data belong to $C^2_{loc}(\mathbb{R}^2\setminus\{0\})$. These stronger weighted bounds are not uniform under approximation of rough data. Instead, the passage to \(L^2_{\rm loc}\) data relies only on the unweighted energy estimates. Finally, we use strong time continuity of the limiting periodic profile to recover the uniform spatial tightness needed to verify the initial data.

发表机构

• University of Macau（澳门大学）
• Zhuhai UM Science and Technology Research Institute（珠海UM科学与技术研究院）
• Shanghai Jiao Tong University（上海交通大学）

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