三维Navier–Stokes方程的一个临界单分量正则性准则

中文摘要 AI 辅助

我们为三维不可压缩Navier–Stokes方程建立了两个临界单分量正则性准则。首先，对于无散度初值$u_0\in H^1(\mathbb R^3)$，我们证明若\\[ u^3\in L^2\bigl(0,T;\dot B^{3/2}_{2,r}(\mathbb R^3)\bigr) \qquad\text{对某个 }2<r\leq\infty,\\]则强解可以延拓过有限时间$T$。其次，对每个$2<p<\infty$，若$u_0\in\dot H^{1/2}(\mathbb R^3)$且对某个固定单位向量$\boldsymbol e\in\mathbb S^2$，\\[ u\cdot\boldsymbol e \in L^p\bigl(0,T;\dot H^{1/2+2/p}(\mathbb R^3)\bigr),\\]则相应的Fujita–Kato解也能延拓过$T$，且无需对初始涡量作任何额外的低可积性假设。这些结果在两个不同方向上推广了Han、Lei、Li和Zhao的临界单分量正则性准则。在时间端点$p=2$处，我们将新的垂直涡量估计与耦合的各向异性能量估计以及依赖于能量的频率分解相结合，将二进平方可和条件\\[ \dot H^{3/2}=\dot B^{3/2}_{2,2} \\]放宽为\\[ \dot B^{3/2}_{2,r}, \qquad 2<r\leq\infty,\\]其中端点情形$r=\infty$通过Osgood型论证封闭。在非端点范围$2<p<\infty$内，我们使用正时间热流分解将解分离为光滑线性部分和非线性余项。余项具有有限能量，并在正时间获得所需的涡量$L^q$-可积性，这使得各向异性能量估计得以应用，同时由热流产生的额外输运、拉伸和压力项保持可积。这去除了对初始涡量的额外$L^{q_0}$-可积性假设，同时保留了自然的临界初值空间$\dot H^{1/2}(\mathbb R^3)$。

英文摘要

We establish two critical one-component regularity criteria for the three-dimensional incompressible Navier--Stokes equations. First, for divergence-free initial data $u_0\in H^1(\mathbb R^3)$, we prove that a strong solution can be continued beyond a finite time $T$ provided that \[ u^3\in L^2\bigl(0,T;\dot B^{3/2}_{2,r}(\mathbb R^3)\bigr) \qquad\text{for some }2<r\leq\infty. \] Second, for every $2<p<\infty$, if $u_0\in\dot H^{1/2}(\mathbb R^3)$ and, for some fixed unit vector $\boldsymbol e\in\mathbb S^2$, \[ u\cdot\boldsymbol e \in L^p\bigl(0,T;\dot H^{1/2+2/p}(\mathbb R^3)\bigr), \] then the corresponding Fujita--Kato solution also extends beyond $T$, without any additional low-integrability assumption on the initial vorticity. These results extend the critical one-component regularity criteria of Han, Lei, Li, and Zhao in two different directions. At the time endpoint $p=2$, we combine a new vertical-vorticity estimate with coupled anisotropic energy estimates and an energy-dependent frequency decomposition to relax the dyadic square-summability condition \[ \dot H^{3/2}=\dot B^{3/2}_{2,2} \] to \[ \dot B^{3/2}_{2,r}, \qquad 2<r\leq\infty, \] with the endpoint case $r=\infty$ closed by an Osgood-type argument. In the nonendpoint range $2<p<\infty$, we use a positive-time heat-flow decomposition to separate the solution into a smooth linear part and a nonlinear remainder. The remainder has finite energy and acquires the required $L^q$-integrability of the vorticity at positive times, which allows the anisotropic energy estimates to be applied while the additional transport, stretching, and pressure terms generated by the heat flow remain integrable. This removes the extra $L^{q_0}$-integrability assumption on the initial vorticity while retaining the natural critical initial-data space $\dot H^{1/2}(\mathbb R^3)$.

发表机构

• Harbin University of Science and Technology（哈尔滨科技大学）
• Shanghai Jiao Tong University（上海交通大学）
• Beijing University of Technology（北京工业大学）

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