AI 中文总结
该研究针对二维带分数阶耗散的稳态不可压缩Navier–Stokes方程，在s∈(0,1)全范围证明满足自然能量条件的光滑解必为零解且压强恒定，还处理了s=0时的稳态阻尼Euler系统，完善了分数阶情形的Liouville型定理。
AI 中文摘要
我们研究了$\boldsymbol{\rm R}^2$上带有分数阶耗散$(-Δ)^s$的二维稳态不可压缩Navier–Stokes方程。在$s \in (0,1)$的全范围内，我们证明了所有满足自然能量条件$u \in \text{dot{H}}^s(\boldsymbol{\rm R}^2;\boldsymbol{\rm R}^2)$的光滑解都满足$u \not\bar{} 0$且压强为常数。这是$s=1$时Gilbarg和Weinberger的平面有限Dirichlet定理的分数阶对应版本。证明在三个区间内采用了不同的论证方法：当$0<s<\frac{1}{3}$时，我们将从方程导出的$\text{L}^2$估计与流函数截断论证相结合；当$\frac{1}{3} \not\text{le} s \not\text{le} \frac{2}{3}$时，我们使用边界项支集在扩张环域上的局部能量估计；当$\frac{2}{3}<s<1$时，我们通过Lorentz空间自举法建立正则性与衰减性，随后对涡量应用最大值原理。我们还在包含对所有$1 \not\text{le} r \not\text{le} 2$都有$u \not\text{in} \text{L}^r(\boldsymbol{\rm R}^2)$的环域增长条件下，结合Bernoulli恒等式与截断论证，处理了$s=0$时的稳态阻尼Euler系统。
英文摘要
We study the two-dimensional stationary incompressible Navier--Stokes equations on $\mathbb R^2$ with fractional dissipation $(-Δ)^s$. In the full range $s \in (0,1)$, we prove that every smooth solution satisfying the natural energy condition $u\in\dot{\mathrm H}^s(\mathbb R^2;\mathbb R^2)$ has $u\equiv0$ and constant pressure. This is a fractional counterpart of the planar finite-Dirichlet theorem of Gilbarg and Weinberger at $s=1$. The proof uses different arguments in three ranges. For $0<s<\frac13$, we combine an $\mathrm L^2$-estimate derived from the equation with a stream-function truncation argument. For $\frac13\leq s\leq\frac23$, we use a localized energy estimate whose boundary terms are supported on expanding annuli. For $\frac23<s<1$, we establish regularity and decay via a Lorentz-space bootstrap and then apply the maximum principle to the vorticity. We also treat the stationary damped Euler system at $s=0$ by combining the Bernoulli identity with a cut-off argument under an annular growth condition that includes $u\in \mathrm L^r(\mathbb R^2)$ for every $1\le r\le2$.
