发表机构
Shanghai Jiao Tong University; Tianjin Normal University; Xiamen University（上海交通大学; 天津师范大学; 厦门大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文证明二维不可压缩欧拉方程中平稳或均匀旋转解在特定涡量类下必为径向对称，且非零旋转时中心为原点，方法基于水平集几何与缺陷测度分析。
AI 中文摘要
我们证明了二维不可压缩欧拉方程中具有涡量 $ \omega_0\in C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2) $ 的平稳和均匀旋转解的刚性定理。若角速度 $\Omega$ 满足 \\[ \Omega\leq\frac12\inf_{\mathbb{R}^2}\omega_0 \qquad\text{或}\qquad \Omega\geq\frac12\sup_{\mathbb{R}^2}\omega_0, \\] 则 $\omega_0$ 是径向对称的；特别地，在类 $C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$ 中，每个单号平稳涡量必定关于某点径向对称。此外，对于 $\Omega\neq0$，中心必定是原点。证明基于归一化流函数的水平集几何分析。分量形式的 Bernoulli 公式，结合等周不等式和 Pohozaev 恒等式，产生一个非负缺陷测度，该测度同时编码几何缺陷和拓扑分支。在无穷远处的分析迫使该测度消失，将问题简化为具有有界非负 Borel 非线性项的半线性椭圆方程。径向对称性随后由 P.-L. Lions 的一个定理得出。
英文摘要
We prove a rigidity theorem for stationary and uniformly rotating solutions of the two-dimensional incompressible Euler equation with vorticity $ ω_0\in C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2). $ If the angular velocity $Ω$ satisfies \[ Ω\leq\frac12\inf_{\mathbb{R}^2}ω_0 \qquad\text{or}\qquad Ω\geq\frac12\sup_{\mathbb{R}^2}ω_0, \] then $ω_0$ is radially symmetric, in particular, every stationary vorticity of one sign in the class $C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$ must be radially symmetric about some point. Furthermore, for $Ω\neq0$, the center is necessarily the origin; The proof is based on the analysis for level-set geometry of a normalized stream function. A componentwise Bernoulli formula, together with the isoperimetric inequality and Pohozaev identities, yields a nonnegative defect measure encoding both geometric defects and topological branching. An analysis at infinity forces this measure to vanish, reducing the problem to a semilinear elliptic equation with a bounded nonnegative Borel nonlinearity. Radial symmetry then follows from a theorem of P.-L.\ Lions.
