AI 中文总结
研究环面上的正压Euler-Riesz系统，通过Caffarelli-Silvestre延拓处理非局部力，引入全局耗散解，证明对任意$\boldsymbol{\beta}\boldsymbol{\beta}(0,2)$和$\boldsymbol{\beta}>1$的弱-强唯一性。
AI 中文摘要
我们研究环面$\boldsymbol{\top}^d$（$d=2,3$）上的Euler-Riesz系统：具有正压压力$p(\boldsymbol{\rho})=a\boldsymbol{\rho}^\boldsymbol{\beta}$（$\boldsymbol{\beta}>1$，$a>0$）的可压缩Euler方程，与阶数为$\boldsymbol{\beta}\boldsymbol{\beta}(0,2)$的Riesz核$K(x)\boldsymbol{\beta}|x|^{\boldsymbol{\beta}-d}$的排斥非局部力（$\boldsymbol{\beta}\boldsymbol{\rho}\nabla_x K \boldsymbol{\beta}\boldsymbol{\rho}$）耦合。由于$K$是分数阶拉普拉斯逆算子$(-\boldsymbol{\beta})^{\boldsymbol{\beta}/2}$的核，我们通过Caffarelli-Silvestre延拓将该力重述为局部应力张量的迹，在额外变量中用局部恒等式替代非局部相互作用。对于排斥核，总能量是强制的，我们利用这一点引入了任意大有限能量数据的全局时间耗散解概念。我们的主要结果是，对每个$\boldsymbol{\beta}\boldsymbol{\beta}(0,2)$和每个$\boldsymbol{\beta}>1$，弱（测度值）-强唯一性成立：在强解存在的任意区间上，具有相同初始数据的每个耗散解都与该强解一致，且所有缺陷消失。该证明依赖于相对能量的适当调整。
英文摘要
We study the Euler--Riesz system on the torus $\mathbb T^d$, $d=2,3$: the compressible Euler equations with barotropic pressure $p(\varrho)=a\varrho^γ$ ($γ>1$, $a>0$), coupled to a repulsive nonlocal force ($\approx \varrho \nabla_x K \ast \varrho$) with Riesz kernel $K(x)\propto|x|^{β-d}$ of order $β\in(0,2)$. Since $K$ is the kernel of the inverse fractional Laplacian $(-Δ)^{-β/2}$, we recast the force through the Caffarelli--Silvestre extension as the trace of a local stress tensor, replacing the nonlocal interaction by a local identity in one extra variable. For a repulsive kernel the total energy is coercive, and we use this to introduce a notion of global-in-time \emph{dissipative solution} for arbitrarily large finite-energy data. Our main result is weak (measure-valued)--strong uniqueness, for every order $β\in(0,2)$ and every $γ>1$ independently: on any interval on which a strong solution exists, every dissipative solution with the same initial data coincides with it and all defects vanish. The proof rests on a suitable adaptation of relative energy.
