Generalized solution and Weak-Strong uniqueness for a barotropic Euler-Riesz system

We study the Euler--Riesz system on the torus $\mathbb T^d$, $d=2,3$: the compressible Euler equations with barotropic pressure $p(\varrho)=a\varrho^\gamma$ ($\gamma>1$, $a>0$), coupled to a repulsive nonlocal force ($\approx \varrho \nabla_x K \ast \varrho$) with Riesz kernel $K(x)\propto|x|^{\beta-d}$ of order $\beta\in(0,2)$. Since $K$ is the kernel of the inverse fractional Laplacian $(-\Delta)^{-\beta/2}$, 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 $\beta\in(0,2)$ and every $\gamma>1$ 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.