2026/08/03 by Nilasis Chaudhuri
Mathematics · #math.AP #msc:35Q31 #msc:35A35 #msc:35B30 #msc:35R11 #msc:76N10
arxiv created 2026/08/03 · arxiv updated 2026/08/04
We study the Euler--Riesz system on the torus \mathbb Td, d=2,3: the compressible Euler equations with barotropic pressure p(\varrho)=a\varrhoγ (γ>1, a>0), coupled to a repulsive nonlocal force (≈ \varrho ∇x K ∗ \varrho) with Riesz kernel K(x)∝|x|β-d of order β∈(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 dissipative solution for arbitrarily large finite-energy data. Our main result is weak (measure-valued)--strong uniqueness, for every order β∈(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.