Regularity theory and low Mach number limit for the fractional Euler-alignment system

RESUMO

We study the compressible Euler-alignment system with pressure under a singular pressure scaling, where the hypersingular communication weight induces a fractional alignment operator of order $2\alpha$, $0<\alpha<1$. The scaling corresponds to a large-time and small-velocity regime and leads to a low Mach number problem in which the density is forced to remain close to a constant state. Our main result is a uniform regularity theory for this scaled pressure system. We establish uniform estimates with respect to the scaling parameter and construct global strong solutions near the constant state. A key feature of the analysis is that, in the low-order fractional regime $0<\alpha\le\frac12$, the estimates close under the lower Sobolev condition $s>\frac d2+1-2\alpha$, gaining $\alpha$ derivatives over the threshold $s>\frac d2 + 1 - \alpha$ arising from a direct use of the fractional alignment dissipation. This is achieved by combining refined commutator estimates for the singular alignment operator with the density dissipation induced by the pressure scaling. As an application of the uniform estimates, we justify the low Mach number limit toward the incompressible Navier--Stokes system with fractional dissipation. For general small, possibly ill-prepared initial data, a Helmholtz decomposition combined with dispersive estimates for the acoustic component yields subsequential strong convergence locally in space-time to a distributional solution of the limiting system. For well-prepared initial data, a relative-energy argument further identifies the limit with a prescribed sufficiently regular solution and yields global strong convergence in the fractional dissipation norm.