High Mach number limit of the compressible Navier--Stokes equations in critical Besov spaces

Abstract: We investigate the high Mach number limit for the scaled compressible Navier--Stokes system in the critical Besov framework. In the scaled momentum equation, the pressure force is represented by the term (\varepsilon^2\nabla a^\varepsilon), where $\varepsilon$ is the inverse Mach number; as (\varepsilon\to0), the formal limiting system is the compressible pressureless Navier--Stokes system. The analysis is complicated by the absence of density dissipation in the limiting model and by the highest-order coupling created by the viscous terms. For (d\geq2), we prove the global well-posedness of the scaled system for small initial data and obtain estimates that are uniform with respect to $\varepsilon$. A crucial ingredient is a parameter-dependent lower-order estimate for (\varepsilon a^\varepsilon), which compensates for the purely transport nature of the density equation and allows the uniform bounds to be closed. Based on these estimates, we justify the high Mach number limit and recover a global strong solution to the pressureless Navier--Stokes system. For (d\geq3), we further derive quantitative error estimates between the scaled solutions and the pressureless limiting solution. More precisely, on each fixed finite time interval, if the initial discrepancy is of order (\mathcal{O}(\varepsilon)), then the corresponding lower-order critical Besov error satisfies the same rate, which yields a quantitative justification of the pressureless limit.