Low Mach number limit of the full compressibleNavier-Stokes-Korteweg equations with general initial data

Abstract

In this paper, the low Mach number limit for the three-dimensional full compressible Navier-Stokes-Korteweg equations with general initial data is rigorously justified within the framework of local smooth solution. Under the assumption of large temperature variations, we first obtain the uniform-in- Mach-number estimates of the solutions in a $\varepsilon$-weighted Sobolev space, which establishes the local existence theorem of the three-dimensional full compressible Navier-Stokes-Korteweg equations on a finite time interval independent of Mach number. Then, the low mach limit is proved by combining the uniform estimates and a strong convergence theorem of the solution for the acoustic wave equations. This result improves that of [K.-J. Sha and Y.-P. Li, Z. Angew. Math. Phys., 70(2019), 169] for well-prepared initial data.