Low Mach number limit of the compressible Navier-Stokes-Smoluchowski equations in multi-dimensions

Abstract

This paper is concerned with the incompressible limit of the compressible Navier-Stokes-Smoluchowski equations with periodic boundary conditions in multidimensions. The authors establish the uniform stability of the local solution family which yields a lifespan of the Navier-Stokes-Smoluchowski system. Then, the local existence of strong solutions for the incompressible system with small initial data is rigorously proved via the incompressible limit. Furthermore, the authors obtain the convergence rates in the case without external force.

Incompressible limit of a fluid-particle interaction model

“…Ballew-Trivisa [1] rigorously established the low Mach number and low stratification limits of the system (1.1) for both bounded and unbounded domains in a weak sense. More recently, Huang-Huang-Wen [17] established the uniform stability of the local solution family, which yields a lifespan of the Navier-Stokes-Smoluchowski system and proves the local existence of strong solutions for the incompressible system with small initial data. Furthermore, they obtained the convergence rates in the case without external force.…”

“…The present paper considers the low Mach number limit of system (1.1) in the scaling form (1.4), which is different from that of [2] and [17]. Under some reasonable assumptions, we prove rigorously that the weak solutions of the fluid-particle interaction model (1.4) correspond to the strong solution of the incompressible Navier-Stokes equations in the time interval (provided the latter exists) by using the refined relative entropy method.…”

Strong solutions to a fluid-particle interaction model with magnetic field in $ \mathbb{R}^2 $

“…al. [17] established the incompressible limit of the compressible system as weil as the convergence rates of the strong solutions under the periodic boundary conditions in two or three dimensions. In a One-dimensional bounded domain, the global well-posedness of strong and classical solutions are considered in [13,26].…”

Remarks on global weak solutions to a two-fluid type model

“…The system (1.1) with nt + (nu)x = 0 replaced by nt + (nu)x = nxx ("nt + ∇ • (nu) = ∆n" in multi-dimensions) is associated to the bubbling regime. See [2,10,1,31] for the global existence of weakly dissipative solutions and weak-strong uniqueness and low Mach number limits in high dimensions, respectively. For the well-posedness results, please refer to [11,14,16,30].…”