Nonlinear Diﬀer. Equ. Appl. (2017) 24:52
2017 Springer International Publishing AG
published online August 5, 2017
Nonlinear Diﬀerential Equations
and Applications NoDEA
Uniqueness of the 1D compressible
to incompressible limit
Rinaldo M. Colombo and Graziano Guerra
Abstract. Consider two compressible immiscible ﬂuids in 1D in the isen-
tropic approximation. The ﬁrst ﬂuid is surrounded and in contact with the
second one. As the Mach number of the ﬁrst ﬂuid vanishes, the coupled
dynamics of the two ﬂuids results as the compressible to incompressible
limit and is known to satisfy an ODE–PDE system. Below, a characteri-
zation of this limit is provided, ensuring its uniqueness.
Mathematics Subject Classiﬁcation. 35L65, 35Q35, 76N99.
Keywords. Compressible to incompressible limit, Hyperbolic conservation
laws, Uniqueness of the zero Mach number limit.
The literature on the compressible to incompressible limit is vast. We refer for
instance to the well known results [15,16,18,19], the more recent [4,21], the
review  and the references therein.
In this paper, following , we consider two compressible immiscible ﬂu-
ids and study the limit as one of the two becomes incompressible. A volume
of a compressible inviscid ﬂuid, say the liquid, is surrounded by another com-
pressible ﬂuid, say the gas. Using the Lagrangian formulation, in the isentropic
case, we assume that the gas obeys a ﬁxed pressure law P
(τ), while for the
liquid we assume a one parameter family of pressure laws P
(τ) such that
(τ) →−∞as κ → 0. The total mass of the liquid is ﬁxed so that in Lagran-
gian coordinates the liquid and gas phases ﬁll the ﬁxed sets (see Fig. 1)
L =]0,m[andG = R \ ]0,m[ .
For an Eulerian description, see .
This article is part of the topical collection “Hyperbolic PDEs, Fluids, Transport and Appli-
cations: Dedicated to Alberto Bressan for his 60th birthday” guest edited by Fabio Ancona,
Stefano Bianchini, Pierangelo Marcati, Andrea Marson.