Nonlinear Diﬀer. Equ. Appl.
2017 Springer International Publishing AG
Nonlinear Diﬀerential Equations
and Applications NoDEA
On the uniqueness of vanishing viscosity
solutions for Riemann problems
for polymer ﬂooding
Abstract. We consider the vanishing viscosity solutions of Riemann prob-
lems for polymer ﬂooding models. The models reduce to triangular sys-
tems of conservation laws in a suitable Lagrangian coordinate, which con-
nects to scalar conservation laws with discontinuous ﬂux. These systems
are parabolic degenerate along certain curves in the domain. A vanishing
viscosity solution based on a partially viscous model is given in a par-
allell paper (Guerra and Shen in Partial Diﬀer Equ Math Phys Stoch
Anal: 2017). In this paper the fully viscous model is treated. Through
several counter examples we show that, as the ratio of the viscosity pa-
rameters varies, inﬁnitely many vanishing viscosity limit solutions can be
constructed. Under some further monotonicity assumptions, the unique-
ness of vanishing viscosity solutions for Riemann problems can be proved.
Mathematics Subject Classiﬁcation. Primary 35L65, Secondary 35L80,
Keywords. Conservation laws, Riemann problems, Vanishing viscosity,
Parabolic degeneracy, Discontinuous ﬂux, Polymer ﬂooding.
In this paper we study the uniqueness of the solutions of Riemann problems for
some systems of conservation laws, obtained as the vanishing viscosity limit.
In particular, we consider the equations for polymer ﬂooding in secondary oil
+ f(s, c)
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.