74 research outputs found
Stability properties of the Euler-Korteweg system with nonmonotone pressures
We establish a relative energy framework for the Euler-Korteweg system with
non-convex energy. This allows us to prove weak-strong uniqueness and to show
convergence to a Cahn-Hilliard system in the large friction limit. We also use
relative energy to show that solutions of Euler-Korteweg with convex energy
converge to solutions of the Euler system in the vanishing capillarity limit,
as long as the latter admits sufficiently regular strong solutions
Existence of global strong solutions for the shallow-water equations with large initial data
This work is devoted to the study of a viscous shallow-water system with
friction and capillarity term. We prove in this paper the existence of global
strong solutions for this system with some choice of large initial data when
in critical spaces for the scaling of the equations. More precisely,
we introduce as in \cite{Hprepa} a new unknown,\textit{a effective velocity}
v=u+\mu\n\ln h ( is the classical velocity and the depth variation of
the fluid) with the viscosity coefficient which simplifies the system and
allow us to cancel out the coupling between the velocity and the depth
variation . We obtain then the existence of global strong solution if
is small in and large in
. In particular it implies that the classical momentum
can be large in , but small when we
project on the divergence field. These solutions are in some sense
\textit{purely compressible}. We would like to point out that the friction term
term has a fundamental role in our work inasmuch as coupling with the pressure
term it creates a damping effect on the effective velocity
Highly rotating viscous compressible fluids in presence of capillarity effects
In this paper we study a singular limit problem for a Navier-Stokes-Korteweg
system with Coriolis force, in the domain and for general
ill-prepared initial data. Taking the Mach and the Rossby numbers to be
proportional to a small parameter \veps going to , we perform the
incompressible and high rotation limits simultaneously. Moreover, we consider
both the constant capillarity and vanishing capillarity regimes. In this last
case, the limit problem is identified as a -D incompressible Navier-Stokes
equation in the variables orthogonal to the rotation axis. If the capillarity
is constant, instead, the limit equation slightly changes, keeping however a
similar structure. Various rates at which the capillarity coefficient can
vanish are also considered: in most cases this will produce an anisotropic
scaling in the system, for which a different analysis is needed. The proof of
the results is based on suitable applications of the RAGE theorem.Comment: Version 2 includes a corrigendum, which fixes errors contained in the
proofs to Theorems 6.5 and 6.
Relative entropy in multi-phase models of 1d elastodynamics: Convergence of a non-local to a local model
In this paper we study a local and a non-local regularization of the system
of nonlinear elastodynamics with a non-convex energy. We show that solutions of
the non-local model converge to those of the local model in a certain regime.
The arguments are based on the relative entropy framework and provide an
example how local and non-local regularizations may compensate for
non-convexity of the energy and enable the use of the relative entropy
stability theory -- even if the energy is not quasi- or poly-convex
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