51 research outputs found
Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities
We consider a diffuse interface model for an incompressible isothermal
mixture of two viscous Newtonian fluids with different densities in a bounded
domain in two or three space dimensions. The model is the nonlocal version of
the one recently derived by Abels, Garcke and Gr\"{u}n and consists in a
Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard
equation. The density of the mixture depends on an order parameter. For this
nonlocal system we prove existence of global dissipative weak solutions for the
case of singular double-well potentials and non degenerate mobilities. To this
goal we devise an approach which is completely independent of the one employed
by Abels, Depner and Garcke to establish existence of weak solutions for the
local Abels et al. model.Comment: 43 page
Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems
A well-known diffuse interface model for incompressible isothermal mixtures
of two immiscible fluids consists of the Navier-Stokes system coupled with a
convective Cahn-Hilliard equation. In some recent contributions the standard
Cahn-Hilliard equation has been replaced by its nonlocal version. The
corresponding system is physically more relevant and mathematically more
challenging. Indeed, the only known results are essentially the existence of a
global weak solution and the existence of a suitable notion of global attractor
for the corresponding dynamical system defined without uniqueness. In fact,
even in the two-dimensional case, uniqueness of weak solutions is still an open
problem. Here we take a step forward in the case of regular potentials. First
we prove the existence of a (unique) strong solution in two dimensions. Then we
show that any weak solution regularizes in finite time uniformly with respect
to bounded sets of initial data. This result allows us to deduce that the
global attractor is the union of all the bounded complete trajectories which
are strong solutions. We also demonstrate that each trajectory converges to a
single equilibrium, provided that the potential is real analytic and the
external forces vanish.Comment: 30 page
Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D
We study a diffuse interface model for incompressible isothermal mixtures of
two immiscible fluids coupling the Navier--Stokes system with a convective
nonlocal Cahn--Hilliard equation in two dimensions of space. We apply recently
proved well-posedness and regularity results in order to establish existence of
optimal controls as well as first-order necessary optimality conditions for an
associated optimal control problem in which a distributed control is applied to
the fluid flow.Comment: 32 page
On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials
We consider a model describing the evolution of a tumor inside a host tissue
in terms of the parameters , (proliferating and dead
cells, respectively), (cell velocity) and (nutrient concentration). The
variables , satisfy a Cahn-Hilliard type system with
nonzero forcing term (implying that their spatial means are not conserved in
time), whereas obeys a form of the Darcy law and satisfies a
quasistatic diffusion equation. The main novelty of the present work stands in
the fact that we are able to consider a configuration potential of singular
type implying that the concentration vector is
constrained to remain in the range of physically admissible values. On the
other hand, in view of the presence of nonzero forcing terms, this choice gives
rise to a number of mathematical difficulties, especially related to the
control of the mean values of and . For the resulting
mathematical problem, by imposing suitable initial-boundary conditions, our
main result concerns the existence of weak solutions in a proper regularity
class.Comment: 41 page
Nonlocal Cahn-Hilliard-Hele-Shaw systems with singular potential and degenerate mobility
We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an
incompressible mixture of two fluids. The relative concentration difference
is governed by a convective nonlocal Cahn-Hilliard equation with
degenerate mobility and logarithmic potential. The volume averaged fluid
velocity obeys a Darcy's law depending on the so-called Korteweg
force , where is the nonlocal chemical potential. In
addition, the kinematic viscosity may depend on . We establish
first the existence of a global weak solution which satisfies the energy
identity. Then we prove the existence of a strong solution. Further regularity
results on the pressure and on are also obtained. Weak-strong
uniqueness is demonstrated in the two dimensional case. In the
three-dimensional case, uniqueness of weak solutions holds if is
constant. Otherwise, weak-strong uniqueness is shown by assuming that the
pressure of the strong solution is -H\"{o}lder continuous in space for
.Comment: 70 page
A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility
We consider a diffuse interface model for incompressible isothermal
mixtures of two immiscible fluids with matched constant densities. This model
consists of the Navier-Stokes system coupled with a convective nonlocal
Cahn-Hilliard equation with non-constant mobility. We first prove the
existence of a global weak solution in the case of non-degenerate mobilities
and regular potentials of polynomial growth. Then we extend the result to
degenerate mobilities and singular (e.g. logarithmic) potentials. In the
latter case we also establish the existence of the global attractor in
dimension two. Using a similar technique, we show that there is a global
attractor for the convective nonlocal Cahn-Hilliard equation with degenerate
mobility and singular potential in dimension three
Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential
In this paper, we consider a two-dimensional diffuse interface model
for the phase separation of an incompressible and isothermal binary fluid
mixture with matched densities. This model consists of the NavierStokes
equations, nonlinearly coupled with a convective nonlocal CahnHilliard
equation. The system rules the evolution of the (volume-averaged) velocity u
of the mixture and the (relative) concentration difference ' of the two
phases. The aim of this work is to study an optimal control problem for such
a system, the control being a time-dependent external force v acting on the
fluid. We first prove the existence of an optimal control for a given
tracking type cost functional. Then we study the differentiability properties
of the control-to-state map v 7! [u; '], and we establish first-order
necessary optimality conditions. These results generalize the ones obtained
by the first and the third authors jointly with E. Rocca in [19]. There the
authors assumed a constant mobility and a regular potential with polynomially
controlled growth. Here, we analyze the physically more relevant case of a
degenerate mobility and a singular (e.g., logarithmic) potential. This is
made possible by the existence of a unique strong solution which was recently
proved by the authors and C. G. Gal in [14]
A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility
We consider a diffuse interface model for incompressible isothermal mixtures
of two immiscible fluids with matched constant densities. This model consists
of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard
equation with non-constant mobility. We first prove the existence of a global
weak solution in the case of non-degenerate mobilities and regular potentials
of polynomial growth. Then we extend the result to degenerate mobilities and
singular (e.g. logarithmic) potentials. In the latter case we also establish
the existence of the global attractor in dimension two. Using a similar
technique, we show that there is a global attractor for the convective nonlocal
Cahn-Hilliard equation with degenerate mobility and singular potential in
dimension three.Comment: 47 page
Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D
We study a diffuse interface model for incompressible isothermal
mixtures of two immiscible fluids coupling the Navier-Stokes system with a
convective nonlocal Cahn-Hilliard equation in two dimensions of space. We
apply recently proved well-posedness and regularity results in order to
establish existence of optimal controls as well as first-order necessary
optimality conditions for an associated optimal control problem in which a
distributed control is applied to the fluid flow
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