57 research outputs found
Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis
We consider a diffuse interface model for tumor growth consisting of a
Cahn--Hilliard equation with source terms coupled to a reaction-diffusion
equation, which models a tumor growing in the presence of a nutrient species
and surrounded by healthy tissue. The well-posedness of the system equipped
with Neumann boundary conditions was found to require regular potentials with
quadratic growth. In this work, Dirichlet boundary conditions are considered,
and we establish the well-posedness of the system for regular potentials with
higher polynomial growth and also for singular potentials. New difficulties are
encountered due to the higher polynomial growth, but for regular potentials, we
retain the continuous dependence on initial and boundary data for the chemical
potential and for the order parameter in strong norms as established in the
previous work. Furthermore, we deduce the well-posedness of a variant of the
model with quasi-static nutrient by rigorously passing to the limit where the
ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is
small.Comment: 33 pages, minor typos corrected, accepted versio
On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition
We study a coupled bulk-surface Allen-Cahn system with an affine linear
transmission condition, that is, the trace values of the bulk variable and the
values of the surface variable are connected via an affine relation, and this
serves to generalize the usual dynamic boundary conditions. We tackle the
problem of well-posedness via a penalization method using Robin boundary
conditions. In particular, for the relaxation problem, the strong
well-posedness and long-time behavior of solutions can be shown for more
general and possibly nonlinear relations. New difficulties arise since the
surface variable is no longer the trace of the bulk variable, and uniform
estimates in the relaxation parameter are scarce. Nevertheless, weak
convergence to the original problem can be shown. Using the approach of Colli
and Fukao (Math. Models Appl. Sci. 2015), we show strong existence to the
original problem with affine linear relations, and derive an error estimate
between solutions to the relaxed and original problems.Comment: 34 page
Convergence to equilibrium for a bulk--surface Allen--Cahn system coupled through a Robin boundary condition
We consider a coupled bulk--surface Allen--Cahn system affixed with a
Robin-type boundary condition between the bulk and surface variables. This
system can also be viewed as a relaxation to a bulk--surface Allen--Cahn system
with non-trivial transmission conditions. Assuming that the nonlinearities are
real analytic, we prove the convergence of every global strong solution to a
single equilibrium as time tends to infinity. Furthermore, we obtain an
estimate on the rate of convergence. The proof relies on an extended
Lojasiewicz--Simon type inequality for the bulk--surface coupled system.
Compared with previous works, new difficulties arise as in our system the
surface variable is no longer the trace of the bulk variable, but now they are
coupled through a nonlinear Robin boundary condition.Comment: 29 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
Phase field modelling of surfactants in multi-phase flow
A diffuse interface model for surfactants in multi-phase flow with three or
more fluids is derived. A system of Cahn-Hilliard equations is coupled with a
Navier-Stokes system and an advection-diffusion equation for the surfactant
ensuring thermodynamic consistency. By an asymptotic analysis the model can be
related to a moving boundary problem in the sharp interface limit, which is
derived from first principles. Results from numerical simulations support the
theoretical findings. The main novelties are centred around the conditions in
the triple junctions where three fluids meet. Specifically the case of local
chemical equilibrium with respect to the surfactant is considered, which allows
for interfacial surfactant flow through the triple junctions
Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach
We consider shape and topology optimization for fluids which are governed by
the Navier--Stokes equations. Shapes are modelled with the help of a phase
field approach and the solid body is relaxed to be a porous medium. The phase
field method uses a Ginzburg--Landau functional in order to approximate a
perimeter penalization. We focus on surface functionals and carefully introduce
a new modelling variant, show existence of minimizers and derive first order
necessary conditions. These conditions are related to classical shape
derivatives by identifying the sharp interface limit with the help of formally
matched asymptotic expansions. Finally, we present numerical computations based
on a Cahn--Hilliard type gradient descent which demonstrate that the method can
be used to solve shape optimization problems for fluids with the help of the
new approach
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