112 research outputs found
Unsaturated deformable porous media flow with phase transition
In the present paper, a continuum model is introduced for fluid flow in a
deformable porous medium, where the fluid may undergo phase transitions.
Typically, such problems arise in modeling liquid-solid phase transformations
in groundwater flows. The system of equations is derived here from the
conservation principles for mass, momentum, and energy and from the
Clausius-Duhem inequality for entropy. It couples the evolution of the
displacement in the matrix material, of the capillary pressure, of the absolute
temperature, and of the phase fraction. Mathematical results are proved under
the additional hypothesis that inertia effects and shear stresses can be
neglected. For the resulting highly nonlinear system of two PDEs, one ODE and
one ordinary differential inclusion with natural initial and boundary
conditions, existence of global in time solutions is proved by means of cut-off
techniques and suitable Moser-type estimates
Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions
We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous
Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions
We show that sweeping processes with possibly non-convex prox-regular
constraints generate a strongly continuous input-output mapping in the space of
absolutely continuous functions. Under additional smoothness assumptions on the
constraint we prove the local Lipschitz continuity of the input-output mapping.
Using the Banach contraction principle, we subsequently prove that also the
solution mapping associated with the state-dependent problem is locally
Lipschitz continuous.Comment: Changes: p. 2 line 10; p. 5 lines 1 to 6; p. 9 line -1;
Acknowledgment section; New References [3] and [23
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A vanishing diffusion limit in a nonstandard system of phase field equations
We are concerned with a nonstandard phase field model of CahnHilliard
type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006),
describes two-species phase segregation and consists of a system of two
highly nonlinearly coupled PDEs. It has been recently investigated by Colli,
Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in
particular, SIAM J. Appl. Math. 2011, and Boll. Unione Mat. Ital. 2012. In
the latter contribution, the authors can treat the very general case in which
the diffusivity coefficient of the parabolic PDE is allowed to depend
nonlinearly on both variables. In the same framework, this paper investigates
the asymptotic limit of the solutions to the initial-boundary value problems
as the diffusion coefficient ơ in the equation governing the evolution of
the order parameter tends to zero. We prove that such a limit actually exists
and solves the limit problem, which couples a nonlinear PDE of parabolic type
with an ODE accounting for the phase dynamics. In the case of a constant
diffusivity, we are able to show uniqueness and to improve the regularity of
the solution
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