139 research outputs found
Global and exponential attractors for the Penrose-Fife system
The Penrose-Fife system for phase transitions is addressed. Dirichlet
boundary conditions for the temperature are assumed. Existence of global and
exponential attractors is proved. Differently from preceding contributions,
here the energy balance equation is both singular at 0 and degenerate at
infinity. For this reason, the dissipativity of the associated dynamical
process is not trivial and has to be proved rather carefully
Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations
A doubly nonlinear parabolic equation of the form , complemented with initial and either Dirichlet or Neumann
homogeneous boundary conditions, is addressed. The two nonlinearities are given
by the maximal monotone function and by the derivative of a
smooth but possibly nonconvex potential ; is a known source. After
defining a proper notion of solution and recalling a related existence result,
we show that from any initial datum emanates at least one solution which gains
further regularity for . Such "regularizing solutions" constitute a
semiflow for which uniqueness is satisfied for strictly positive times and
we can study long time behavior properties. In particular, we can prove
existence of both global and exponential attractors and investigate the
structure of -limits of single trajectories
On a Navier-Stokes-Allen-Cahn model with inertial effects
A mathematical model describing the flow of two-phase fluids in a bounded
container is considered under the assumption that the phase transition
process is influenced by inertial effects. The model couples a variant of the
Navier-Stokes system for the velocity with an Allen-Cahn-type equation for
the order parameter relaxed in time in order to introduce inertia.
The resulting model is characterized by second-order material derivatives which
constitute the main difficulty in the mathematical analysis. Actually, in order
to obtain a tractable problem, a viscous relaxation term is included in the
phase equation. The mathematical results consist in existence of weak solutions
in 3D and, under additional assumptions, existence and uniqueness of strong
solutions in 2D. A partial characterization of the long-time behavior of
solutions is also given and in particular some issues related to dissipation of
energy are discussed.Comment: 24 page
On the viscous Cahn-Hilliard equation with singular potential and inertial term
We consider a relaxation of the viscous Cahn-Hilliard equation induced by the
second-order inertial term~. The equation also contains a semilinear
term of "singular" type. Namely, the function is defined only on a
bounded interval of corresponding to the physically admissible
values of the unknown , and diverges as approaches the extrema of that
interval. In view of its interaction with the inertial term , the term
is difficult to be treated mathematically. Based on an approach
originally devised for the strongly damped wave equation, we propose a suitable
concept of weak solution based on duality methods and prove an existence
result.Comment: 11 page
Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations
A nonlinear parabolic equation of the fourth order is analyzed. The equation
is characterized by a mobility coefficient that degenerates at 0. Existence of
at least one weak solution is proved by using a regularization procedure and
deducing suitable a-priori estimates. If a viscosity term is added and
additional conditions on the nonlinear terms are assumed, then it is proved
that any weak solution becomes instantaneously strictly positive. This in
particular implies uniqueness for strictly positive times and further
time-regularization properties. The long-time behavior of the problem is also
investigated and the existence of trajectory attractors and, under more
restrictive conditions, of strong global attractors is shown
Well-posedness and long-time behavior for a class of doubly nonlinear equations
This paper addresses a doubly nonlinear parabolic inclusion of the form
. Existence of a solution is proved under suitable
monotonicity, coercivity, and structure assumptions on the operators and
, which in particular are both supposed to be subdifferentials of
functionals on . Moreover, under additional hypotheses on ,
uniqueness of the solution is proved. Finally, a characterization of
-limit sets of solutions is given and we investigate the convergence of
trajectories to limit points
Singular limit of a transmission problem for the parabolic phase-field model
summary:A transmission problem describing the thermal interchange between two regions occupied by possibly different fluids, which may present phase transitions, is studied in the framework of the Caginalp-Fix phase field model. Dirichlet (or Neumann) and Cauchy conditions are required. A regular solution is obtained by means of approximation techniques for parabolic systems. Then, an asymptotic study of the problem is carried out as the time relaxation parameter for the phase field tends to 0 in one of the domains. It is also proved that the limit formulation admits a unique solution in a suitable weak sense
Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories
We consider a nonlinear reaction-diffusion equation settled on the whole
euclidean space. We prove the well-posedness of the corresponding Cauchy
problem in a general functional setting, namely, when the initial datum is
uniformly locally bounded in L^2. Then we adapt the short trajectory method to
establish the existence of the global attractor and, if the space dimension is
at most 3, we also find an upper bound of its Kolmogorov's entropy
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