1,721,128 research outputs found
Singular limits of a conserved Penrose-Fife phase field model with special heat flux laws and memory effects
No full textA phase-field model of Penrose-Fife type for diffusive phase transitions with conserved order parameter is introduced. A Cauchy-Neumann problem is considered for the related parabolic system which couples a nonlinear Volterra integro-differential equation for the temperature T with a fourth order relation describing the evolution of the phase variable h. The latter equation contains a relaxation parameter w related to the speed of the transition process, which happens to be very small in the applications. Existence and uniqueness for this model as w>0 have been recently proved by the first author. Here, the asymptotic behaviour of the model is studied as w is let tend to zero. By a priori estimates and compactness arguments, the convergence of the solutions is shown. The approximating initial data have to be properly chosen. The problem obtained at the limit turns out to couple the original energy balance equation with an elliptic fourth order inclusion
Asymptotic analysis of a conserved phase-field model with memory for vanishing time relaxation
No full text
The conserved Penrose–Fife system with temperature-dependent memory
No full textAbstractA nonlinear parabolic system of Penrose–Fife type with a singular evolution term, arising from modelling dynamic phenomena of the nonisothermal diffusive phase separation, is studied. Here, we consider the evolution of a material in which the heat flux is a superposition of two different contributions: one part is proportional to the spacial gradient of the inverse of the absolute temperature ϑ, while the other agrees with the Gurtin–Pipkin law, introduced in the theory of materials with thermal memory. The phase transition here is described through the evolution of the conserved order parameter χ, which may represent the density or concentration of some substance. It is shown that an initial-boundary value problem for the resulting state equations has a unique solution
On the long-time behavior of some mathematical models for nematic liquid crystals
Full text linkA model describing the evolution of a liquid crystal substance in the nematic
phase is investigated in terms of two basic state variables: the velocity field u and the director field d, representing the preferred orientation of molecules in a neighborhood of any point in a reference domain. After recalling a known existence result, we investigate the long-time behavior of weak solutions. In particular, we show that any solution trajectory admits a non-empty ω-limit set containing only stationary solutions. Moreover, we give a number of sufficient conditions in order that the ω-limit set contains a single point. Our approach
improves and generalizes existing results on the same problem
Energy-variational solutions for viscoelastic fluid models
No full textIn this article, we introduce the concept of energy-variational solutions for a class of nonlinear dissipative evolutionary equations, which turns out to be especially suited to treat viscoelastic fluid models. Under certain convexity assumptions, the existence of such solutions can be shown constructively by an adapted minimizing movement scheme in a general framework. Weak-strong uniqueness follows by a suitable relative energy inequality. Our main motivation is to apply the general framework to viscoelastic fluid models. Therefore, we give a short overview on different versions of such models and their derivation. The abstract result is applied to two of these viscoelastic fluid models in full detail. In the conclusion, we comment on further applications of the general theory and its possible impact
Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids
Full text linkWe consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. The model was recently introduced in [11] where existence of weak solutions was proved in three space dimensions. Here, we aim to study the properties of solutions in the two-dimensional case. In particular, we can show existence of global in time solutions satisfying a stronger formulation of the model with respect to the one considered in [11]
The Oberbeck–Boussinesq approximation and Rayleigh–Bénard convection revisited
No full textWe consider the Oberbeck–Boussinesq approximation driven by an inhomogeneous temperature distribution on the boundary of a bounded fluid domain. The relevant boundary conditions are perturbed by a non–local term arising in the incompressible limit of the Navier–Stokes–Fourier system. The long time behaviour of the resulting initial/boundary value problem is investigated
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