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Guide Models of Phase Transitions (Progress in Nonlinear Differential Equations and Their Applications)

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A recent review of alternative energy density functions for interface tracking applications has proposed a modified form of the double-obstacle function which avoids the spontaneous drop shrinkage phenomena and limits on mobility, [14] with comparative results provide for a number of benchmark simulations using the double-well function and the VOF sharp interface technique. A phase field model can be constructed to purposely reproduce a given interfacial dynamics as represented by a sharp interface model. In such a case the sharp interface limit i.

These expansions are performed both in the interfacial region inner expansion and in the bulk outer expansion , and then are asymptotically matched order by order. The result gives a partial differential equation for the diffusive field and a series of boundary conditions at the interface, which should correspond to the sharp interface model and whose comparison with it provides the values of the parameters of the phase field model.

For example, this technique has permitted to cancel kinetic effects, [11] to treat cases with unequal diffusivities in the phases, [15] to model viscous fingering [2] and two-phase Navier—Stokes flows, [16] to include fluctuations in the model, [17] etc. In multi-phase field models, microstructure is described by set of order parameters, each of which is related to a specific phase or crystallographic orientation. This model is mostly used for solid state phase transformations where multiple grains evolve e. Besides allowing the description of multiple grains in a microstructure, multi-phase field models especially allow for consideration of multiple thermodynamic phases occurring e.

From Wikipedia, the free encyclopedia. Annual Review of Materials Research. Numerical study". Physical Review E.

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Bibcode : PhRvE.. April Journal of the Mechanics and Physics of Solids. Interfaces and Free Boundaries : — Journal of Elasticity.

Physical Review Letters. Bibcode : PhRvL.. Fasano and M. Primicerio, p. Directions in Condensed Matter Physics. Series on Directions in Condensed Matter Physics. Singapore: World Scientific. Reviews of Modern Physics. Bibcode : RvMP Journal of Computational Physics. Bibcode : JCoPh. International Journal of Multiphase Flow. The vector quasi-static Stefan problem. I Math. Private communication. David Gilbarg and Neil S.


  1. Gieri Simonett?
  2. Progress in Nonlinear Differential Equations and Their Applications.
  3. Guidetti , Lorenzi : A mixed type identification problem related to a phase-field model with memory?

Trudinger , Elliptic partial differential equations of second order , 2nd ed. John E. Partial Differential Equations 10 , no. Differential Geom. Robert V. Kohn and Peter Sternberg , Local minimisers and singular perturbations , Proc. Edinburgh Sect. A , no. Luciano Modica , The gradient theory of phase transitions and the minimal interface criterion , Arch.

Luciano Modica , Monotonicity of the energy for entire solutions of semilinear elliptic equations , Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl. Lecture notes on general relativity. Stanford University, Peter Sternberg , The effect of a singular perturbation on nonconvex variational problems , Arch. Peter Sternberg , Vector-valued local minimizers of nonconvex variational problems , Rocky Mountain J. Current directions in nonlinear partial differential equations Provo, UT, Variational methods for phase transitions, an approach via -convergence.

Ambrosio and N. Dancer, Calculus of variations and partial differential equations , edited by G. Buttazzo, A. Marino, M. Springer, Berlin, , pp.

Some highlights

MR 2. On the connection problem for potentials with several global minima. MR h 4. Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. MR 91j 5. Bethuel, H. Brezis, and F. Ginzburg-Landau vortices. MR 95c 6. Brezis, and G. Asymptotics of the Ginzburg-Landau equation in arbitrary dimensions.

References

MR ka 7. Bronsard and F. On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation. MR 94h 9. Caffarelli, N. Garofalo, and F. A gradient bound for entire solutions of quasi-linear equations and its consequences. MR 95k De Giorgi and T. Su un tipo di convergenza variazionale. Atti Accad. Regularity theory for mean curvature flow. MR b Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in.

I, Math. Two results on entire solutions of Ginzburg-Landau system in higher dimensions.

Linearize a Differential Equation

MR g Gilbarg and N. Elliptic partial differential equations of second order. Springer-Verlag, Berlin, second edition, MR 86c