Front Propagation and Phase Field Theory

Barles, Guy; Soner, H. Meté; Sougandis, P. E.

doi:10.1137/0331021

Cited by 306 publications

(380 citation statements)

References 47 publications

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“…In particular, F is geometric (see Barles, Soner and Souganidis [12]), because M → F (M, p) is linear and…”

Section: Resultsmentioning

confidence: 99%

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Lio¹,

Forcadel²,

Monneau³

2008

J. Eur. Math. Soc.

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Abstract. We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.

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“…In particular, F is geometric (see Barles, Soner and Souganidis [12]), because M → F (M, p) is linear and…”

Section: Resultsmentioning

confidence: 99%

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Lio¹,

Forcadel²,

Monneau³

2008

J. Eur. Math. Soc.

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“…The theory of viscosity solutions has been first introduced by Crandall, Lions [19]. We also refer to the article of Barles, Soner, Souganidis [8] and to its references for a detailed presentation of the theory of geometric equations.…”

Section: Resultsmentioning

confidence: 99%

“…When the initial data is discontinuous, the natural notion of uniqueness is that any two discontinuous solutions must have the same semicontinuous envelopes because the definition of viscosity solutions only view these envelopes. But, as is well-known, uniqueness in this sense may be untrue for non-stationary Hamilton-Jacobi equations (see Barles, Soner, Souganidis [8]). This issue will be discussed in a more detailed manner later in the context of geometric equations.…”

Section: Solvability Of Local Hamilton-jacobi Equationsmentioning

confidence: 99%

“…However, after a certain time, the hypersurface may loose its smoothness and develop topological singularities (such as the appearance of an interior). The phase field method, which was introduced by Barles, Soner, Souganidis [8] in the context of Hamilton-Jacobi equations as a limiting case of the level set approach, allows to define an evolution that is global in time by exploiting the theory of discontinuous viscosity solutions. It relies on the extension to an arbitrary closed set of the notion of an oriented hypersurface.…”

Section: Geometric Viscosity Solutionsmentioning

confidence: 99%

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Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

Alvarez

Hoch

Bouar

et al. 2006

Arch Rational Mech Anal

117

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We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation whose velocity is a non-local quantity depending on the whole shape of the dislocation line. We study the special cases where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for HamiltonJacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.

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“…Step 1 yields, a<(t;s,y) = J p(t,x;s,y)[n ( (t;dx) + i(^(t, Hence (7.1) holds for all R > e. 4. In this step we study the case 0 <R<e. (3.2) yields, that for 0 < r < T, »<(r;B R (x 0 )) = Since R < 6, i?^" 1 < ii^"" 1 .…”

Section: Clearing-outmentioning

confidence: 99%

Convergence of the Phase-Field Equations to the Mullins-Sekerka Problem with Kinetic Undercooling

Soner

1999

Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids

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I prove that the solutions of the phase field equations, on a subsequence, converge to a weak solution of the Mullins-Sekerka problem with kinetic undercooling. The method is based on energy estimates, a monotonicity formula, and the equipartition of the energy at each time. I also show that the limiting interface is (d -1)-rectifiable for almost all t with a square integrable mean curvature vector.

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Front Propagation and Phase Field Theory

Cited by 306 publications

References 47 publications

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

Convergence of the Phase-Field Equations to the Mullins-Sekerka Problem with Kinetic Undercooling

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