On the approximation of front propagation problems with nonlocal terms

Cardaliaguet, Pierre; Pasquignon, Denis

doi:10.1051/m2an:2001120

Cited by 21 publications

(59 citation statements)

References 21 publications

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“…This work is inspired by [8] where Barles and Souganidis proved a general convergence result for monotone, stable and consistent schemes in the local framework. We also refer to the works of Cardaliaguet and Pasquignon [11] and Slepčev [21] on the approximation of moving fronts in the nonlocal but monotone case. This paper is organized as follows: in Section 2, we define a class of approximation schemes and prove the general convergence result.…”

Section: H[χ](x T P A) ≤ H[χ](x T P B) If a ≤ Bmentioning

confidence: 99%

Convergence of approximation schemes for nonlocal front propagation equations

Monteillet

2010

Math. Comp.

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Section: H[χ](x T P A) ≤ H[χ](x T P B) If a ≤ Bmentioning

confidence: 99%

Convergence of approximation schemes for nonlocal front propagation equations

Monteillet

2010

Math. Comp.

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“…To verify this, we further show that if ∂E 0 is a smooth hypersurface, then there is a unique smooth solution for small times of the evolution law (1.1), and that any minimizing movement E coincides with this smooth evolution as long as the latter exists. This uses the notions of lower/upper limits mentioned above and of sub/super pairs of solutions of Cardaliaguet and Pasquignon [14].…”

Section: {U(· T) > 0} ⊂ E(t) ⊂ {U(· T) ≥ 0}mentioning

confidence: 99%

“…Following Cardaliaguet [12], we formulate this statement in terms of test functions: let us first define the classical mean curvature operator…”

Section: Velocity Of E * and E *mentioning

confidence: 99%

Minimizing movements for dislocation dynamics with a mean curvature term

Forcadel¹,

Monteillet²

2009

ESAIM: COCV

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Abstract. We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.Mathematics Subject Classification. 53C44, 49Q15, 49L25, 28A75, 58A25.

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“…By convention, if u ∈ H 1 0 (Ω), then we extend u by setting u = 0 on R N \ Ω. Let S be as in (6). For an open bounded subset Ω of R N such that S ⊂⊂ Ω, the capacity of Ω with respect to S is defined by…”

Section: Capacity and Capacity Potentialmentioning

confidence: 99%

“…Without loss of generality we can assume that K * ∩ ∂K r = {(t 0 , x 0 )}. Then by standard stability arguments (see [7]), one can find a sequence of smooth regular tubes K k r converging to K r in the C 1,b sense (see Section 2 for the definition), and sequences h k → 0 and…”

mentioning

confidence: 99%