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Probabilistic Approximation of a Nonlinear Parabolic Equation Occurring in Rheology

  • Mohamed Ben Alaya (a1) and Benjamin Jourdain (a2)

Abstract

In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

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Copyright

Corresponding author

Postal address: LAGA, Institut Galilée, Université Paris 13, 93430 Villetaneuse, France. Email address: mba@math.univ-paris13.fr
∗∗ Postal address: CERMICS, École des Ponts, ParisTech, 6–8 avenue Blaise Pascal, 77455 Marne la Vallée, France. Email address: jourdain@cermics.enpc.fr

References

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[7] Méléard, S. (1995). Asymptotic behaviour of some interacting particle systems; McKean–Vlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations (Lecture Notes Math. 1627; Montecatini Terme, 1995), Springer, Berlin, pp. 4295.
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Probabilistic Approximation of a Nonlinear Parabolic Equation Occurring in Rheology

  • Mohamed Ben Alaya (a1) and Benjamin Jourdain (a2)

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