121 research outputs found
Stochastic homogenization of -convex gradient flows
In this paper we present a stochastic homogenization result for a class of
Hilbert space evolutionary gradient systems driven by a quadratic dissipation
potential and a -convex energy functional featuring random and rapidly
oscillating coefficients. Specific examples included in the result are
Allen-Cahn type equations and evolutionary equations driven by the -Laplace
operator with . The homogenization procedure we apply is based
on a stochastic two-scale convergence approach. In particular, we define a
stochastic unfolding operator which can be considered as a random counterpart
of the well-established notion of periodic unfolding. The stochastic unfolding
procedure grants a very convenient method for homogenization problems defined
in terms of (-)convex functionals.Comment: arXiv admin note: text overlap with arXiv:1805.0954
Stochastic unfolding and homogenization
The notion of periodic two-scale convergence and the method of periodic
unfolding are prominent and useful tools in multiscale modeling and analysis of
PDEs with rapidly oscillating periodic coefficients. In this paper we are
interested in the theory of stochastic homogenization for continuum mechanical
models in form of PDEs with random coefficients, describing random
heterogeneous materials. The notion of periodic two-scale convergence has been
extended in different ways to the stochastic case. In this work we introduce a
stochastic unfolding method that features many similarities to periodic
unfolding. In particular it allows to characterize the notion of stochastic
two-scale convergence in the mean by mere weak convergence in an extended
space. We illustrate the method on the (classical) example of stochastic
homogenization of convex integral functionals, and prove a new result on
stochastic homogenization for a non-convex evolution equation of Allen-Cahn
type. Moreover, we discuss the relation of stochastic unfolding to previously
introduced notions of (quenched and mean) stochastic two-scale convergence. The
method described in the present paper extends to the continuum setting the
notion of discrete stochastic unfolding, as recently introduced by the second
and third author in the context of discrete-to-continuum transition.Comment: 46 page
Stochastic unfolding and homogenization
The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogenization result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method introduced in this paper extends extitdiscrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition
Stochastic unfolding and homogenization
The notion of periodic two-scale convergence and the method of periodic un- folding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coe cients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coe cients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in di erent ways to the stochastic case. In this work we introduce a stochastic unfolding method that fea- tures many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogeniza- tion result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method descibed in the present paper extends to the continuum setting the notion of discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition
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Stochastic two-scale convergence and Young measures
In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence
Stochastic homogenization of Λ-convex gradient flows
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the p-Laplace operator with p∈(1,∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals
Stochastic homogenization of Lambda-convex gradient flows
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals
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