Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales

Woukeng, Jean Louis

doi:10.1007/s10231-009-0112-y

Cited by 30 publications

(29 citation statements)

References 5 publications

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“…a i (x; t; x="; t="; v 0 (x; t); v(x; t)) on Q, as an element of L 1 (Q), denoted by a " i ( ; v 0 ; v). We have the following result whose proof is exactly the same as that of [45,Proposition 3.1] and is therefore omitted. …”

Section: Y;mentioning

confidence: 99%

Generalized Besicovitch spaces and applications to deterministic homogenization

Sango

Svanstedt

Woukeng

2011

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. The purposes of this paper is to introduce a framework which enables us to study nonlinear homogenization problems. The starting point for this work is the theory of algebras with mean value. Very often in physics, from very simple experimental data, one gets sometimes complicated structure phenomena. These phenomena are represented by functions which are permanent in mean, but complicated in detail, which functions are subject to verify a functional equation often nonlinear. The problem is therefore to give an interpretation of these phenomena by functions having the following qualitative properties: they are functions that represent a phenomenon on a large scale, which vary irregularly, undergoing many nonperiodic oscillations. In this work we study the qualitative properties of spaces of such functions, say generalized Besicovitch spaces, and we prove general compactness results related to these spaces. We then apply these results to study some new homogenization problems. One important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo monotone parabolic-type operator. We also give the answer to the question raised by Frid and Silva in their paper [26] (Homogenization of nonlinear pde's in the Fourier-Stieltjes algebras, SIAM J. Math. Anal., Vol. 41, No. 4, pp. 1589-1620 to know whether there exist or not ergodic algebras that are not subalgebras of the Fourier-Stieltjes algebra, and further we show that the theory of homogenization algebras by Nguetseng in [36] (Homogenization structures and applications I, Zeit. Anal. Anw., 22 (2003) 73-107) cannot handle weakly almost periodic homogenization problems.

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Section: Y;mentioning

confidence: 99%

Generalized Besicovitch spaces and applications to deterministic homogenization

Sango

Svanstedt

Woukeng

2011

Nonlinear Analysis: Theory, Methods & Applications

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“…We prove by means of very weak multiscale convergence [2] that the corrector 2 associated with the gradient for the second rapid spatial scale 2 actually vanishes. Already, in [3,4], it was observed that having more than one rapid temporal scale in parabolic problems does not generate a reiterated problem and in this paper we can see that nor does the addition of a spatial scale if it is contained in a coefficient that is multiplied with the time derivative of .…”

Section: Introductionmentioning

confidence: 84%

“…Parabolic homogenization problems for ≡ 1 have been studied for different combinations of spatial and temporal scales in several papers by means of techniques of two-scale convergence type with approaches related to the one first introduced in [5], see, for example, [2,3,[6][7][8], and in, for example, [9][10][11], techniques not of two-scale convergence type are applied. Concerning cases where, as in (1) above, we do not have ≡ 1, Nandakumaran and Rajesh [12] studied a nonlinear parabolic problem with the same frequency of oscillation in time and space, respectively, in the elliptic part of the equation and an operator oscillating in space with the same frequency appearing in the temporal differentiation term.…”

Section: Introductionmentioning

confidence: 99%

A Note on Parabolic Homogenization with a Mismatch between the Spatial Scales

Flodén

Holmbom

Lindberg

et al. 2013

Abstract and Applied Analysis

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We consider the homogenization of the linear parabolic problem ( / 2 ) ( , ) − ∇ ⋅ ( ( / 1 , / 2 1 )∇ ( , )) = ( , ) which exhibits a mismatch between the spatial scales in the sense that the coefficient ( / 1 , / 2 1 ) of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient ( / 2 ) of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

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“…It then generated a great number of research activities that increases over time as shown by the vast existing literature to date; see, e.g., [2,3,5,6,20,37,38,41] and the references therein. However, being strictly limited to periodic structures, it quickly showed its inadequacy as far as the non periodic phenomena are concerned, since in nature, few physical phenomena have in fact a periodic behaviour.…”

Section: Introductionmentioning

confidence: 99%

Stochastic two-scale convergence of an integral functional

Sango

Woukeng

2011

Asymptotic Analysis

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In this paper we discuss the concept of stochastic two-scale convergence, which is appropriate to solve coupledperiodic and stochastic homogenization problems. This concept is a combination of both well-known two-scale convergence and stochastic two-scale convergence in the mean schemes, and is a generalization of the said previous methods. By way of illustration we apply it to solve a homogenization problem related to an integral functional with convex integrand. This problematic relies on the notion of dynamical system which is our basic tool.

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Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales

Cited by 30 publications

References 5 publications

Generalized Besicovitch spaces and applications to deterministic homogenization

Generalized Besicovitch spaces and applications to deterministic homogenization

A Note on Parabolic Homogenization with a Mismatch between the Spatial Scales

Stochastic two-scale convergence of an integral functional

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