Homogenization of a Wilson–Cowan model for neural fields

Svanstedt, Nils; Woukeng, Jean Louis

doi:10.1016/j.nonrwa.2012.11.006

Cited by 12 publications

(44 citation statements)

References 20 publications

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“…The set Q being arbitrary here, the fundamental difference between the previous proof, and the upcoming one are at the level of the proof of assertion (P2) in the succeeding text. Thus, the following proof generalizes the one in . Proof Appealing to , we observe that the sequence ( u ϵ * v ϵ ) ϵ > 0 is bounded in L m ( Q ). Now, let η > 0 and let

ψ_{0} MathClass-rel\in scriptK ((), {double-struckR}^{N} MathClass-punc; AP ((), {double-struckR}^{N}))

(the space of continuous functions from

{double-struckR}^{N}

into

AP ({double-struckR}^{N})

with compact support in

{double-struckR}^{N}

) be such that

{(∥∥, {\overset{v}{false}}_{0} MathClass-bin- {\overset{ψ}{false}}_{0})}_{L^{q} ({double-struckR}^{N} MathClass-bin\times scriptK)} MathClass-rel\leq \frac{η}{2}

.…”

Section: σ‐Convergence and Related Convolution Resultsmentioning

confidence: 64%

“…Assume that, as ϵ → 0 , u ϵ → u 0 in L p ( Q )‐weak Σ and v ϵ → v 0 in

L^{q} ((), {double-struckR}^{N})

‐strong Σ, where u 0 and v 0 are in

L^{p} ((), Q MathClass-punc; {scriptB}_{AP}^{p} ((), {double-struckR}^{N}))

and

L^{q} ((), {double-struckR}^{N} MathClass-punc; {scriptB}_{AP}^{q} ((), {double-struckR}^{N}))

, respectively. Then, as ϵ → 0,

u_{ϵ} MathClass-bin* v_{ϵ} MathClass-rel\to u_{0} MathClass-bin* MathClass-bin* v_{0} in L^{m} (Q) ‐weak Σ

Remark Before proving the result, let us first of all precise that, in the case where the open set Q is bounded and further p = 2 and q = 1, we have provided a full proof of this result in (see especially the proof of Theorem 2 therein). Although we will follow the same lines of reasoning as in , let us note however that the situation will be different in the succeeding text because in the proof of ,Theorem 2, we heavily rely on the assumption that Q is bounded.…”

Section: σ‐Convergence and Related Convolution Resultsmentioning

confidence: 91%

“…Moreover, the mapping

j MathClass-punc: {double-struckR}^{N} MathClass-rel\to scriptK

given by j ( y ) = δ y is a continuous group homomorphism satisfying the property

\overset{τ_{y} u}{false} MathClass-rel= τ_{j (y)} \overset{u}{false}, all u MathClass-rel\in AP ((), {double-struckR}^{N}) and all y MathClass-rel\in {double-struckR}^{N}

where

\overset{MathClass-bin\cdot}{false}

denotes the Gelfand transformation on

AP ((), {double-struckR}^{N})

, τ y u = u ( ⋅ + y ) and

τ_{j (y)} \overset{u}{false} MathClass-rel= \overset{u}{false} (MathClass-bin\cdot MathClass-bin+ δ_{y})

. Proof The fact that K can be provided with a group operation is classically known; see, for example, . For the proof of , see the proof of Proposition 1 in .□…”

Section: σ‐Convergence and Related Convolution Resultsmentioning

confidence: 99%

“…Theorem Let ( u ϵ ) ϵ > 0 be a sequence in L p ( Q ) ( 1 ≤ p < ∞ ), and let the sequence ( v ϵ ) ϵ > 0 be defined by

v_{ϵ} (x) MathClass-rel= u_{ϵ} (x MathClass-bin+ t) (x MathClass-rel\in Q MathClass-bin- t)

Finally, let

u_{0} MathClass-rel\in L^{p} ((), Q MathClass-punc; {scriptB}_{AP}^{p} ((), {double-struckR}^{N}))

and assume that u ϵ → u 0 in L p ( Q )‐weak Σ. If holds true, then

v_{ϵ} MathClass-rel\to v_{0} in L^{p} (Q MathClass-bin- t) ‐weak Σ

where

v_{0} MathClass-rel\in L^{p} ((), Q MathClass-bin- t MathClass-punc, {scriptB}_{AP}^{p} ((), {double-struckR}^{N}))

is defined by

{\overset{v}{false}}_{0} (x MathClass-punc, s) MathClass-rel= {\overset{u}{false}}_{0} (x MathClass-bin+ t MathClass-punc, s MathClass-bin+ r)

for

(x MathClass-punc, s) MathClass-rel\in (Q MathClass-bin- t) MathClass-bin\times scriptK

. Proof We recall that the aforementioned result has been proved in ,Proposition 3. However, we give a simplified proof here.…”

Section: σ‐Convergence and Related Convolution Resultsmentioning

confidence: 99%

“…The fact that K can be provided with a group operation is classically known; see, for example, [15]. For the proof of (4), see the proof of Proposition 1 in [10].…”

Section: Proofmentioning

confidence: 99%

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Linearized viscoelastic Oldroyd fluid motion in an almost periodic environment

Woukeng

2013

Math Methods in App Sciences

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In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes-Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our "-problem converges in L 2 to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma-convergence of convolution sequences.

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ψ_{0} MathClass-rel\in scriptK ((), {double-struckR}^{N} MathClass-punc; AP ((), {double-struckR}^{N}))

(the space of continuous functions from

{double-struckR}^{N}

into

AP ({double-struckR}^{N})

with compact support in

{double-struckR}^{N}

) be such that

{(∥∥, {\overset{v}{false}}_{0} MathClass-bin- {\overset{ψ}{false}}_{0})}_{L^{q} ({double-struckR}^{N} MathClass-bin\times scriptK)} MathClass-rel\leq \frac{η}{2}

.…”

Section: σ‐Convergence and Related Convolution Resultsmentioning

confidence: 64%

“…Assume that, as ϵ → 0 , u ϵ → u 0 in L p ( Q )‐weak Σ and v ϵ → v 0 in

L^{q} ((), {double-struckR}^{N})

‐strong Σ, where u 0 and v 0 are in

L^{p} ((), Q MathClass-punc; {scriptB}_{AP}^{p} ((), {double-struckR}^{N}))

and

L^{q} ((), {double-struckR}^{N} MathClass-punc; {scriptB}_{AP}^{q} ((), {double-struckR}^{N}))

, respectively. Then, as ϵ → 0,

u_{ϵ} MathClass-bin* v_{ϵ} MathClass-rel\to u_{0} MathClass-bin* MathClass-bin* v_{0} in L^{m} (Q) ‐weak Σ

Section: σ‐Convergence and Related Convolution Resultsmentioning

confidence: 91%

“…Moreover, the mapping

j MathClass-punc: {double-struckR}^{N} MathClass-rel\to scriptK

given by j ( y ) = δ y is a continuous group homomorphism satisfying the property

\overset{τ_{y} u}{false} MathClass-rel= τ_{j (y)} \overset{u}{false}, all u MathClass-rel\in AP ((), {double-struckR}^{N}) and all y MathClass-rel\in {double-struckR}^{N}

where

\overset{MathClass-bin\cdot}{false}

denotes the Gelfand transformation on

AP ((), {double-struckR}^{N})

, τ y u = u ( ⋅ + y ) and

τ_{j (y)} \overset{u}{false} MathClass-rel= \overset{u}{false} (MathClass-bin\cdot MathClass-bin+ δ_{y})

. Proof The fact that K can be provided with a group operation is classically known; see, for example, . For the proof of , see the proof of Proposition 1 in .□…”

Section: σ‐Convergence and Related Convolution Resultsmentioning

confidence: 99%

“…Theorem Let ( u ϵ ) ϵ > 0 be a sequence in L p ( Q ) ( 1 ≤ p < ∞ ), and let the sequence ( v ϵ ) ϵ > 0 be defined by

v_{ϵ} (x) MathClass-rel= u_{ϵ} (x MathClass-bin+ t) (x MathClass-rel\in Q MathClass-bin- t)

Finally, let

u_{0} MathClass-rel\in L^{p} ((), Q MathClass-punc; {scriptB}_{AP}^{p} ((), {double-struckR}^{N}))

and assume that u ϵ → u 0 in L p ( Q )‐weak Σ. If holds true, then

v_{ϵ} MathClass-rel\to v_{0} in L^{p} (Q MathClass-bin- t) ‐weak Σ

where

v_{0} MathClass-rel\in L^{p} ((), Q MathClass-bin- t MathClass-punc, {scriptB}_{AP}^{p} ((), {double-struckR}^{N}))

is defined by

{\overset{v}{false}}_{0} (x MathClass-punc, s) MathClass-rel= {\overset{u}{false}}_{0} (x MathClass-bin+ t MathClass-punc, s MathClass-bin+ r)

for

(x MathClass-punc, s) MathClass-rel\in (Q MathClass-bin- t) MathClass-bin\times scriptK

. Proof We recall that the aforementioned result has been proved in ,Proposition 3. However, we give a simplified proof here.…”

Section: σ‐Convergence and Related Convolution Resultsmentioning

confidence: 99%

“…The fact that K can be provided with a group operation is classically known; see, for example, [15]. For the proof of (4), see the proof of Proposition 1 in [10].…”

Section: Proofmentioning

confidence: 99%

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Linearized viscoelastic Oldroyd fluid motion in an almost periodic environment

Woukeng

2013

Math Methods in App Sciences

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Asymptotic behaviour of viscoelastic composites with almost periodic microstructures

Woukeng

2017

Math Methods in App Sciences

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In this paper, we study the acoustic properties of porous media saturated by an incompressible viscoelastic fluid. The model considered here consists of a linear deformable porous skeleton having memory that is surrounded by a viscoelastic Oldroyd fluid. Assuming the microstructures to be almost periodically distributed and under the almost periodicity hypothesis on the coefficients of the governing equations, we determine the macroscopic equivalent medium. To achieve our goal, we use some very recent tools about the sigma convergence of convolution sequences.Let Y D .0, 1/ N (N D 2 or 3 being fixed once for all here and in the sequel) be the reference cell, and let Y 1 and Y 2 be two open disjoint subsets of Y representing the local structure of the skeleton (which is a linear elastic material) and the local structure of the fluid, respectively. We assume that Y 1 Y, Y D Y 1 [ Y 2 , Y 2 is connected and that the boundary @Y 1 of Y 1 is Lipschitz continuous. Moreover, we assume that Y 1 and Y 2 have positive Lebesgue measure. Let S Z N be an infinite subset of Z N , and let

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