
Stress has been demonstrated to be a key modulator in learning and memory processes, in which the hippocampus plays a central role. A great number of neuropeptides have been reported to modulate learning and memory under stressful conditions. Neuropeptidases are proteolytic enzymes capable of regulating the function of neuropeptides in the central and peripheral nervous system. In this regard, a number of neuropeptidases, i.e. angiotensinases, oxytocinase, or enkephalinases, have received attention. Their involvement in stress and memory processes is a promising perspective, as it is possible to influence their activities through various activators or inhibitors and, consequently, to pharmacologically modulate the functions of the endogenous substrates that are involved. The present review describes the key findings showing the involvement of neuropeptides and neuropeptidases in stress and memory and highlights the role of the hippocampus in these processes.
Citation: I. Prieto, A.B. Segarra, M. de Gasparo, M. Ramírez-Sánchez. Neuropeptidases, Stress, and Memory—A Promising Perspective[J]. AIMS Neuroscience, 2016, 3(4): 487-501. doi: 10.3934/Neuroscience.2016.4.487
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Stress has been demonstrated to be a key modulator in learning and memory processes, in which the hippocampus plays a central role. A great number of neuropeptides have been reported to modulate learning and memory under stressful conditions. Neuropeptidases are proteolytic enzymes capable of regulating the function of neuropeptides in the central and peripheral nervous system. In this regard, a number of neuropeptidases, i.e. angiotensinases, oxytocinase, or enkephalinases, have received attention. Their involvement in stress and memory processes is a promising perspective, as it is possible to influence their activities through various activators or inhibitors and, consequently, to pharmacologically modulate the functions of the endogenous substrates that are involved. The present review describes the key findings showing the involvement of neuropeptides and neuropeptidases in stress and memory and highlights the role of the hippocampus in these processes.
We consider a system of reaction-diffusion equations in a domain
In the limit, the thin layer reduces to an interface
The rigorous derivation of interface conditions for multi-scale and multi-physics problems is a field that is just at the beginning. More and better analytical multi-scale tools are required to treat the arising nonlinear systems in many applications (e.g. biology, material sciences or geosciences, to mention just few of them). Our paper is one of the first steps in this process. The effective model derived here is a non-standard micro-macro strongly coupled system of nonlinear equations, involving the bulk-regions, the separating interface and the so called cell problem which takes care of the microscopic processes in the thin layer.
The derivation of effective interface conditions is based on two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the layer. Compared to previous contributions (see [14,15] for
The investigation of processes in domains separated by thin layers with periodic microstructure can also be found in elasticity problems, see e.g., [11,13] where the heterogeneous structure is described by periodically varying constitutive properties, or [12], where two domains separated by a thin layer made of periodic vertical beams are considered. Further applications can be encountered in fluid flow through thin filters, built up by an array of obstacles, see e.g., [3]. In [17], a reactive transport model with an additional convective contribution in the thin layer and a nonlinear transmission condition of Dirichlet type at the bulk-layer interface was considered. In the latter, however, a thin homogeneous layer was considered. In [4], long and horizontally arranged inclusions, only connected in one direction, were considered for a linear reaction-diffusion problem, and a concept of two-scale convergence adapted to this special structure was introduced.
This paper is organized as follows: In Section 2, we introduce the microscopic model, and establish existence and uniqueness of a weak solution. In Section 4, we derive estimates for the microscopic solutions necessary for the derivation of strong compactness results. The averaged function is defined in Section 5 and some basic properties are established. In Section 6 the averaging operator for thin domains is defined and the commuting property of the generalized time derivative and the unfolding operator is proved. Section 7 contains our main results. First, we prove compactness results for the microscopic solutions, especially the strong convergences in the thin layer. These are then used for the derivation of macroscopic models, including the effective interface conditions. In the A, we briefly recapitulate the concepts of two-scale convergence and unfolding operator for thin domains, together with related basic results.
We consider the domain
Ω+ϵ:=Σ×(ϵ,H),ΩMϵ:=Σ×(−ϵ,ϵ),Ω−ϵ:=Σ×(−H,−ϵ), |
see Figure 1. The domains
S+ϵ:=Σ×{ϵ} and S−ϵ:=Σ×{−ϵ}, |
hence, we have
Ω+:=Σ×(0,H) and Ω−:=Σ×(−H,0). |
To describe the microscopic structure of
Z:=Y×(−1,1) with Y:=(0,1)n−1, |
and denote the upper and lower boundary of
S+:=Y×{1}=(0,1)n−1×{1} and S−:=Y×{−1}=(0,1)n−1×{−1}. |
The vector valued functions
∂tu±i,ϵ−D±iΔu±i,ϵ=f+i(u±ϵ) in (0,T)×Ω±ϵ, | (1a) |
−D±i∇u±i,ϵ⋅ν=−h±i(u±ϵ,uMϵ) on (0,T)×S±ϵ, | (1b) |
−D±i∇u±i,ϵ⋅ν=0 on (0,T)×∂Ω±ϵ∖S±ϵ, | (1c) |
u±ϵ(0)=U±0 in Ω±ϵ, | (1d) |
for
1ϵ∂tuMi,ϵ−ϵγ∇⋅(DMi(xϵ)∇uMi,ϵ)=1ϵgi(xϵ,uMϵ) in (0,T)×ΩMϵ, | (1e) |
−ϵγDMi(⋅ϵ)∇uMi,ϵ⋅ν=−hM,±i(ˉxϵ,uMϵ,u±ϵ) on (0,T)×S±ϵ, | (1f) |
−ϵγDMi(⋅ϵ)∇uMi,ϵ⋅ν=0 on (0,T)×∂ΩMϵ∖S±ϵ, | (1g) |
uMϵ(0)=UM0(ˉ⋅,⋅xnϵ) in ΩMϵ, | (1h) |
for
Thin heterogeneous layers occur in many applications, e.g., biology and engineering, and for every specific situation appropriate transmission conditions between the layer domain and the bulk regions are required. In previous contributions (see [14,15] for
Assumptions on the data:
A1) For
A2) The function
|f±i(z)|≤C(1+|z|) for all z∈Rm. |
A3) The function
|gi(ˉy,yn,z)|≤C(1+|z|)for all (ˉy,yn,z)∈¯Y×[−1,1]×Rm. |
We shortly write
A4) The function
|h±(z±,zM)|≤C(1+|z±|+|zM|)∀ (z±,zM)∈Rm×Rm. |
A5) The function
|hM,±(ˉy,zM,z±)|≤C(1+|z±|+|zM|)∀ (ˉy,zM,z±)∈¯Y×Rm×Rm. |
A6) For the initial functions we assume
In the following, we denote for an arbitrary open set
Definition 2.1. We call
u±ϵ∈L2((0,T),H1(Ω±ϵ))m∩H1((0,T),H1(Ω±ϵ)′)m,uMϵ∈L2((0,T),H1(ΩMϵ))m∩H1((0,T),H1(ΩMϵ)′)m, |
and for all test-functions
⟨∂tu±i,ϵ,ϕ±⟩Ω±ϵ+D±i(∇u±i,ϵ,∇ϕ±)Ω±ϵ=(f±i(u±ϵ),ϕ±)Ω±ϵ+(h±i(u±ϵ,uMϵ),ϕ±)S±ϵ,1ϵ⟨∂tuMi,ϵ,ϕM⟩ΩMϵ+ϵγ(DMi(⋅ϵ)∇uMi,ϵ,∇ϕM)ΩMϵ=1ϵ(gi(⋅ϵ,uMϵ),ϕM)ΩMϵ+(hM,+i(⋅ˉxϵ,uMϵ,u+ϵ),ϕM)S+ϵ+(hM,−i(⋅ˉxϵ,uMϵ,u−ϵ),ϕM)S−ϵ, | (2) |
together with the initial conditions (1d) and (1h).
First of all, we establish the existence of a unique solution using a fix-point argument. The idea is standard, therefore we only give a short sketch of the proof.
Proposition 1. For every
Proof. Uniqueness follows by standard energy estimates. For the existence, we use Schäfer's fixed point theorem on the space
X:=X+×XM×X− |
with
L2((0,T),H1(Ω∗ϵ))∩H1((0,T),H1(Ω∗ϵ)′)↪L2((0,T),Hβ(Ω∗ϵ)) |
and similar estimates as in Lemma 4.2 below.
Our aim is to derive macroscopic approximations for the microscopic solutions
In this section, we point out main steps in this process, and indicate the challenging aspects together with our original contributions. Eventually, we present the macroscopic models (which differ for different values of the parameter
To pass to the asymptotic limit for
‖vϵ‖2Hϵ,γ:=1ϵ‖vϵ‖2L2(ΩMϵ)+ϵγ‖∇vϵ‖2L2(ΩMϵ). | (3) |
on
Our next step is to prove compactness results for the microscopic solutions, especially in the thin layer. We treat differently the cases
ˉuϵ(t,ˉx):=12ϵ∫ϵ−ϵuϵ(t,ˉx,xn)dxn. |
In Section 5, we provide estimates for the averaged function in thin domains and estimate the difference between the averaged function and the microscopic solution
In the critical case
Using the results from the previous sections, in Section 7 we give the proofs for our main results: The convergences of the microscopic solutions, see Theorem 7.1, 7.3, and 7.5, as well as the macroscopic models. The latter consist of equations in the bulk regions
Theorem 3.1. Let
u±0∈L2((0,T),H1(Ω±))m∩H1((0,T),H1(Ω±)′)m,uM0∈L2((0,T),H1(Σ))m∩H1((0,T),H1(Σ)′)m, |
and
∂tu±i,0−D±iΔu±i,0=f±i(u±0)in(0,T)×Ω±,−D±i∇u±i,0⋅ν=0on(0,T)×∂Ω±∖Σ,−D±i∇u±i,0⋅ν=−h±i(u±0,uM0)on(0,T)×Σ,u±0(t)=U±0inΩ±, |
for
|Z|∂tuMi,0−∇ˉx⋅(DM,∗i∇ˉxuMi,0)=∫Zgi(y,uM0(⋅t,⋅ˉx))dy+∑α∈{±}∫YhM,αi(ˉy,uM0(⋅t,⋅ˉx),uα0(⋅t,⋅ˉx,0))dˉyin(0,T)×Σ,−DMi∇yuMi,0⋅ν=0on(0,T)×∂Σ,uM0(0,⋅ˉx)=∫ZUM0(⋅ˉx,yn)dyinΣ, |
where the homogenized diffusion matrix
(DM,∗i)kl=∫ZDMi(y)(∇wi,k+ek)⋅(∇wi,l+el)dy, |
and the
−∇⋅(DMi(∇wi,j+ej))=0inZ,−DMi(∇wi,j+ej)⋅ν=0onS+∪S−,wi,jisY−periodicwith∫Zwi,jdy=0. | (4) |
Theorem 3.2. For
u±0∈L2((0,T),H1(Ω±))m∩H1((0,T),H1(Ω±)′)m,uM0∈H1((0,T),L2(Σ))m, |
and
∂tu±i,0−D±iΔu±i,0=f±i(u±0)in(0,T)×Ω±,−D±i∇u±i,0⋅ν=0on(0,T)×∂Ω±∖Σ,−D±i∇u±i,0⋅ν=−h±i(u±0,uM0)on(0,T)×Σ,u±0(0)=U±0inΩ±, |
for
|Z|∂tuMi,0=∫Zgi(y,uM0(⋅t,⋅ˉx))dy+∑α∈{±}∫YhM,αi(ˉy,uM0(⋅t,⋅ˉx),uα0(⋅t,⋅ˉx,0))dˉyin(0,T)×Σ,uM0(0,⋅ˉx)=∫1−1UM0(⋅ˉx,yn)dyninΣ. |
Theorem 3.3. Let
u±0∈L2((0,T),H1(Ω±))m∩H1((0,T),H1(Ω±)′)m,uM0∈L2((0,T),L2(Σ,Hper))m∩H1((0,T),L2(Σ,Hper)′)m, |
and
∂tu±i,0−D±iΔu±i,0=f±i(u±0)in(0,T)×Ω±,−D±i∇u±i,0⋅ν=0on(0,T)×∂Ω±∖Σ,−D±i∇u±i,0⋅ν=−∫Yh±i(u±0,uM0(⋅t,⋅ˉx,⋅ˉy,±1)dˉyon(0,T)×Σ,u±0(t)=U±0inΩ±, |
for
∂tuMi,0−∇y⋅(DMi∇yuMi,0)=gi(⋅y,uM0)in(0,T)×Σ×Z,−DMi∇yuMi,0⋅ν=−hM,±i(⋅ˉy,uM0(⋅t,⋅ˉx,⋅ˉy,±1),u±0(⋅t,⋅ˉx,0))on(0,T)×Σ×S±,−DMi∇yuMi,0⋅ν=0on(0,T)×Σ×∂±Z,uM0(0,⋅ˉx,⋅y)=UM0(⋅ˉx,⋅yn)inΣ×Z,uM0isY−periodicwithrespecttothelastvariable. |
Our aim is to derive macroscopic approximations for the microscopic solutions
Throughout this paper, we will frequently use the following trace estimate for thin domains.
Lemma 4.1. For
‖uϵ‖L2(S±ϵ)≤C(θ)√ϵ‖uϵ‖L2(ΩMϵ)+θ√ϵ‖∇uϵ‖L2(ΩMϵ), |
with a constant
Proof. There exists an extension
‖˜uϵ‖L2(Rn−1×(−ϵ,ϵ))≤C∗‖uϵ‖L2(ΩMϵ),‖˜uϵ‖H1(Rn−1×(−ϵ,ϵ))≤C∗‖uϵ‖H1(ΩMϵ), |
with a constant
‖uϵ‖L2(S±ϵ)≤‖˜uϵ‖L2(R×{±ϵ})≤C(θ)√ϵ‖˜uϵ‖L2(Rϵ)+θ√ϵ‖∇˜uϵ‖L2(Rϵ)≤C(θ)√ϵ‖uϵ‖L2(ΩMϵ)+θC∗√ϵ‖∇uϵ‖L2(ΩMϵ). |
In a first step, we obtain the following estimates:
Lemma 4.2. The solution
‖u±i,ϵ‖L∞((0,T),L2(Ω±ϵ))+‖∇u±i,ϵ‖L2((0,T),L2(Ω±ϵ))≤C, | (5a) |
1√ϵ‖uMi,ϵ‖L∞((0,T),L2(ΩMϵ))+ϵγ2‖∇uMi,ϵ‖L2((0,T),L2(ΩMϵ))≤C, | (5b) |
1√ϵ‖∂tuMi,ϵ‖L2((0,T),H1(ΩMϵ)′)+‖∂tu±i,ϵ‖L2((0,T),H1(Ω±ϵ)′)≤C, | (5c) |
for
Proof. Test the variational equation (2) for
⟨∂tu±i,ϵ,u±i,ϵ⟩Ω±ϵ+D±i‖∇u±i,ϵ‖2L2(Ω±ϵ)=(f±i(u±ϵ),u±i,ϵ)Ω±ϵ+(h±i(u±ϵ,uMϵ),u±i,ϵ)S±ϵ≤C(1+‖u±i,ϵ‖2L2(Ω±ϵ)+‖u±i,ϵ‖2L2(S±ϵ)+‖uMϵ‖2L2(S±ϵ))≤C(1+‖u±ϵ‖2L2(Ω±ϵ)+1ϵ‖uMϵ‖2L2(ΩMϵ))+δϵ‖∇uMi,ϵ‖2L2(ΩMϵ)+δ‖∇u±ϵ‖2L2(Ω±ϵ)), |
for an arbitrary
1ϵ⟨∂tuMi,ϵ,uMi,ϵ⟩ΩMϵ+ϵγ(DMi(⋅ϵ)∇uMi,ϵ,∇uMi,ϵ)ΩMϵ=1ϵ(gi(⋅ϵ,uMϵ),uMi,ϵ)ΩMϵ+(hM,+i(⋅ˉxϵ,uMϵ,u+ϵ),uMi,ϵ)S+ϵ+(hM,−i(⋅ˉxϵ,uMϵ,u−ϵ),uMi,ϵ)S−ϵ≤C(1+1ϵ‖uMϵ‖2L2(ΩMϵ)+‖uMϵ‖2L2(S+ϵ∪S−ϵ)+‖u+ϵ‖2L2(S+ϵ)+‖u−ϵ‖2L2(S−ϵ))≤C(1+1ϵ‖uMϵ‖2L2(ΩMϵ)+‖u+ϵ‖2L2(Ω+ϵ)+‖u−ϵ‖2L2(Ω−ϵ))+δ(ϵ‖∇uMϵ‖2L2(ΩMϵ)+‖∇u+ϵ‖2L2(Ω+ϵ)+‖∇u−ϵ‖2L2(Ω−ϵ)) |
for an arbitrary
ddt(1ϵ‖uMϵ(t)‖2L2(ΩMϵ)+‖u+ϵ(t)‖2L2(Ω+ϵ)+‖u−ϵ(t)‖2L2(Ω−ϵ))+ϵγ‖∇uMϵ(t)‖2L2(ΩMϵ)+‖∇u+ϵ(t)‖2L2(Ω+ϵ)+‖∇u−ϵ(t)‖2L2(Ω−ϵ)≤C(1+1ϵ‖uMϵ(t)‖2L2(ΩMϵ)+‖u+ϵ(t)‖2L2(Ω+ϵ)+‖u−ϵ(t)‖2L2(Ω−ϵ)). |
Integrating with respect to time from
1ϵ‖uMϵ(t)‖2L2(ΩMϵ)+‖u+ϵ(t)‖2L2(Ω+ϵ)+‖u−ϵ(t)‖2L2(Ω−ϵ)+ϵγ‖∇uMϵ‖2L2((0,t)×ΩMϵ)+‖∇u+ϵ‖2L2((0,t)×Ω+ϵ)+‖∇u−ϵ‖2L2((0,t)×Ω−ϵ)≤C(1+1ϵ‖uMϵ‖2L2((0,t)×ΩMϵ)+‖u+ϵ‖2L2((0,t)×Ω+ϵ)+‖u−ϵ‖2L2((0,t)×Ω−ϵ))+1ϵ‖UM0(⋅ˉx,⋅xnϵ)‖2L2(ΩMϵ)+‖U+0‖2L2(Ω+ϵ)+‖U−0‖2L2(Ω−ϵ). |
The assumptions for the initial conditions and Gronwall's inequality imply
1ϵ‖uMϵ‖2L∞((0,T),L2(ΩMϵ))+‖u+ϵ‖2L∞((0,T),L2(Ω+ϵ))+‖u−ϵ‖2L∞((0,T),L2(Ω−ϵ))≤C, |
and together with the inequality above, we obtain the estimates for the gradients in (5a) and (5b). It remains to prove the estimates for the time derivative. Therefore we test the variational equation (2) for
⟨∂tu±i,ϵ,v±⟩Ω±ϵ=−D±i(∇u±i,ϵ,∇v±)Ω±ϵ+(f±i(u±ϵ),v±)Ω±ϵ+(h±i(u±ϵ,uMϵ),v±)S±ϵ≤C(‖∇u±i,ϵ‖L2(Ω±ϵ)‖∇v±‖L2(Ω±ϵ)+‖v±‖L2(Ω±ϵ)+‖u±ϵ‖L2(Ω±ϵ)‖v±‖L2(Ω±ϵ)+‖v±‖L2(S±ϵ)+‖u±ϵ‖L2(S±ϵ)‖v±‖L2(S±ϵ)+‖uMϵ‖L2(S±ϵ)‖v±‖L2(S±ϵ)). |
Using again the trace estimate from Lemma 4.1, the boundedness of
‖∂tu±i,ϵ(t)‖H1(Ω±ϵ)′≤C(1+‖u±ϵ‖H1(Ω±ϵ)+1√ϵ‖uMϵ‖L2(ΩMϵ)+√ϵ‖∇uMϵ‖L2(ΩMϵ)). |
Integration with respect to time and the inequalities (5a) and (5b) imply the second inequality in (5c). For the first inequality, we choose
1ϵ⟨∂tuMi,ϵ,vϵ⟩ΩMϵ=−ϵγ(DMi(⋅ϵ)∇uMi,ϵ,∇vϵ)ΩMϵ+1ϵ(gi(⋅ϵ,uMϵ),vϵ)ΩMϵ+(hM,+i(⋅ˉxϵ,uMϵ,u+ϵ),vϵ)S+ϵ+(hM,−i(⋅ˉxϵ,uMϵ,u−ϵ),vϵ)S−ϵ≤C(ϵγ‖∇uMi,ϵ‖L2(ΩMϵ)‖∇vϵ‖L2(ΩMϵ)+1√ϵ‖vϵ‖L2(ΩMϵ)+1ϵ‖uMϵ‖L2(ΩMϵ)‖vϵ‖L2(ΩMϵ)+‖vϵ‖L2(S+ϵ)+‖uMϵ‖L2(S+ϵ)‖vϵ‖L2(S+ϵ)+‖u+ϵ‖L2(S+ϵ)‖vϵ‖L2(S+ϵ)+‖vϵ‖L2(S−ϵ)+‖uMϵ‖L2(S−ϵ)‖vϵ‖L2(S−ϵ)+‖u−ϵ‖L2(S−ϵ)‖vϵ‖L2(S−ϵ)). | (6) |
Then, using the trace inequality and the boundedness of
‖∂tuMi,ϵ‖2H1(ΩMϵ)′≤C(ϵ+ϵ2+2γ‖∇uMi,ϵ‖2L2(ΩMϵ)+‖uMϵ‖2L2(ΩMϵ)+ϵ‖u+ϵ‖2L2(Ω+ϵ)+ϵ‖u−ϵ‖2L2(Ω−ϵ)). |
Integration with respect to time and the a priori estimates (5a) and (5b) give
‖∂tuMi,ϵ‖2L2((0,T),H1(ΩMϵ)′)≤C(ϵ+ϵ2+2γ‖∇uMi,ϵ‖2L2((0,T)×ΩMϵ))≤C(ϵ+ϵ2+γ)≤Cϵ. |
This gives us the last inequality and the proof is complete.
The above estimates are not sufficient for the derivation of appropriate strong convergence results for the solution
Since we are operating in bounded domains, we have to make sure that the shifts are well-defined. For an arbitrary domain
Uh:={x∈U:dist(∂U,x)>h}. | (7) |
Further, we write
ΩMϵh:=Σh×(−ϵ,ϵ),Ω+ϵh:=Σh×(ϵ,H),Ω−ϵh:=Σh×(−H,−ϵ). | (8) |
In the same way as in (21) (see Appendix A), we define the sets
δlv(x):=v(x+ϵ(l,0))−v(x). | (9) |
In the following, we suppress the index
Lemma 4.3. Let
1ϵ‖δuMϵ(t)‖2L2(ΩMϵth)+ϵγ‖∇δuMi,ϵ‖2L2((0,T)×ΩMϵth)≤C(‖δu+ϵ‖2L2(0,T)×Ω+ϵh)+‖δu−ϵ‖2L2((0,T)×Ω−ϵh)+ϵγ+1+1ϵ‖δuMϵ(0)‖2L2(ΩMϵh)+‖δu+ϵ(0)‖2L2(Ω+ϵh)+‖δu−ϵ(0)‖2L2(Ω−ϵh)). |
Proof. Let
1ϵ⟨∂tδuMi,ϵ,η2ϕ⟩ΩMϵ+ϵγ∫ΩMϵDMi(xϵ)∇δuMi,ϵ⋅∇(η2ϕ)dx=1ϵ∫ΩMϵδgi(xϵ,uMϵ)η2ϕdx+∑α∈±∫SαϵδhM,αi(ˉxϵ,uMϵ,uαϵ)η2ϕdσ, |
with
δgi(xϵ,uMϵ):=gi(xϵ,uMϵl)−gi(xϵ,uMϵ),δhM,±i(ˉxϵ,uMϵ,u±ϵ):=hM,±i(ˉxϵ,uMϵl,u±ϵl)−hM,±i(ˉxϵ,uMϵ,u±ϵ), |
where
12ϵddt‖ηδuMi,ϵ‖2L2(ΩMϵ)+c0ϵγ‖η∇δuMi,ϵ‖2L2(ΩMϵ)≤1ϵ∫ΩMϵδgi(xϵ,uMϵ)η2δuMi,ϵdx−2ϵγ∫ΩMϵηδuMi,ϵDMi(xϵ)∇δuMi,ϵ⋅∇ηdx+∑α∈±∫SαϵδhM,αi(ˉxϵ,uMϵ,uαϵ)η2δuMi,ϵdσ=:I1ϵ+I2ϵ+I+ϵ+I−ϵ. |
Now, we integrate with respect to time and estimate the terms on the right-hand side. From the Lipschitz continuity of
∫t0I1ϵdt≤Cϵ‖ηδuMϵ‖2L2((0,t)×ΩMϵ). |
For the second term, our a priori estimates imply
∫t0I2ϵdt≤Cϵγ‖ηδuMi,ϵ‖L2((0,t)×ΩMϵ)‖∇δuMi,ϵ‖L2((0,t)×ΩMϵh)≤C(1ϵ‖ηδuMi,ϵ‖2L2((0,t)×ΩMϵ)+ϵγ+1). |
Using the Lipschitz continuity of
∫t0I±ϵdt≤C∫t0∫S±ϵ|ηδu+ϵm|2+|ηδuMϵ|2dσdt≤C(‖ηδu+ϵm‖2L2((0,t)×Ω±ϵ)+1ϵ‖ηδuMϵ‖2L2((0,t)×ΩMϵ)+‖δu+ϵm‖2L2((0,t)×Ω±ϵ,h)+ϵ2)+θ(‖η∇δu+ϵm‖2L2((0,t)×Ω±ϵ)+ϵ‖η∇δuMϵ‖2L2((0,t)×ΩMϵ)). |
Altogether, we obtain for arbitrary
1ϵ‖ηδuMi,ϵ(t)‖2L2(ΩMϵ)−1ϵ‖ηδuMi,ϵ(0)‖2L2(ΩMϵ)+ϵγ‖η∇δuMi,ϵ‖2L2((0,t)×ΩMϵ)≤C(∑α∈±‖ηδu+ϵm‖2L2((0,t)×Ω±ϵ)+1ϵ‖ηδuMϵ‖2L2((0,t)×ΩMϵ)+‖δu+ϵm‖2L2((0,t)×Ω±ϵ,h)+ϵγ+1)+θ(∑α∈±‖η∇δu+ϵm‖2L2((0,t)×Ω±ϵ)+ϵ‖η∇δuMϵ‖2L2((0,t)×ΩMϵ))=:Δ. |
In a similar way, we get
‖ηδu±i,ϵ(t)‖2L2(Ω±ϵ)−‖ηδu±i,ϵ(0)‖2L2(Ω±ϵ)+‖η∇δu±i,ϵ‖2L2((0,t)×Ω±ϵ)≤Δ. |
Adding up all these inequalities, choosing
Finally, we give further estimates for the solution in the thin layer and its time derivative with respect to equivalent norms, which are introduced on
Hϵ,γ:={vϵ∈L2(ΩMϵ):∇vϵ∈L2(ΩMϵ)n}, |
together with the inner product
(vϵ,wϵ)Hϵ,γ:=1ϵ(vϵ,wϵ)ΩMϵ+ϵγ(∇vϵ,∇wϵ)ΩMϵ, |
i.e., the norm
Remark 1. We consider the Gelfand-triple
For the solution
Lemma 4.4. Let
‖uMϵ‖L2((0,T),Hϵ,γ)≤C, | (10a) |
‖∂tuMi,ϵ‖L2((0,T),H′ϵ,γ)≤Cϵ. | (10b) |
Proof. The first inequality follows directly from Lemma 4.2. For the second inequality, we choose
‖vϵ‖L2(ΩMϵ)≤√ϵ,‖∇vϵ‖L2(ΩMϵ)≤ϵ−γ2,‖vϵ‖L2(S±ϵ)≤C. |
Inequality (6) from the proof of Lemma 4.2 is still valid for this
|⟨∂tuMi,ϵ,vϵ⟩H′ϵ,γ,Hϵ,γ|≤C(ϵ1+γ2‖∇uMϵ‖L2(ΩMϵ)+√ϵ‖uMϵ‖L2(ΩMϵ)+ϵ‖u±ϵ‖H1(Ω±ϵ)+ϵ). |
Squaring, integration with respect to time, and Lemma 4.2 give us the desired result.
The aim of this section is to provide general results for the averaged function in thin domains, obtained by taking the average over the
ˉuϵ(t,ˉx):=12ϵ∫ϵ−ϵuϵ(t,ˉx,xn)dxn. |
Here, we consider a sequence
1√ϵ‖uϵ‖L2((0,T)×ΩMϵ)+ϵγ2‖∇uϵ‖L2((0,T)×ΩMϵ)+1√ϵ‖∂tuϵ‖L2((0,T),H1(ΩMϵ)′)≤C. | (11) |
The following estimates hold for the averaged sequence
Lemma 5.1. It holds that
‖ˉuϵ‖L2((0,T)×Σ)≤1√2ϵ‖uϵ‖L2((0,T)×ΩMϵ)≤C, | (12a) |
‖∇ˉxˉuϵ‖L2((0,T)×Σ)≤1√2ϵ‖∇ˉxuϵ‖L2((0,T)×ΩMϵ)≤Cϵ−γ−12, | (12b) |
‖∂tˉuϵ‖L2((0,T),H1(Σ)′)≤1√2ϵ‖∂tuϵ‖L2((0,T),H1(ΩMϵ)′)≤C. | (12c) |
Further, we have
1√ϵ‖uϵ−ˉuϵ‖L2(ΩMϵ)≤2√ϵ‖∂nuϵ‖L2((0,T)×ΩMϵ)≤Cϵ1−γ2. | (12d) |
Proof. It is clear that
‖ˉuϵ‖2L2((0,T)×Σ)=∫T0∫Σ|12ϵ∫ϵ−ϵuϵ(t,ˉx,xn)dxn|2dˉxdt=1(2ϵ)2∫T0∫Σ|∫ϵ−ϵuϵ(t,ˉx,xn)dxn|2dˉxdt≤12ϵ‖uϵ‖2L2((0,T)×ΩMϵ)(11)≤C. |
In a similar way, we obtain
‖∇ˉxˉuϵ‖2L2(Σ)=1(2ϵ)2∫T0∫Σ|∫ϵ−ϵ∇ˉxuϵ(t,ˉx,xn)dxn|2dˉxdt≤12ϵ‖∇ˉxuϵ‖2L2((0,T)×ΩMϵ)(11)≤Cϵ−γ−1. |
Now, we consider the time derivative of
∫T0∫Σˉuϵ(t,ˉx)ϕ(ˉx)ψ′(t)dˉxdt=12ϵ∫T0∫ΩMϵuϵ(t,x)ϕ(ˉx)ψ′(t)dxdt=−12ϵ∫T0⟨∂uϵ(t),ϕ⟩ΩMϵψ(t)dt, |
i.e., we have
⟨∂tˉuϵ,ϕ⟩Σ=12ϵ⟨∂tuϵ,ϕ⟩ΩMϵ∀ϕ∈H1(Σ). |
Additionally, since
|⟨∂tˉuϵ,ϕ⟩Σ|≤12ϵ‖∂tuϵ‖H1(ΩMϵ)′‖ϕ‖H1(ΩMϵ)≤1√2ϵ‖∂tuϵ‖H1(ΩMϵ)′‖ϕ‖H1(Σ). |
Squaring, integration with respect to time, and (11) gives inequality (12c).
It remains to prove estimate (12d). We obtain with the fundamental theorem of calculus
‖uϵ−ˉuϵ‖2L2((0,T)×ΩMϵ)=1(2ϵ)2∫T0∫ΩMϵ|∫ϵ−ϵuϵ(t,ˉx,xn)−uϵ(t,ˉx,˜xn)d˜xn|2dxdt≤∫T0∫ΩMϵ(∫ϵ−ϵ|∂nuϵ(t,ˉx,s)|ds)2dxdt≤(2ϵ)2‖∂nuϵ‖2L2((0,T)×ΩMϵ)(11)≤Cϵ2−γ. |
In the following proposition, we compare the
Lemma 5.2. It holds that
‖uϵ|S±ϵ−ˉuϵ‖L2(Σ)≤Cϵ1−γ2. |
Proof. We only consider the trace on
‖uϵ|S+ϵ−ˉuϵ‖2L2((0,T)×Σ)=1(2ϵ)2∫T0∫Σ|∫ϵ−ϵuϵ(ˉx,ϵ)−uϵ(ˉx,xn)dxn|2dˉxdt≤Cϵ‖∂nuϵ‖2L2((0,T)×Σ)(11)≤Cϵ1−γ. |
Remark 2. From Lemma 5.1, we immediately obtain the existence of a function
ˉuϵ⇀ˉuo weakly in L2((0,T)×Σ),∂tˉuϵ⇀∂tˉuo weakly in L2((0,T),H1(Σ)′). |
In Section 7, we will use the averaged function to prove convergence results for the solution
Let
Proposition 2. The following relation holds between the weak limit
ˉuo(t,ˉx)=1|Z|∫Zu0(t,ˉx,y)dy. | (13) |
Proof. For all
∫T0∫Σ∫Zu0(t,ˉx,y)ϕ(ˉx)ψ(t)dydˉxdt=limϵ→01ϵ∫T0∫ΩMϵuϵ(t,x)ϕ(ˉx)ψ(t)dxdt=limϵ→0|Z|∫T0∫Σˉuϵ(t,ˉx)ϕ(ˉx)ψ(t)dˉxdt=|Z|∫T0∫Σˉuo(t,ˉx)ϕ(ˉx)ψ(t)dˉxdt. |
From
Proposition 3. We have
⟨∂t(1|Z|∫Zu0(t,⋅ˉx,y)dy),ϕ⟩Σ=⟨∂tˉuo(t),ϕ⟩Σ∀ϕ∈H1(Σ),a.e.t∈(0,T). |
Proof. This follows easily from partial integration with respect to time. In fact, for all
∫T0⟨∂tˉuo,ϕ⟩Σψ(t)dt=limϵ→0∫T0⟨∂tˉuϵ(t),ϕ⟩Σψ(t)dt=−limϵ→012ϵ∫T0∫ΩMϵuϵ(t,x)ϕ(ˉx)ψ′(t)dxdt=−1|Z|∫T0∫Σ∫Zu0(t,ˉx,y)ϕ(ˉx)ψ′(t)dydˉxdt. |
A straightforward consequence of Proposition 2 and Proposition 3 is given in the following
Corollary 1. If the two-scale limit
ˉuo=u0,∂tˉuo=∂tu0. |
For
Proposition 4. Let
‖uϵ‖L2((0,T)×ΩMϵ)+‖∇uϵ‖L2((0,T)×ΩMϵ)+‖∂tuϵ‖L2((0,T),H1(ΩMϵ)′)≤C√ϵ. |
Then, there exist
uϵ→u0stronglyinthetwo−scalesense∇uϵ→∇ˉxu0+∇yu1inthetwo−scalesense,uϵ|S±ϵ→u0inL2((0,T)×Σ),∂tˉuϵ⇀∂tu0weaklyinL2((0,T),H1(Σ)′). |
Proof. Proposition 10 implies that, up to a subsequence,
Lemma A.2 shows that the unfolded sequence
For
UMϵ:L2((0,T)×Σ×Z)→L2((0,T)×ΩMϵ),UMϵ(ϕ)(t,x)={∫Yϕ(t,ϵ(ˉz+[ˉxϵ]),({ˉxϵ},xnϵ))dˉz for x∈ˆΩMϵ,0 for x∈ΛMϵ. |
The following Lemma shows that
Proposition 5. Let
∫T0∫Σ∫ZTMϵuϵ(t,ˉx,y)ϕ(t,ˉx,y)dydˉxdt=1ϵ∫T0∫ΩMϵuϵ(t,x)UMϵ(ϕ)(t,x)dxdt. |
Further, we have the identity
Proof. The proof uses the same arguments as in [6,Proposition 4.3] and is skipped here.
As a corollary we immediately obtain the following inequality:
Corollary 2. For
‖UMϵ(ϕ)‖L2((0,T)×ΩMϵ)≤√ϵ‖ϕ‖L2((0,T)×Σ×Z). |
From the definition of the operator
H0:=¯C∞0(Y×[−1,1])‖⋅‖H1(Z), | (14) |
together with the usual
Proposition 6. It holds
ϵ∇UMϵ(ϕ)(t,x)=UMϵ(∇yϕ)(t,x). |
Proof. Let
∫T0∫ΩMϵUMϵ(ϕ)(t,x)∂xiuϵ(t,x)dxdt=∫T0∫Σ∫Zϕ(t,ˉx,y)∂yiTMϵuϵ(t,ˉx,y)dydˉxdt=−∫T0∫Σ∫Z∂yiϕ(t,ˉx,y)TMϵuϵ(t,ˉx,y)dydˉxdt+∫T0∫Σ∫∂Zϕ(t,ˉx,y)TMϵuϵ(t,ˉx,y)νidσydˉxdt. |
The last term is equal to
∫T0∫ΩMϵUMϵ(ϕ)(t,x)∂xiuϵ(t,x)dxdt=−1ϵ∫T0∫ΩMϵUMϵ(∂yiϕ)(t,x)uϵ(t,x)dxdt, |
which gives us the desired result.
We are now able to state our regularity result with respect to time for the unfolding operator.
Proposition 7. Let
TMϵuϵ∈L2((0,T),L2(Σ×Z))∩H1((0,T),L2(Σ,H0)′), |
and it holds that
⟨∂tTMϵuϵ(t),ϕ⟩L2(Σ,H0)′,L2(Σ,H0)=1ϵ⟨∂tuϵ(t),UMϵ(ϕ)⟩ΩMϵ |
for almost every
Proof. Let
∫T0∫Σ∫ZTMϵuϵ(t,ˉx,y)ϕ(ˉx,y)ψ′(t)dydˉxdt=1ϵ∫T0∫ΩMϵuϵ(t,x)UMϵ(ϕ)(x)ψ′(t)dxdt=−1ϵ∫T0⟨∂tuϵ(t),UMϵ(ϕ)⟩ΩMϵψ(t)dt, |
what gives us the claim.
Our aim now is to estimate the norm of
Lemma 6.1. Let
‖UMϵ(ϕ)‖L2((0,T),Hϵ,1)≤‖ϕ‖L2((0,T)×Σ,H0). |
Proof. The claim follows immediately from Corollary 2 and Proposition 6.
As an easy consequence, we obtain the following estimate for the time derivative
Proposition 8. Let
‖∂tTMϵuϵ‖L2((0,T),L2(Σ,H0)′)≤1ϵ‖∂tuϵ‖L2((0,T),H′ϵ,1). |
Proof. Due to Lemma 6.1, we have
⟨∂tTMϵuϵ(t),ϕ⟩L2(Σ,H0)′,L2(Σ,H0)=1ϵ⟨∂tuϵ(t),UMϵ(ϕ)⟩H′ϵ,1,Hϵ,1≤1ϵ‖∂tuϵ(t)‖H′ϵ,1‖UMϵ(ϕ)‖Hϵ,1≤1ϵ‖∂tuϵ(t)‖H′ϵ,1, |
i.e.,
In this section, we derive convergence results for the sequences
For the derivation of the convergence results for
Let us start with the convergence results in the bulk domains. These are similar to those in [10,15], except for the time derivative, which is in this case only a functional pointwise in time. For the convergence of the time derivative, we transform the fixed domain
Φ±ϵ:Ω±→Ω±ϵ,Φ±ϵ(x)=(ˉx,H−ϵHxn±ϵ), |
and define
⟨∂t˜u±i,ϵ(t),ϕ⟩Ω±=⟨HH−ϵ∂tu±i,ϵ(t),ϕ∘(Φ±ϵ)−1⟩Ω±ϵ, |
and especially we obtain with our a priori estimates from Lemma 4.2
‖∂t˜u±i,ϵ‖L2((0,T),H1(Ω±)′)≤C‖∂tu±i,ϵ‖L2((0,T),H1(Ω±ϵ)′)≤C. |
Proposition 9. Let
χΩ±ϵu±i,ϵ→u±i,0stronglyinL2((0,T)×Ω±),χΩ±ϵ∇u±i,ϵ⇀∇u±i,0weaklyinL2((0,T)×Ω±),˜u±i,ϵ|Σ→u±i,0stronglyinL2((0,T)×Σ),∂t˜u±i,ϵ⇀∂tu±i,0weaklyinL2((0,T),H1(Ω±)′). |
Further, for all
limϵ→0∫T0⟨∂tu±i,ϵ(t),ϕ|Ω±ϵ(t)⟩Ω±ϵdt=∫T0⟨∂tu±i,0(t),ϕ(t)⟩Ω±dt. |
Proof. The convergences of
After having established the convergence results for the sequences
Remark 3. In the following, we consider the traces
To prove the convergence results for the sequence
ˉuMϵ(t,ˉx):=12ϵ∫ϵ−ϵuMϵ(t,ˉx,xn)dxn, | (15) |
introduced in Section 5.
Theorem 7. For
uMi,ϵ→uMi,0stronglyinthetwo−scalesense∇uMi,ϵ→∇ˉxuMi,0+∇yuMi,1inthetwo−scalesense,uMi,ϵ|S±ϵ→uMi,0inL2((0,T)×Σ),∂tˉuMi,ϵ⇀∂tuMi,0weaklyinL2((0,T),H1(Σ)′). |
Proof. This follows directly from Lemma 4.2 and Proposition 4.
From the strong convergences above, we immediately obtain the two-scale convergences of the nonlinear terms:
Corollary 3. For
f±(χΩ±ϵu±ϵ)→f±(u±0)inL2((0,T)×Ω±),g(⋅ϵ,uMϵ)→g(⋅,uM0)inthetwo−scalesense,h±(u±ϵ|S±ϵ,uMϵ|S±ϵ)→h±(u±0|Σ,uM0)inthetwo−scalesenseonΣ,hM,±(⋅ϵ,uMϵ|S±ϵ,u±ϵ|S±ϵ)→hM,±(⋅,uM0,u±0|Σ)inthetwo−scalesenseonΣ. |
Proof. The first convergence follows from Proposition 9. To show the second convergence, we start from
1ϵ∫T0∫ΩMϵg(xϵ,uMϵ)ϕ(t,ˉx,xϵ)dxdt=1ϵ∫T0∫ΩMϵ[g(xϵ,uMϵ)−g(xϵ,ˉuMϵ)]ϕ(t,ˉx,xϵ)dxdt+1ϵ∫T0∫ΩMϵ[g(xϵ,ˉuMϵ)−g(xϵ,uM0)]ϕ(t,ˉx,xϵ)dxdt+1ϵ∫T0∫ΩMϵg(xϵ,uM0)ϕ(t,ˉx,xϵ)dxdt=:Iϵ1+Iϵ2+Iϵ3, |
with
Iϵ2=1ϵ∫T0∫ΩMϵ[g(xϵ,ˉuMϵ)−g(xϵ,uM0)]ϕ(t,ˉx,xϵ)dxdt≤Cϵ∫T0∫ΩMϵ|g(xϵ,ˉuMϵ)−g(xϵ,uM0)|dxdt≤Cϵ∫ΩMϵ|ˉuMϵ−uM0|dxdt≤C||ˉuMϵ−uM0||L1((0,T)×Σ). | (16) |
Thus, the strong convergence of
Now, based on the above convergence results, we derive the macroscopic model from Theorem 3.1.
Proof of Theorem 3.1. To obtain the equations in the bulk domains, we test the variational equation (2) in the bulk with
Testing the variational equation (2) of
1ϵ∫T0⟨∂tuMi,ϵ(t),ϕϵ(t)⟩ΩMϵdt+∫T0∫ΩMϵDMi(xϵ)∇uMi,ϵ(t,x)⋅[θ(xϵ)∇ˉxϕ(ˉx)+1ϵϕ(ˉx)∇yθ(xϵ)]ψ(t)dxdt=1ϵ∫T0∫ΩMϵgi(xϵ,uMϵ)ϕϵ(t,x)dxdt+∑α∈±∫T0∫SαϵhM,αi(ˉxϵ,uMϵ,uαϵ)ϕϵ(t,ˉx,α1)dˉxdt. |
Due to our a priori estimates, all the terms except the diffusion therm involving
∫T0∫Σ∫ZDMi(y)[∇ˉxuMi,0(t,ˉx)+∇yuMi,1(t,ˉx,y)]⋅ψ(t)ϕ(ˉx)∇yθ(y)dydˉxdt=0. |
This implies almost everywhere in
uMi,1(t,ˉx,y)=n−1∑j=1∂juMi,0(t,ˉx)wi,j(y), | (17) |
where
Now, we choose as a test function
∫T0⟨∂tˉuMi,ϵ(t),ϕ⟩Σψ(t)dt+1ϵ∫T0∫ΩMϵDMi(xϵ)∇uMi,ϵ(t,x)⋅∇ˉxϕ(ˉx)ψ(t)dxdt=1ϵ∫T0∫ΩMϵgi(xϵ,uMϵ)ϕ(ˉx)ψ(t)dxdt+∑α∈±∫T0∫SαϵhM,αi(ˉxϵ,uMϵ,uαϵ)ϕ(ˉx)ψ(t)dˉxdt. | (18) |
From Theorem 7.1, Corollary 3, and the identity (17), we get for
|Z|∫T0⟨∂tuMi,0(t),ϕ⟩Σψ(t)dt+∫T0∫ΣDM,∗i∇ˉxuMi,0(t,ˉx)⋅∇ˉxϕ(ˉx)ψ(t)dˉxdt.=∫T0∫Σ(∫Zgi(y,uM0)dy+∑α∈{±}∫YhM,αi(ˉy,uM0,uα0)dˉy)ϕ(ˉx)ψ(t)dˉxdt. |
Since
Again, we use the averaged function
Lemma 7.2. Let
|(v(t+h)−v(t),ϕ)H|≤√|h|‖ϕ‖V‖∂tv‖L2((t,t+h),V′). |
Especially, it holds that
‖v(t+h)−v(t)‖2H≤√|h|‖v(t+h)−v(t)‖V‖∂tv‖L2((t,t+h),V′). |
Proof. The proof follows the same lines as the proof for the special case
Theorem 7.3. For
uMi,ϵ→uMi,0inthetwo−scalesenseˉuMi,ϵ→uMi,0inLp((0,T)×Σ),uMi,ϵ|S±ϵ→uMi,0inLp((0,T)×Σ),∂tˉuMi,ϵ⇀∂tuMi,0weaklyinL2((0,T),H1(Σ)′). |
Proof. As in Proposition 7.1, Lemma 4.2 and Proposition 10 imply that, up to a subsequence,
We extend all functions outside their domain of definition by zero. Since
‖ˉuMϵ(⋅+s,⋅+ˉξ)−ˉuMϵ‖Lp((0,T)×Σ)→0for (s,ˉξ)→0 | (19) |
uniformly with respect to
(ⅰ) For all
supϵ‖ˉuMϵ(⋅+s,⋅+ˉξ)−ˉuMϵ‖Lp((0,T)h×Σh)→0for (s,ˉξ)→0. |
(ⅱ) It holds that
supϵ‖ˉuMϵ‖Lp((0,T)∖(0,T)h×Σ∖Σh)→0for h→0. |
For the definition of the domains
‖ˉuMϵ‖Lp((0,T)∖(0,T)h×Σ∖Σh)≤C|(0,T)∖(0,T)h×Σ∖Σh|2−p2ph→0⟶0. |
For condition (ⅰ), due to the triangle inequality, it is enough to consider shifts separately with respect to the variable
‖ˉuMϵ(⋅,⋅+ˉξ)−ˉuMϵ‖L2((0,T)×Σh)≤1√2ϵ‖uMϵ(⋅,⋅+(ˉξ,0))−uMϵ‖L2((0,T)×ΩMϵh)≤1√2ϵ‖uMϵ(⋅,⋅+(ˉξ,0))−uMϵ(⋅,⋅+ϵ([ˉξϵ],0))‖L2((0,T)×ΩMϵh)+1√2ϵ‖uMϵ(⋅,⋅+ϵ([ˉξϵ],0))−uMϵ‖L2((0,T)×ΩMϵh). |
Due to the mean-value theorem and the a priori estimates for
Now, we have to consider shifts with respect to time. From Lemma 4.4, we have
‖uMϵ‖L2((0,T),Hϵ,γ)≤C,‖∂tuMi,ϵ‖L2((0,T),H′ϵ,γ)≤Cϵ. |
Hence, Lemma 7.2 with
‖ˉuMϵ(⋅+s,⋅)−ˉuMϵ‖2L2((0,T)h×Σ)≤12ϵ‖uMϵ(⋅+s,⋅)−uMϵ‖2L2((0,T)h×ΩMϵ)≤√h2ϵ‖uMϵ(⋅+s,⋅)−uMϵ‖L2((0,T)h,Hϵ,γ)‖∂tuMϵ‖L2((0,T),H′ϵ,γ)≤C√h. |
This gives us the strong convergence of
Again, we obtain the convergences for the nonlinearities.
Corollary 4. For
f±(χΩ±ϵu±ϵ)→f±(u±0)inL2((0,T)×Ω±),g(⋅ϵ,uMϵ)→g(⋅,uM0)inthetwo−scalesense,h±(u±ϵ|S±ϵ,uMϵ|S±ϵ)→h±(u±0|Σ,uM0)inthetwo−scalesenseonΣ,hM,±(⋅ϵ,uMϵ|S±ϵ,u±ϵ|S±ϵ)→hM,±(⋅,uM0,u±0|Σ)inthetwo−scalesenseonΣ. |
Proof. We use the same arguments as in Corollary 3. The only difference is that terms like (16) are estimated by the
Passing to the limit
Proof of Theorem 3.2. The arguments are quite similar to those in Theorem 3.1 for the case
|ϵγ∫T0∫ΩMϵDMi(xϵ)∇uMi,ϵ(t,x)⋅∇ˉxϕ(ˉx)ψ(t)dxdt|≤Cϵ1+γ2. |
Finally, the
Here, we treat the critical case
Lemma 7.4. For all
‖TMϵϕMϵ(⋅,⋅+ˉξ,⋅)−TMϵϕMϵ‖2L2((0,T)×Σ2h×Z)≤1ϵ∑j∈{0,1}n−1‖δlϕMϵ‖2L2((0,T)×ΩMϵh) |
with
Proof. The proof is based on a special decomposition of
Theorem 7.5. For
uMi,ϵ→uMi,0inthetwo−scalesense,ϵ∇uMi,ϵ→∇yuMi,0inthetwo−scalesense,TMϵuMi,ϵ→uMi,0inLp(Σ,L2((0,T),Hβ(Z))),TΣϵ(uMi,ϵ|S±ϵ)→uMi,0|S±inLp(Σ,L2((0,T)×S±)) |
Remark 4. Due to the boundedness of the sequence
Proof of Theorem 7.5. The two-scale convergences of
We prove the strong convergence of
(ⅰ) For every
vϵA(t,y):=∫ATMϵuMϵ(t,ˉx,y)dˉx |
is relatively compact in
(ⅱ) It holds that
supϵ‖TMϵuMϵ(⋅,⋅+ˉξ,⋅)−TMϵuMϵ‖Lp(Σ,L2((0,T),Hβ(Z)))→0for ˉξ→0. |
First of all, we have
⟨∂tvϵA(t),ϕ⟩H′0,H0=⟨∂tTMϵuMi,ϵ(t),χA(⋅ˉx)ϕ(⋅y)⟩L2(Σ,H0)′,L2(Σ,H0), |
since for every
∫T0∫ZvϵA(t,y)ϕ(y)ψ′(t)dydt=∫T0∫Σ∫ZTMϵuMi,ϵ(t,ˉx,y)ϕ(y)χA(ˉx)ψ′(t)dydˉxdt=−∫T0⟨∂tTMϵuMi,ϵ(t),χA(⋅ˉx)ϕ(⋅ˉy)⟩L2(Σ,H0)′,L2(Σ,H0)ψ(t)dt. |
Hence, Lemma 4.2, 4.4, and Proposition 8 imply the boundedness of
‖TMϵuMϵ(⋅,⋅+ˉξ,⋅)−TMϵuMϵ‖Lp(Σ∖Σ2h,L2((0,T),Hβ(Z)))≤C|h|2−p2p‖TMϵuMϵ‖L2((0,T)×Σ,H1(Z))≤Ch2−p2p, |
by the same arguments as in the proof of Theorem 7.3, it is enough to show that for all
supϵ‖TMϵuMϵ(⋅,⋅+ˉξ,⋅)−TMϵuMϵ‖Lp(Σ2h,L2((0,T),Hβ(Z)))→0for ˉξ→0. |
So, we fix
‖TMϵuMϵ(⋅,⋅+ˉξ,⋅)−TMϵuMϵ‖L2(Σ2h,L2((0,T),Hβ(Z)))≤C‖TMϵuMϵ(⋅,⋅+ˉξ,⋅)−TMϵuMϵ‖L2(Σ2h,L2((0,T),H1(Z)))≤C∑j∈{0,1}n−1(1ϵ‖δuMϵ‖2L2((0,T)×ΩMϵh)+ϵ‖∇δuMϵ‖2L2((0,T)×ΩMϵh)). |
Due to Lemma 4.3, the right-hand side converges to zero for
The last statement of the theorem follows from the strong convergence of
Corollary 5. For
f±(χΩ±ϵu±ϵ)→f±(u±0)inL2((0,T)×Ω±)m,g(⋅ϵ,uMϵ)→g(⋅,uM0)inthetwo−scalesense,h±(u±ϵ|S±ϵ,uMϵ|S±ϵ)→h±(u±0|Σ,uM0|S±)inthetwo−scalesenseonΣ,hM,±(⋅ϵ,uMϵ|S±ϵ,u±ϵ|S±ϵ)→hM,±(⋅,uM0|S±,u±0|Σ)inthetwo−scalesenseonΣ. |
Proof. We only show the last convergence. From the strong convergence of
TΣϵ(hM,±(⋅ϵ,uMϵ|S±ϵ,u±ϵ|S±ϵ))=hM,±(⋅ˉy,TΣϵ(uMϵ|S±ϵ),TΣϵ(u±ϵ|S±ϵ)). |
The strong convergences of
Finally, the compactness results from Theorem 7.5 and Corollary 5 allow us to pass to the limit
Proof of Theorem 3.3. The equations for the bulk-domains
To derive the limit equation for the membrane, we test the variational equation of
−1ϵ∫T0∫ΩMϵuMi,ϵ(t,x)ϕ(ˉx,xϵ)ψ′(t)dxdt+ϵ∫T0∫ΩMϵDMi(xϵ)∇uMϵ(t,x)⋅[∇ˉxϕ(ˉx,xϵ)+1ϵ∇yϕ(ˉx,xϵ)]ψ(t)dxdt=1ϵ∫T0∫ΩMϵgi(xϵ,uMϵ)ϕ(ˉx,xϵ)ψ(t)dxdt+∑α∈±∫T0∫SαϵhM,αi(ˉxϵ,uMϵ,uαϵ)ϕ(ˉx,xϵ)ψ(t)dˉxdt. |
For
−∫T0∫Σ∫ZuM0(t,ˉx,y)ϕ(ˉx,y)ψ′(t)dydˉxdt=−∫T0∫Σ∫ZDMi(y)∇yuMi,0(t,ˉx,y)⋅∇yϕ(ˉx,y)ψ(t)dydˉxdt+∫T0∫Σ∫Zgi(y,uM0)ϕ(ˉx,y)ψ(t)dydˉxdt+∑α∈±∫T0∫Σ∫YhM,αi(ˉy,uM0(t,ˉx,ˉy,α),uα0(t,ˉx,0))ϕ(ˉx,ˉy,α)ψ(t)dˉydˉxdt. |
By a density argument the equation holds for all
Remark 5.
(ⅰ) Due to the uniqueness of the solutions of the effective models in Theorems 3.1, 3.2, and 3.3, it follows that the whole sequence converges.
(ⅱ) The results also hold, if the parameter
In this paper, we derived effective models for reaction-diffusion processes through thin layers with nonlinear transmission conditions at the bulk-layer interface, and diffusivities of order
In the critical case
For
The methods developed in this paper can also be used for more complex multi-physics processes. Furthermore, the effective models obtained here rise new challenges also to numerical approaches, which have to take into account the special micro-macro features of the model.
We repeat the definition of the two-scale convergence and the unfolding operator for thin domains, and briefly summarize some results related to these approaches, which are needed frequently throughout the paper. The two-scale convergences for domains was introduced in [1,16], and extended to thin layers with heterogeneous structure in [15].
Definition A.1. We say that a sequence
limϵ→01ϵ∫T0∫ΩMϵuϵ(t,x)ϕ(t,ˉx,xϵ)dxdt=∫T0∫Σ∫Zu0(t,ˉx,y)ϕ(t,ˉx,y)dydˉxdt. |
Further, we say that a (weakly) two-scale convergent sequence
limϵ→01√ϵ‖uϵ‖L2((0,T)×ΩMϵ)=‖u0‖L2((0,T)×Σ×Z). |
We remark that in [4] a definition of the two-scale convergence was given for a thin domain with a more particular geometry.
In the following, we consider functions
1√ϵ‖uϵ‖L2((0,T)×ΩMϵ)+ϵγ2‖∇uϵ‖L2((0,T)×ΩMϵ)≤C, |
with
Hper:={u∈H1(Z):u is Y-periodic},Hpero:={u∈Hper:∫Zudy=0}, | (20) |
equipped with the
Proposition 10. Let
1√ϵ‖uϵ‖L2((0,T)×ΩMϵ)+ϵγ2‖∇uϵ‖L2((0,T)×ΩMϵ)≤C, |
with a constant
(i) For
uϵ→u0inthetwo−scalesense,ϵ∇uϵ→∇yu0inthetwo−scalesense. |
(ii) For
uϵ→u0inthetwo−scalesense,ϵγ+12∇uϵ→∇yu1inthetwo−scalesense. |
(iii) For
uϵ→u0inthetwo−scalesense,∇uϵ→∇ˉxu0+∇yu1inthetwo−scalesense, |
where
Proof. The proof of (ⅰ) can be found in [15], and the proof of (ⅱ) and (ⅲ) in [10].
Next, we define the unfolding operator
Kϵ:={ˉk∈Zn−1:ϵ(ˉk+Y)⊂Σ},ˆΣϵ:=int⋃ˉk∈Kϵϵ(ˉk+¯Y),Λϵ:=int(Σ∖ˆΣϵ),ˆΩMϵ:=ˆΣϵ×(−ϵ,ϵ),ΛMϵ:=Λϵ×(−ϵ,ϵ). | (21) |
Then, we define the unfolding operator as follows:
TMϵ:L2((0,T)×ΩMϵ)→L2((0,T)×Σ×Z),TMϵuϵ(t,ˉx,y)={uϵ(t,ϵ([ˉxϵ],0)+ϵy) for ˉx∈ˆΣϵ,0 for ˉx∈Λϵ. |
The unfolding operator
Lemma A.2. [15] For
(TMϵuϵ,TMϵvϵ)L2((0,T)×Σ×Z)=1ϵ(uϵ,vϵ)L2((0,T)׈ΩMϵ),‖TMϵuϵ‖L2((0,T)×Σ×Z)≤1√ϵ‖uϵ‖L2((0,T)×ΩMϵ). |
If additionally it holds that
∇yTMϵuϵ=ϵTMϵ(∇uϵ). |
Finally, we have the following relation between the unfolding operator
Lemma A.3. [15] Let
‖uϵ‖L2((0,T)×ΩMϵ)≤C√ϵ, |
with a constant
(i)
(ii)
[1] |
Goldstein DS, Kopin IJ (2007) Evolution of concepts of stress. Stress 10: 109-120. doi: 10.1080/10253890701288935
![]() |
[2] | Sterling P (2012) Allostasis: a model of predictive regulation. Physiol Behav 106: 5-15. |
[3] |
Davidson RJ, McEwen BS (2012) Social influences on neuroplasticity: stress and interventions to promote well-being. Nat Neurosci 15: 689-695. doi: 10.1038/nn.3093
![]() |
[4] |
Godsil BP, Kiss JP, Spedding M, et al. (2013) The hippocampal-prefrontal pathway: the weak link in psychiatric disorders? Eur Neuropsychopharmacol 23: 1165-1181. doi: 10.1016/j.euroneuro.2012.10.018
![]() |
[5] |
McEwen BS (2007) Physiology and neurobiology of stress and adaptation: central role of the brain. Physiol Rev 87: 873-904. doi: 10.1152/physrev.00041.2006
![]() |
[6] |
Ulrich-Lai YM, Herman JP (2009) Neural regulation of endocrine and autonomic stress responses. Nat Rev Neurosci 10: 397-409. doi: 10.1038/nrn2647
![]() |
[7] |
Hernández J, Prieto I, Segarra AB, et al. (2015) Interaction of neuropeptidase activities in cortico-limbic regions after acute restraint stress. Behav Brain Res 287: 42-48. doi: 10.1016/j.bbr.2015.03.036
![]() |
[8] | Segarra AB, Hernández J, Prieto I, et al. (2016) Neuropeptidase activities in plasma after acute restraint stress. Interaction with cortico-limbic areas. Acta Neuropsychiatr 28: 239-243. |
[9] | Sandi C, Pinelo-Nava MT (2007) Stress and memory: behavioral effects and neurobiological mechanisms. Neural Plast 2007: 78970. |
[10] |
Schwabe L, Joëls M, Roozendaal B, et al. (2012) Stress effects on memory: an update and integration. Neurosci Biobehav Rev 36: 1740-1749. doi: 10.1016/j.neubiorev.2011.07.002
![]() |
[11] |
Richter-Levin G, Akirav I (2000) Amygdala-hippocampus dynamic interaction in relation to memory. Mol Neurobiol 22: 11-20. doi: 10.1385/MN:22:1-3:011
![]() |
[12] | Kim JJ, Diamond DM (2002) The stressed hippocampus, synaptic plasticity and lost memories. Nat Rev Neurosci 3: 453-462. |
[13] |
Sebastian V, Estil JB, Chen D, et al. (2013) Acute physiological stress promotes clustering of synaptic markers and alters spine morphology in the hippocampus. PLoS One 8: e79077. doi: 10.1371/journal.pone.0079077
![]() |
[14] |
Moss RA (2016) A Theory on the Singular Function of the Hippocampus: Facilitating the Binding of New Circuits of Cortical Columns. AIMS Neurosci 3: 264-305. doi: 10.3934/Neuroscience.2016.3.264
![]() |
[15] |
Toth I, Neumann ID (2013) Animal models of social avoidance and social fear. Cell Tissue Res 354: 107-118. doi: 10.1007/s00441-013-1636-4
![]() |
[16] |
Campos AC, Fogaça MV, Aguiar DC, et al. (2013) Animal models of anxiety disorders and stress. Rev Bras Psiquiatr 35: S101-111. doi: 10.1590/1516-4446-2013-1139
![]() |
[17] | Sandi C, Pinelo-Nava MT (2007) Stress and memory: behavioral effects and neurobiological mechanisms. Neural Plast 2007: 78970. |
[18] |
Gülpinar MA, Yegen BC (2004) The physiology of learning and memory: role of peptides and stress. Curr Protein Pept Sci 5: 457-473. doi: 10.2174/1389203043379341
![]() |
[19] | Bilkei-Gorzo A, Racz I, Michel K, et al. (2008) Control of hormonal stress reactivity by the endogenous opioid system. Psychoneuroendocrinology 33: 425-436. |
[20] | Narita M, Kaneko C, Miyoshi K, et al. (2006) Chronic pain induces anxiety with concomitant changes in opioidergic function in the amygdala. Neuropsychopharmacology 3:739-750. |
[21] | Neumann ID, Torner L, Wigger A (2000) Brain oxytocin: differential inhibition of neuroendocrine stress responses and anxiety-related behaviour in virgin, pregnant and lactating rats. Neuroscience 95: 567-575. |
[22] |
Neumann ID (2007) Stimuli and consequences of dendritic release of oxytocin within the brain. Bioch Soc Trans 35: 1252-1257. doi: 10.1042/BST0351252
![]() |
[23] |
Wright JW, Yamamoto BJ, Harding JW (2008) Angiotensin receptor subtype mediated physiologies and behaviors: new discoveries and clinical targets. Prog Neurobiol 84: 157-181. doi: 10.1016/j.pneurobio.2007.10.009
![]() |
[24] |
Saavedra JM, Benicky J (2007) Brain and peripheral angiotensin II play a major role in stress. Stress 10: 185-193. doi: 10.1080/10253890701350735
![]() |
[25] | Nomura S, Ito T, Mizutani S (2004) Placental leucine aminopeptidase. Aminopeptidases in biology and disease. Kluwer Academic/Plenum, New York, Hooper NM and Lendeckel U Eds. pp 45-59. |
[26] |
Solhonne B, Gros C, Pollard H, et al. (1987) Major localization of aminopeptidase M in rat brain. Neuroscience 22: 225-232. doi: 10.1016/0306-4522(87)90212-0
![]() |
[27] | Thompson MW, Hersh LB (2004) The puromycin-sensitive aminopeptidase. Aminopeptidases in biology and disease. Kluwer Academic/Plenum, New York, Hooper NM and Lendeckel U Eds. pp 1-15. |
[28] |
Ramírez-Sánchez M, Prieto I, Wangensteen R, et al. (2013) The renin-angiotensin system: new insight into old therapies. Curr Med Chem 20: 1313-1322. doi: 10.2174/0929867311320100008
![]() |
[29] | Prieto I, Villarejo AB, Segarra AB, et al. (2015) Tissue distribution of CysAP activity and its relationship to blood pressure and water balance. Life Sci 134: 173-178. |
[30] |
Albiston AL, Mustafa T, McDowall SG, et al. (2003) AT4 receptor is insulin-regulated membrane aminopeptidase: potential mechanisms of memory enhancement. Trends Endocrinol Metab 14: 72-77. doi: 10.1016/S1043-2760(02)00037-1
![]() |
[31] |
De Bundel D, Smolders I,Vanderheyden P, et al. (2008) Ang II and Ang IV: unraveling the mechanism of action on synaptic plasticity, memory, and epilepsy. CNS Neurosci Ther 14: 315-339. doi: 10.1111/j.1755-5949.2008.00057.x
![]() |
[32] |
De Bundel D, Demaegdt H, Lahoutte T, et al. (2010) Involvement of the AT1 receptor subtype in the effects of angiotensin IV and LVV-haemorphin 7 on hippocampal neurotransmitter levels and spatial working memory. J Neurochem 112: 1223-1234. doi: 10.1111/j.1471-4159.2009.06547.x
![]() |
[33] |
Drolet G, Dumont EC, Gosselin I, et al. (2001) Role of endogenous opioid system in the regulation of the stress response. Prog Neuropsychopharmacol Biol Psychiatry 25: 729-741. doi: 10.1016/S0278-5846(01)00161-0
![]() |
[34] |
McCubbin JA (1993) Stress and endogenous opioids: behavioral and circulatory interactions. Biol Psychol 35: 91-122. doi: 10.1016/0301-0511(93)90008-V
![]() |
[35] |
Bodnar RJ (2014) Endogenous opiates and behavior: 2013. Peptides 62: 67-136. doi: 10.1016/j.peptides.2014.09.013
![]() |
[36] |
Bodnar RJ (2016) Endogenous opiates and behavior: 2014. Peptides 75: 18-70. doi: 10.1016/j.peptides.2015.10.009
![]() |
[37] | Bali A, Randhawa PK, Jaggi AS (2015) Stress and opioids: role of opioids in modulating |
[38] | stress-related behavior and effect of stress on morphine conditioned place preference. Neurosci Biobehav Rev 51: 1150. |
[39] |
38. Olff M, Frijling JL, Kubzansky LD, et al. (2013) The role of oxytocin in social bonding, stress regulation and mental health: an update on the moderating effects of context and interindividual differences. Psychoneuroendocrinology 38: 1883-1894. doi: 10.1016/j.psyneuen.2013.06.019
![]() |
[40] |
39. Okimoto N, Bosch OJ, Slattery DA, et al. (2012) RGS2 mediates the anxiolytic effect of oxytocin. Brain Res 1453: 26-33. doi: 10.1016/j.brainres.2012.03.012
![]() |
[41] |
40. Neumann ID, Slattery DA (2016) Oxytocin in General Anxiety and Social Fear: A Translational Approach. Biol Psychiatry 79: 213-221. doi: 10.1016/j.biopsych.2015.06.004
![]() |
[42] |
41. Light KC, Grewen KM, Amico JA, et al. (2004) Deficits in plasma oxytocin responses and increased negative affect, stress, and blood pressure in mothers with cocaine exposure during pregnancy. Addict Behav 29: 1541-1564. doi: 10.1016/j.addbeh.2004.02.062
![]() |
[43] |
42. Linnen AM, Ellenbogen MA, Cardoso C, et al. (2012) Intranasal oxytocin and salivary cortisol concentrations during social rejection in university students. Stress 15: 393-402. doi: 10.3109/10253890.2011.631154
![]() |
[44] |
43. Norman GJ, Cacioppo JT, Morris JS, et al. (2011) Oxytocin increases autonomic cardiac control: moderation by loneliness. Biol Psychol 86: 174-180. doi: 10.1016/j.biopsycho.2010.11.006
![]() |
[45] |
44. Windle RJ, Kershaw YM, Shanks N, et al. (2004) Oxytocin attenuates stress-induced c-fos mRNA expression in specific forebrain regions associated with modulation of hypothalamo-pituitary-adrenal activity. J Neurosci 24: 2974-2982. doi: 10.1523/JNEUROSCI.3432-03.2004
![]() |
[46] |
45. Domes G, Heinrichs M, Gläscher J, et al. (2007) Oxytocin attenuates amygdala responses to emotional faces regardless of valence. Biol Psychiatry 62: 1187-1190. doi: 10.1016/j.biopsych.2007.03.025
![]() |
[47] |
46. Kirsch P, Esslinger C, Chen Q, et al. (2005) Oxytocin modulates neural circuitry for social cognition and fear in humans. J Neurosci 25: 11489-11493. doi: 10.1523/JNEUROSCI.3984-05.2005
![]() |
[48] |
47. Viviani D, Charlet A, van den Burg E, et al. (2011) Oxytocin selectively gates fear responses through distinct outputs from the central amygdala. Science 333: 104-107. doi: 10.1126/science.1201043
![]() |
[49] |
48. Sripada CS, Phan KL, Labuschagne I, et al. (2013) Oxytocin enhances resting-state connectivity between amygdala and medial frontal cortex. Int J Neuropsychopharmacol 16: 255-260. doi: 10.1017/S1461145712000533
![]() |
[50] |
49. Ferguson JN, Young LJ, Hearn EF, et al. (2000) Social amnesia in mice lacking the oxytocin gene. Nat Genet 25: 284-288. doi: 10.1038/77040
![]() |
[51] |
50. Wirth MM (2015) Hormones, stress, and cognition: The effects of glucocorticoids and oxytocin on memory. Adapt Human Behav Physiol 1: 177-201. doi: 10.1007/s40750-014-0010-4
![]() |
[52] | 51. McDonald JK, Barrett AJ (1986) Mammalian proteases: a glossary and bibliography (Academic Press, London) vol 2. |
[53] | 52. Checler F (1993) Methods in neurotransmitter and neuropeptide research, eds Parvez SH, Naoi M, Nagatsu T, Parvez S (Elsevier, Amsterdam). |
[54] |
53. Ramírez M, Prieto I, Banegas I, et al. (2011) Neuropeptidases. Methods Mol Biol 789: 287-294. doi: 10.1007/978-1-61779-310-3_18
![]() |
[55] |
54. Morales-Mulia M, de Gortari P, Amaya MI, et al. (2012) Activity and expression of enkephalinase and aminopeptidase N in regions of the mesocorticolimbic system are selectively modified by acute ethanol administration. J Mol Neurosci 46: 58-67. doi: 10.1007/s12031-011-9623-2
![]() |
[56] |
55. Ramírez M, Prieto I, Alba F, et al. (2008) Role of central and peripheral aminopeptidase activities in the control of blood pressure: a working hypothesis. Heart Fail Rev 13: 339-353. doi: 10.1007/s10741-007-9066-6
![]() |
[57] |
56. Reid KJ, McGee-Koch LL, Zee PC (2011) Cognition in circadian rhythm sleep disorders. Prog Brain Res 190: 3-20. doi: 10.1016/B978-0-444-53817-8.00001-3
![]() |
[58] | 57. Wright KP, Lowry CA, Lebourgeois MK (2012) Circadian and wakefulness-sleep modulation of cognition in humans. Front Mol Neurosci 5: 50. |
[59] |
58. Ramírez M, Prieto I, Vives F, et al. (2004) Neuropeptides, neuropeptidases and brain asymmetry. Curr Protein Pept Sci 5: 497-506. doi: 10.2174/1389203043379350
![]() |
[60] | 59. Turner AJ (2004) Neprilysin, In Handbook of Proteolytic Enzymes, eds Barrett AJ, Rawlings ND, Woessner JF (Elsevier, London) 419-426. |
[61] |
60. Iwata N, Takaki Y, Fukami S, et al. (2002) Region-specific reduction of A beta-degrading endopeptidase, neprilysin, in mouse hippocampus upon aging. J Neurosci Res 70: 493-500. doi: 10.1002/jnr.10390
![]() |
[62] |
61. Iwata N, Mizukami H, Shirotani K, et al. (2004) Presynaptic localization of neprilysin contributes to efficient clearance of amyloid-beta peptide in mouse brain. J Neurosci 24: 991-998. doi: 10.1523/JNEUROSCI.4792-03.2004
![]() |
[63] |
62. Li L, Tang BL (2005) Environmental enrichment and neurodegenerative diseases. Biochem Biophys Res Commun 334: 293-297. doi: 10.1016/j.bbrc.2005.05.162
![]() |
[64] |
63. Deweerdt S (2011) Prevention: activity is the best medicine. Nature 475: S16-17. doi: 10.1038/475S16a
![]() |
[65] |
64. Albiston AL, Fernando R, Ye S, et al. (2004) Alzheimer's, angiotensin IV and an aminopeptidase. Biol Pharm Bull 27: 7767. doi: 10.1248/bpb.27.765
![]() |
[66] |
65. Marvar PJ, Goodman J, Fuchs S, et al. (2014) Angiotensin type 1 receptor inhibition enhances the extinction of fear memory. Biol Psychiatry 75: 864-872. doi: 10.1016/j.biopsych.2013.08.024
![]() |
[67] | 66. Hmazzou R, Flahault A, Marc Y, et al. (2016) [OP.6D.03] Mode of action of rb150, an aminopeptidase a inhibitor prodrug as a centrally-acting antihypertensive agent in doca-salt hypertensive rats. J Hypertens 34 Suppl 2: e75. |
[68] |
67. Hernández J, Segarra AB, Ramírez M, et al. (2009) Stress influences brain enkephalinase, oxytocinase and angiotensinase activities: a new hypothesis. Neuropsychobiology 59: 184-189. doi: 10.1159/000219306
![]() |
[69] |
68. Tuppy H, Nesbadva H (1957) The aminopeptidase acitvity of serum in pregnancy and its relationship to the potential for inactivating oxytocin. Monatsh Chem 88: 977-988. doi: 10.1007/BF00905420
![]() |
[70] |
69. Tsujimoto M, Mizutani S, Adachi H, et al. (1992) Identification of human placental leucine aminopeptidase as oxytocinase. Arch Biochem Biophys 292: 388-392. doi: 10.1016/0003-9861(92)90007-J
![]() |
[71] |
70. Keller SR (2003) The insulin-regulated aminopeptidase: a companion and regulator of GLUT4. Front Biosci 8: s410-420. doi: 10.2741/1078
![]() |
[72] |
71. Stragier B, De Bundel D, Sarre S, et al. (2008) Involvement of insulin-regulated aminopeptidase in the effects of the renin-angiotensin fragment angiotensin IV: a review. Heart Fail Rev 13: 321-337. doi: 10.1007/s10741-007-9062-x
![]() |
[73] | 72. Gard PR (2008) Cognitive-enhancing effects of angiotensin IV. BMC Neurosci 9 Suppl 2: S15. |
[74] |
73. Banegas I, Prieto I, Vives F, et al. (2010) Lateralized response of oxytocinase activity in the medial prefrontal cortex of a unilateral rat model of Parkinson's disease. Behav. Brain Res 213: 328-331. doi: 10.1016/j.bbr.2010.05.030
![]() |
[75] | 74. Ramírez M, Banegas I, Segarra AB, et al. (2012) Bilateral Distribution of Oxytocinase Activity in the Medial Prefrontal Cortex of Spontaneously Hypertensive Rats with Experimental Hemiparkinsonism, Mechanisms in Parkinson's Disease—Models and Treatments, Dr. Juliana Dushanova (Ed.), ISBN: 978-953-307-876-2. |
[76] |
75. Prieto I, Villarejo AB, Segarra AB, et al. (2014) Brain, heart and kidney correlate for the control of blood pressure and water balance: role of angiotensinases. Neuroendocrinology 100: 198-208. doi: 10.1159/000368835
![]() |
[77] | 76. Maroun M, Richter-Levin G (2003) Exposure to acute stress blocks the induction of long-term potentiation of the amygdala-prefrontal cortex pathway in vivo. J Neurosci 23: 4406-4409. |
[78] |
77. Richardson MP, Strange BA, Dolan RJ (2004) Encoding of emotional memories depends on amygdala and hippocampus and their interactions. Nat Neurosci 7: 2285. doi: 10.1038/nn1190
![]() |
[79] |
78. Bass DI, Nizam ZG, Partain KN, et al. (2014) Amygdala-mediated enhancement of memory for specific events depends on the hippocampus. Neurobiol Learn Mem 107: 37-41. doi: 10.1016/j.nlm.2013.10.020
![]() |
[80] |
79. Thayer JF, Lane RD (2009) Claude Bernard and the heart-brain connection: further elaboration of a model of neurovisceral integration. Neurosci Biobehav Rev 33:81-88. doi: 10.1016/j.neubiorev.2008.08.004
![]() |
[81] |
80. Banegas I, Prieto I, Vives F, et al. (2009) Asymmetrical response of aminopeptidase A and nitric oxide in plasma of normotensive and hypertensive rats with experimental hemiparkinsonism. Neuropharmacology 56:573-579. doi: 10.1016/j.neuropharm.2008.10.011
![]() |
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