This question is about equation (2.16) of Lecture 2 on Homogenization in Porous Media_ by Allaire page 28.

There are two spacial scales: $x$ being macroscopic and $y=\dfrac{x}{\varepsilon}$ being microscopic.

As well there are two time scales: $t$ the short time scale and $\tau=\varepsilon^2t$ the long time scale.

Let $(\lambda, w(y))$ be the first eigenvalue/-function pair of the cell spectral problem (2.9, page 25). The first function of the asymptotic expansion (2.13, page 27), $u_0(t,\tau,x,y)$, can then be written in terms of the long-time behaviour of the solution $u(\tau,x)e^{-\lambda t}w(y)$.

Deriving the cascade of equations with different powers of $\varepsilon$ from plugging series (2.13) into the initial problem equation (2.8) is easy. Thus equation (2.15) is clear: $$c(y)\frac{\partial u_1}{\partial t} - \mathrm{div}_y(A(y)\nabla_y u_1)=\sigma(y)u_1+g_1$$ with $$g_1=\mathrm{div}_x(A(y)\nabla_y u_0)+\mathrm{div}_y(A(y)\nabla_x u_0)$$

Though the argumentation, that $g_1$ must be orthogonal to the first eigenfunction $w(y)$ is reasonable, I'm struggling with solving (2.16, page 28) $$\int_Y g_1(y)w(y)dy=0$$ into $$\int_Y g_1(y)w(y)dy \\=e^{-\lambda t}\int_Y A(y)\nabla_y w(y)\cdot\nabla_x u(x)w(y)-A(y)w(y)\nabla_x u(x)\cdot\nabla_y w(y)dy=0$$ withverwarnung the symmetric diffusion tensor $A(y)$.

Especially I'm confused by the change of sign. As well, I would assume, that the first term can be written as $$A(y)\nabla_y w(y)u(x)\cdot\nabla_xw(y),$$ which would equal to zero as $\nabla_xw(y)=0$. And following it would read $$\int_Y A(y)w(y)\nabla_xu(x)\cdot\nabla_yw(y)dy=0,$$
which I don't see to be true in general.

Thank you for any help and hints.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.