Singular perturbations for PDEs

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4.4. Singular perturbations for PDEs

  1. Introduction
  2. Boundary layer type solutions. 1
  3. Boundary layer type solutions. 2
  4. Final remarks

Introduction

Here we consider PDEs. We do not consider regular perturbation as their theory is an obvious modification of what we studied in Section 4.1

Boundary layer type solution. 1

Let $L$ be a $0$-order and $M$ be a second-order operators as in Section 4.2: \begin{align} &Lu= \alpha(x)u, \label{eq-4.4.1}\\ &Mu=\sum_{j,k} \partial_j(\beta^{jk}(x)\partial_k u) \label{eq-4.4.2} \end{align} with $\alpha >0$ and $(\beta^{jk})$ real symmetric and positive definite matrix.

Again we are looking for solution \begin{align} &L_\varepsilon u:= (L+\varepsilon^2M)u =f&&\text{in } \mathcal{D} \label{eq-4.4.3}\\ &u|_\Gamma =g \label{eq-4.4.4} \end{align} where $\Gamma=\partial \mathcal{D}$.

Again like in Section 4.2 we can assume that $f=0$ as we can get rid off it ignoring boundary condition and finding some solution by methods of the regular perturbation theory; see again Section 4.1. Again we are looking for solution in the form $u=e^{-\varepsilon^{-1}\phi(x)}w$ and we get considering only terms with $\varepsilon^0$ \begin{align} &\sum_j\beta^{jk}(\partial_j\phi)(\partial_k\phi)=\alpha &&\text{in } \mathcal{D}, \label{eq-4.4.5}\\ &\phi|_\Gamma=0. \label{eq-4.4.6} \end{align} This is non-linear first-order PDE. The theory see in Subsection 2.2.2 of online Textbook for PDEs. Such equations do not necessarily have global solutions but here in contrast to what we will encounter in the future we do not need them: we need only solution in the vicinity of $\Gamma$ and this solution is exactly the distance to $\Gamma$ in the metrics \begin{equation} ds^2=\sum _{j,k} \alpha^{-1}\beta_{jk}dx_jdx_k \label{eq-4.4.7} \end{equation} where $(\beta_{jk})$ is an inverse matrix to $(\beta^{jk})$. Another solution would be negative in $\mathcal{D}$ near $\Gamma$.

Then \begin{equation} L_\varepsilon (e^{-\varepsilon^{-1}\phi(x)}w)=e^{-\varepsilon^{-1}\phi(x)} \bigl(\varepsilon P w +\varepsilon^2 Mw\bigr) \label{eq-4.4.8} \end{equation} where \begin{align} &Pw= \sum _{j,k} 2\beta^{jk} (\partial_j\phi) (\partial_kw)+ \sigma w, \label{eq-4.4.9}\\ &\sigma= \sum _{j,k} \beta^{jk} (\partial_j\partial_k\phi)+ \sum_j \gamma^j\partial_j \phi , \label{eq-4.4.10} \end{align} $Q=-M$. Now we can find locally near $\Gamma$ $w\sim\sum_{n\ge 0} w_n\varepsilon^n$ from \begin{equation} Pw_n + Qw_{n-1}=0, \qquad w_n|_\Gamma=g_n. \label{eq-4.4.11} \end{equation} Important is that differentiation in $P$ is not tangent to $\Gamma$, it is along normal (in metrics (\ref{eq-4.4.7})).

Remark 1.

  1. Again the is a boundary layer (a.k.a. inner zone) $\{x\in D\colon \phi(x) \le \varepsilon ^{1-\delta}\}$ and a regular zone (a.k.a. inner zone) $\{x\in D\colon \phi(x) \ge \varepsilon ^{1-\delta}\}$ ; see Remark 4.2.3.
  2. Introducing near $\Gamma$ coordinates $x_1:=\phi(x)$ and $x'$ coordinates on $\Gamma$ we see that in the inner zone there is a fast coordinate $x_1$ and slow coordinates (actually normal) $x'$.

Boundary layer type solution. 2

Let $L$ be a first-order and $M$ be a second-order operators as in Subsection 4.3.2: \begin{equation} Lu=\sum_j \frac{1}{2}(\alpha_j \partial_j u +\partial _j (\alpha_j u)) \label{eq-4.4.12} \end{equation} and $M$ is defined by (\ref{eq-4.4.2}). Then $L_\varepsilon u=(L+\varepsilon M)u=f$ and we consider BVP (\ref{eq-4.4.2}). Here we assume that

Condition 1. Vector-field $\vec{\alpha}=(\alpha^1,\ldots,\alpha^d)$ does not vanish anywhere in $\mathcal{D}$ and is not tangent to $\Gamma$.

Again plugging $u=(e^{-\varepsilon^{-1}\phi(x)}w)$ and considering only terms with $\varepsilon^{-1}$ we get \begin{equation} \sum_j\alpha^j\partial_j\phi = \sum_{j,k}\beta^{jk}(\partial_j\phi)(\partial_k\phi) \label{eq-4.4.13} \end{equation} which again is non-linear equation. With $\phi_\Gamma=0$ it has the unique solution (with $\nabla\phi\ne 0$) but there is a twist: this solution is positive in $\mathcal{D}$ only near $\Gamma_+$ where $\Gamma_+$ is a part of $\Gamma$ where $\vec{\alpha}$ is directed inside $\mathcal{D}$. The rest of the construction is the same: we define $w\sim\sum_{n\ge 0}w_n\varepsilon^n$ solving \begin{equation} Pw_n + Qw_{n-1}=0, \qquad w_n|_{\Gamma_+}=g_n \label{eq-4.4.14} \end{equation} with \begin{align} &Pw= \sum _{j} \eta^k (\partial_kw)+ \sigma w, \label{eq-4.4.15}\\ &\eta^k =-2\sum_j \beta^{jk}(\partial_j \phi)-\alpha^k. \label{eq-4.4.16} \end{align} Observe that vector field $\vec{\eta}$ is not tangent to $\Gamma$. Indeed, one can see easily that $\sum_k\eta^k \partial_k\phi \ne 0$.

Therefore boundary layer type solution exists only near $\Gamma_+$. For another part of $\Gamma$ the regular perturbation theory takes care. Indeed, we take $U\sim \sum_{n\ge 0}U_n \varepsilon ^n$ where $U_n$ are found from \begin{align} &LU_n+Mu_{n-1}=f_n \label{eq-4.4.17}\\ &U_n|_{\Gamma_-}=g_{*n} \label{eq-4.3.18} \end{align} where $\Gamma_-$ is a part of $\Gamma$ where $\vec{\alpha}$ is directed outside $\mathcal{D}$.

Remark 2. If Condition 1 does not hold then under some restrictions one can derive different asymptotics near points of $\Gamma$ where $\vec{\alpha}$ is tangent to $\Gamma$. However it requires much more advanced technique.

Final remarks

Remark 3. In Sections 4.2--4.4 one can consider more general operators and we cannot always guarantee that the original problem is well-posed and its solution exists. Then we are looking for formal asymptotic solutions satisfying \begin{align} &L_\varepsilon u_\varepsilon \sim f,\label{eq-4.3.189}\\ &u|_\Gamma \sim g \label{eq-4.3.20}. \end{align} Here errors are $O(\varepsilon^\infty)$.

Remark 4.

  1. Consider again (\ref{eq-4.4.3})-\ref{eq-4.4.4}) with $L,M$ defined by (\ref{eq-4.4.1}) and (\ref{eq-4.4.2}), however with $g$ having jump at $\Gamma_0\subset \Gamma$ is a smooth subsurface, separating $\Gamma$ into $\Gamma_+=\{x\in \Gamma \colon \psi(x)>0\}$ and $\Gamma_-=\{x\in \Gamma \colon \psi(x)<0\}$, $\nabla_\Gamma \psi $ disjoint from $0$ at $\Gamma_0$. Scaling we can construct an asymptotic in the outer zone $\{x\in D\colon \phi (x) \ge \varepsilon^{1-\delta}\}$ and in the inner zone $\{x\in D\colon \phi (x) \le \varepsilon^{1-\delta},\ |\psi (x)|\ge \varepsilon^{1-\delta}\}$, but there remains zone $\{x\in D\colon \phi (x) \le \varepsilon^{1-\delta},\ |\psi (x)|\le \varepsilon^{1-\delta}\}$ with respect to each two previous zones are outer zones, and this one is an inner zone. So, we have a hierarchy of inner zones. In the "inner inner zone" there are two fast coordinates $x_1=\phi(x)$ and $x_2=\psi(x)$ and other coordinates $x''$ (which are coordinates along $\Gamma_0$ are slow.
  2. Similar situation appears when there are different boundary conditions on $\Gamma_+$ and $\Gamma_-$: \begin{gather*} u|_{\Gamma_-}=g,\qquad \sum_{j,k} \nu_k \beta^{jk} \partial_k u|_{\Gamma_+}=h. \end{gather*}
  3. Similar situation appears as $L$ is defined by (\ref{eq-4.4.12}) but on $\Gamma_0$ vector field $\vec{\alpha}$ is tangent to $\Gamma$ (but not to $\Gamma_0$).

However all these cases are much more difficult and we do not consider them.

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