Singular perturbations for PDEs

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$

Singular perturbations for PDEs

  1. Introduction
  2. Boundary layer type solution. 1
  3. Boundary layer type solution. 2


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} \beta^{jk}(x)\partial_j\partial_k u +\sum_j \gamma^j\partial_j u + \gamma^0 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^2)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 regular perturbation theory. Again we are looking for solution in the form $u=e^{-\phi(x)/\varepsilon}w$ 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 equation. 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^{-\phi(x)/\varepsilon}w)=e^{-\phi(x)/\varepsilon} \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 that differentiation in $P$ is not tangent to $\Gamma$, it is along normal (in metrics (\ref{eq-4.4.7})).

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 \alpha_j \partial_j u +\alpha^0 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 $\vec{\alpha}=(\alpha^,\ldots,\alpha^d)$ does not vanish anywhere in $\mathcal{D}$ and is not tangent to $\Gamma$.

Again plugging $u=(e^{-\phi(x)/\varepsilon}w)$ 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 $w\sim\sum_{n\ge 0}w_nvarepsilon^n$ 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 $\mathcal{D}$.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$