[Paper Review] A nonlinear singular perturbation problem

А. Г. Рамм|ArXiv.org|May 3, 2004

Differential Equations and Numerical Methods7 references3 citations

TL;DR

This paper studies a nonlinear singular perturbation problem in a Hilbert space, where the Fréchet derivative of the nonlinear operator at the solution of the limiting equation is not Fredholm or invertible. By assuming a specific resolvent growth condition $\|(A + \varepsilon)^{-1}\| \leq c\varepsilon^{-1}$, it proves existence and convergence $\|u_\varepsilon - y\| \to 0$ as $\varepsilon \to 0$, with $\|u_\varepsilon - y\| = O(\varepsilon)$, and establishes uniqueness in a small $O(\varepsilon)$-neighborhood. The result is applied to a nonlinear integral equation with a Newtonian-type kernel.

ABSTRACT

Let F(u_\ve)+\ve(u_\ve-w)=0 \eqno{(1)} where $F$ is a nonlinear operator in a Hilbert space $H$, $w\in H$ is an element, and $\ve>0$ is a parameter. Assume that $F(y)=0$, and $F'(y)$ is not a boundedly invertible operator. Sufficient conditions are given for the existence of the solution to \eqref{e1.1} and for the convergence $\lim_{\ve o 0}\|u_\ve-y\|=0$. An example of applications is considered. In this example $F$ is a nonlinear integral operator.

Motivation & Objective

• To analyze the convergence of solutions $u_\varepsilon$ to a nonlinear operator equation $F(u_\varepsilon) + \varepsilon(u_\varepsilon - w) = 0$ as $\varepsilon \to 0$, when the Fréchet derivative $F'(y)$ is not invertible or Fredholm.
• To establish existence and convergence $\|u_\varepsilon - y\| \to 0$ under a weaker condition than standard invertibility, specifically $\|(A + \varepsilon)^{-1}\| \leq c\varepsilon^{-1}$, where $A = F'(y)$.
• To extend singular perturbation theory beyond the classical case where $F'(y)$ is an isomorphism or Fredholm operator, particularly to cases where $F'(y)$ is compact or has a singular resolvent growth.
• To demonstrate the applicability of the theory to a nonlinear integral equation involving the Newtonian potential, where $F(u) = \int_D \frac{u^3(s)}{4\pi|x-s|} ds$.

Proposed method

• Transform the equation $F(u_\varepsilon) + \varepsilon(u_\varepsilon - w) = 0$ into a fixed-point problem via the substitution $z_\varepsilon = u_\varepsilon - y$, leading to $\phi(z_\varepsilon) + \varepsilon z_\varepsilon + \varepsilon(y - w) = 0$, where $\phi(z) = F(y + z)$.
• Use the Taylor expansion $\phi(z) = Az + K(z)$ with $\|K(z)\| \leq \frac{M_2\|z\|^2}{2}$, and rewrite the equation as $z_\varepsilon = -A_\varepsilon^{-1}K(z_\varepsilon) - \varepsilon A_\varepsilon^{-1}(y - w)$, where $A_\varepsilon = A + \varepsilon I$.
• Apply Schauder’s fixed-point theorem in a ball $B(0, R)$ with $R = O(\varepsilon)$, under the assumption $\|(A + \varepsilon)^{-1}\| \leq c\varepsilon^{-1}$ and $y - w = Av$ with $\|v\|$ small enough.
• Prove uniqueness by showing the fixed-point map is a contraction in a small ball, using the estimate $\|K(z) - K(v)\| \leq \left(\frac{M_3 R^2}{6} + M_2 R\right)\|z - v\|$, leading to a contraction factor less than 1 for small $\varepsilon$.
• Apply the abstract result to a specific nonlinear integral equation $\int_D \frac{u^3(s)}{4\pi|x-s|} ds + \varepsilon(u_\varepsilon - w) = f$, with $f = 0$, by verifying the resolvent condition and using Sobolev space estimates in $H^1(D)$ for $n=3$.
• Establish existence and convergence in $H^1(D)$ by showing the operator $T(u) = -\varepsilon^{-1}F(u) + \varepsilon h$ maps a ball of radius $R = \varepsilon^{2/3}$ into itself and is a contraction for small $\varepsilon$.

Experimental results

Research questions

• RQ1 Under what conditions does the solution $u_\varepsilon$ of the perturbed equation $F(u_\varepsilon) + \varepsilon(u_\varepsilon - w) = 0$ converge to $y$ as $\varepsilon \to 0$ when $F'(y)$ is not invertible or Fredholm?
• RQ2 Can the classical singular perturbation theory be extended to cases where the linearization $F'(y)$ is a compact operator or has a singular resolvent growth $\|(A + \varepsilon)^{-1}\| \leq c\varepsilon^{-1}$?
• RQ3 What conditions on $w$ ensure the existence and convergence of $u_\varepsilon$ in the absence of standard invertibility assumptions on $F'(y)$?
• RQ4 How can the abstract convergence result be applied to a specific nonlinear integral equation with a Newtonian kernel?
• RQ5 What is the rate of convergence $\|u_\varepsilon - y\|$ in such cases, and can uniqueness be established in a small $O(\varepsilon)$-neighborhood?

Key findings

• The solution $u_\varepsilon$ to the perturbed equation exists for all sufficiently small $\varepsilon > 0$ under the condition $\|(A + \varepsilon)^{-1}\| \leq c\varepsilon^{-1}$, where $A = F'(y)$, and $F$ is compact and three times continuously Fréchet differentiable in a neighborhood of $y$.
• The convergence $\|u_\varepsilon - y\| \to 0$ holds as $\varepsilon \to 0$, and the rate is $\|u_\varepsilon - y\| = O(\varepsilon)$, which is stronger than mere convergence.
• Uniqueness of the solution holds in a ball of radius $R = O(\varepsilon)$ centered at $y$, provided $\|v\| < \frac{1}{2M_2 c(1 + c)}$ where $y - w = Av$ and $c$ is the constant in the resolvent bound.
• For the specific integral equation $\int_D \frac{u^3(s)}{4\pi|x-s|} ds + \varepsilon(u_\varepsilon - w) = 0$, with $w = \varepsilon h$ and $\|h\|_{H^1(D)} = 1$, a unique solution $u_\varepsilon$ exists in $H^1(D)$ for all $\varepsilon \in (0, \varepsilon_0)$ with $\varepsilon_0$ sufficiently small.
• The solution satisfies $\|u_\varepsilon\|_{H^1(D)} = O(\varepsilon^{2/3})$, and $\|u_\varepsilon\|_{H^1(D)} \to 0$ as $\varepsilon \to 0$, confirming convergence to the trivial solution $y = 0$.
• The contraction mapping principle is successfully applied in $H^1(D)$ by choosing $R = \varepsilon^{2/3}$, and the contraction factor is $O(\varepsilon^{1/3})$, which is less than 1 for small $\varepsilon$.