[Paper Review] A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant

Barbara Niethammer, Sebastian Throm+1 more|arXiv (Cornell University)|Oct 12, 2015

Stochastic processes and statistical mechanics5 references5 citations

TL;DR

This paper provides a corrected and complete proof of the uniqueness of self-similar profiles for Smoluchowski's coagulation equation with kernels close to constant, under homogeneity of degree zero and analyticity assumptions. Using Laplace transform techniques and perturbation analysis, it establishes uniqueness for sufficiently small perturbations of the constant kernel, resolving a gap in a prior proof and extending the result to a broad class of physically relevant kernels.

ABSTRACT

In this article we correct the proof of a uniqueness result for self-similar solutions to Smoluchowski's coagulation equation for kernels $K=K(x,y)$ that are homogeneous of degree zero and close to constant in the sense that \begin{equation*} -\varepsilon \leq K(x,y)-2 \leq \varepsilon \left( \Big(\frac{x}{y}\Big)^α + \Big(\frac{y}{x}\Big)^α ight) \end{equation*} for $α\in [0,\frac 1 2)$. Assuming in addition that $K$ has an analytic extension to $\mathbb{C}\setminus(-\infty,0]$ and prescribing the precise asymptotic behaviour of $K$ at the origin, we prove that self-similar solutions with given mass are unique if $\varepsilon$ is sufficiently small.

Motivation & Objective

• To correct a critical gap in a prior proof of uniqueness for self-similar solutions to Smoluchowski’s coagulation equation with kernels near constant.
• To establish the uniqueness of self-similar profiles with finite mass for kernels that are homogeneous of degree zero and close to the constant kernel in a specific weighted norm.
• To prove uniqueness under analyticity and boundedness conditions on the perturbation of the kernel, particularly for kernels with homogeneity zero and $α \in [0, \frac{1}{2})$.
• To extend the applicability of uniqueness results to non-solvable kernels, such as the classical kernel for three-dimensional coagulating particles.

Proposed method

• Transform the coagulation equation into an integral equation for the self-similar profile using the scaling ansatz $\phi(\xi,t) = t^{-2}f(\xi/t)$.
• Use the Laplace transform of the profile $f$ to convert the integral equation into a differential equation in the Laplace variable $p$.
• Apply perturbation analysis by writing the kernel as $K = 2 + \varepsilon W$, with $W$ bounded by a power-law function of $x/y$ and $y/x$.
• Use analyticity of $W(\cdot,1)$ on $\mathbb{C} \setminus (-\infty,0]$ and its boundedness to control the Laplace transform of the perturbation term.
• Derive a linearized equation for the difference $m = \mu - c_\varepsilon$ between two candidate profiles and estimate its $L^1$ and weighted $L^1$ norms via $p$-dependent bounds.
• Use the resulting differential equation $M'(p) + \frac{2c_\varepsilon}{p}M(p) = R(p)$ and integrability arguments to show $m = 0$ for small $\varepsilon$, implying uniqueness.

Experimental results

Research questions

• RQ1 Does the self-similar profile for Smoluchowski’s coagulation equation remain unique when the kernel is a small perturbation of the constant kernel?
• RQ2 Can the uniqueness result be rigorously established for kernels homogeneous of degree zero and close to constant, under analyticity and growth conditions?
• RQ3 What role does the parameter $\alpha \in [0, \frac{1}{2})$ play in ensuring the uniqueness of self-similar solutions?
• RQ4 How can the gap in the earlier proof of uniqueness be corrected using Laplace transform and perturbation techniques?

Key findings

• The self-similar profile with finite mass is unique for kernels $K$ that are homogeneous of degree zero and satisfy $|K(x,y) - 2| \leq \varepsilon \left( (x/y)^\alpha + (y/x)^\alpha \right)$ with $\alpha \in [0, \frac{1}{2})$.
• Uniqueness holds for sufficiently small $\varepsilon > 0$, provided $K$ has an analytic extension to $\mathbb{C} \setminus (-\infty, 0]$ and satisfies the stated growth and positivity bounds on $W$.
• The proof establishes that any two self-similar profiles with the same mass must be identical, by showing the difference $m = \mu - c_\varepsilon$ vanishes in the limit $\varepsilon \to 0$.
• The Laplace transform method successfully controls the perturbation term via bounds on $R(p)$, leading to $\|m\|_{(0)} \leq C\delta \|m\|_{(0)}$ with $\delta \to 0$ as $\varepsilon \to 0$, forcing $m = 0$.
• The result confirms the dynamic scaling hypothesis for this class of kernels, implying convergence to a unique self-similar profile as $t \to \infty$.