Inertial manifolds for a Smoluchowski equation on the unit sphere
type: Item
instance of: scholarly article
title: Inertial manifolds for a Smoluchowski equation on the unit sphere
zbMATH Open document ID: 1177.37079
published in: Communications in Mathematical Physics
publication date: 2009-10-02T00:00:00Z
review text: The aim of the author is to construct inertial manifolds for the following equation of Smoluchowski type: \[ \partial_t\psi=\Delta_m\psi+\nabla_m\cdot(\psi\nabla_mV).\tag{1} \] Here \(m=S^2\), i.e., \(m=(m_1,m_2,m_3)\) where \(m_1=\sin\theta\cos\varphi\), \(m_2=\sin\theta\sin\varphi\), \(m_3=\cos\theta\). Moreover, \(\Delta_m\) is the Laplace-Beltrami operator on \(S^2\) while \(\nabla_m\) is the gradient on \(S^2\). Moreover, \(V=V(m,t)\) is a mean field interaction potential expressed in terms of \(\psi(m,t)\) as follows: \[ V(m,t)\equiv-b(m\cdot m-I/3):S[\psi(t)]\text{ with }S[\psi(t)]=\int_{S^2}(m\cdot m-I/3)\psi(m,t)\,dm\tag{2} \] where \(I=(\delta_{jk})\) and \(m\cdot m=(m_jm_k)\). Under these stipulations, (1) becomes a nonlinear, nonlocal equation for \(\psi(m,t)\) which is assumed to satisfy \[ \int_{S^2}\psi(m,t)\,dm\equiv 1\text{ and }0\leq\psi(m,t).\tag{3} \] It is known that if \(\psi_0\in C^0(S^2)\) satisfies \(\psi_0>0\) and \(\int_{S^2}\psi_0\,dm=1\) then there exists a unique smooth solution \(\psi(m,t)\) of (1) which satisfies (3) and \(\psi(m,0)=\psi_0(m)\). It is noted that (1) preserves certain symmetries. This allows the author to restrict attention to solutions invariant with respect to reflections at the coordinate planes. Based on a digression on spherical harmonics, the author proves Thm. 2, which displays an estimate guaranteeing the existence of an absorbing set for (1). The author then rephrases (1)+(2) as an abstract evolution equation on a Hilbert space \(H\) with scalar product (\,,\,) and norm \(\|\;\|\): \[ \partial_tu+Au=N(u)\tag{4} \] with \(A\) positive, selfadjoint and having compact inverse, while \(N:H\to H\) is a locally Lipschitz function. An inertial manifold \(\mathcal M\) for (4) is a finite dimensional Lipschitz manifold such that \({\mathcal S}(t){\mathcal M}\subset {\mathcal M}\), \(t\geq 0\) where \({\mathcal S}(t)\), \(t\geq 0\) is the nonlinear evolution semigroup associated with (4). The author then recalls a resul…
reviewed by: Bruno Scarpellini
Mathematics Subject Classification ID: 37L25
Mathematics Subject Classification ID: 37N20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B42
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 5610529
author: Jesenko Vukadinović
MaRDI profile type: Publication
full work available at URL: s00220-008-0460-2
OpenAlex ID: W2065657475
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DOI: 10.1007/S00220-008-0460-2
description: scientific article; zbMATH DE number 5610529