Inertial manifolds for a Smoluchowski equation on the unit sphere
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 result (Thm. 3) from the literature which gives sufficient conditions for \(A\) and \(N(\;)\) so that (4) admits an inertial manifold. Thus \(N(\;)\) is required to have compact support, to be bounded and globally Lipschitz. In addition, a spectral gap condition is assumed to hold. Now an inspection shows that (1)+(2) and hence (4) violate the above assumptions. In order to remedy the situation the author performs the substitution \[ u=\psi\exp(V/2)\tag{5} \] with the aim to eliminate the gradient from (1). After a series of transformations followed by a smoothening process the author obtains a regularized equation for \(u\) in (5), i.e.: \[ \partial_pu+Au=N_p(u).\tag{6} \] In the proof of the last result, i.e. Thm. 4, it is shown that (6) satisfies all the requirements listed above (Thm. 3) and hence admits an inertial manifold \({\mathcal M}_p\) and an inertial manifold \(\mathcal M\) for (4) and hence for (1)+(2) can be obtained.
scientific article; zbMATH DE number 5610529
scientific article