Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models (Q2737920)

In the main body of the paper the authors consider the two-dimensional initial-boundary value problem for the nonlocal Swift-Hohenberg equation \(u_t= \mu u-(1+ \Delta)^2 u- uG* u^2\), in a bounded planar domain \(D\) with a smooth boundary. \(G\) is either a positive kernel such that \(0< b\leq G\leq a\) and \(\nabla G,\Delta G\in L^\infty(D)\) or it is a special `mollifier'-like function \(J_{\delta_0}\), in which case the lower bound \(b\) is zero.NEWLINENEWLINENEWLINEThe authors prove that this problem possesses a compact global attractor \({\mathcal A}\) and that the upper bound on the Hausdorff dimension of \({\mathcal A}\) is \(m\sim C(1+\sqrt{{2a\mu\over b}})\). In the case of the two-dimensional local Swift-Hohenberg equation \(u_t=\mu u-(1+ \Delta)^2- u^3\), which is discussed by the authors in a final part of the paper, the Hausdorff dimension of the global attractor is bounded by \(m_1\sim C(1+ \sqrt\mu)\). Thus, even if \(G= a= b\) (i.e., the kernel is a constant function) the dimension estimate for a nonlocal model still differs from the dimension estimate for the local model by a constant, which depends on the Rayleigh number.