Diffusion and mixing in fluid flow. (English) Zbl 1180.35084

Constantin, Peter; Kiselev, Alexander; Ryzhik, Lenya; Zlatoš, Andrej

Ann. Math. (2) 168, No. 2, 643-674 (2008).

An enhancement of diffusive mixing on a compact Riemannian manifold \(M\) by a fast incompressible flow is studied in the paper. The incompressible flow \(u\) on \(M\) and positive parameter \(A\) define the scalar function \(\varphi^A(x,t)\) satisfying the problem
\[ \frac{\partial}{\partial t}\varphi^A(x,t)+Au\cdot\nabla\varphi^A(x,t)-\Delta\varphi^A(x,t)=0,\quad \varphi^A(x,0)=\varphi_0(x), \]
where \(\Delta\) is the Laplace-Beltrami operator on \(M\), \(\nabla\) is the covariant derivative. The behavior of the function \(\varphi^A(x,t)\) for large \(A\) at a fixed time \(\tau>0\) is the topic of the paper. It is proved that for every \(\tau>0\) and \(\delta>0\) the inequality
\[ \| \varphi^A(\cdot,\tau)-\bar{\varphi}\|_{L^2(M)}<\delta \]
holds for a large number \(A\) if and only if the operator \(u\cdot\nabla\) has no eigenfunctions in \(H^1(M)\), other than the constant function. Here \(\varphi_0\in L^2(M)\), \(\|\varphi_0\|_{L^2(M)}=1\),
\[ \bar{\varphi}=\frac{1}{|M|}\int\limits_M\varphi_0(x)\,d\mu. \]
The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form \(\Gamma+iAL\) with a negative unbounded self-adjoint operator \(\Gamma\) and a self-adjoint operator \(L\).

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

MSC:

35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K15	Initial value problems for second-order parabolic equations
35K57	Reaction-diffusion equations
35Q35	PDEs in connection with fluid mechanics
58J35	Heat and other parabolic equation methods for PDEs on manifolds

Keywords:

diffusive mixing; incompressible flow; Riemannian manifold; compact Riemannian manifold; fast incompressible flow