时间周期流的弛豫增强
Relaxation Enhancement by Time-Periodic Flows
摘要：
我们研究了快速不可压缩时间周期流对扩散混合的增强作用。在文献[2]中已经介绍了在加速混合方面特别有效的弛豫增强流的类别。流的弛豫增强特性已被证明与它所产生的动力系统的特性密切相关。特别地，使得算子$u \cdot \nabla$具有足够平滑的本征函数的时间无关流$u$不是松弛增强的。这里我们将[2]的结果推广到时间周期流$u(x,t)$，并特别证明了存在这样的流：对于每个固定的时间，流是Hamilton的，但由此产生的时间相关流是松弛增强的。因此，我们确认的物理直觉，时间依赖性的流动可能有助于混合。我们还提供了我们的结果的非线性扩散模型的情况下的一个扩展。证明是基于一个半群的衰变的一般准则所产生的算子的形式$Γ+iAL(t)$与一个负的无界自伴算子$Γ$，一个时间周期的自伴算子值函数$L(t)$，和一个参数$A>>1$。
英文摘要：
We study enhancement of diffusive mixing by fast incompressible time-periodic flows. The class of relaxation-enhancing flows that are especially efficient in speeding up mixing has been introduced in [2]. The relaxation-enhancing property of a flow has been shown to be intimately related to the properties of the dynamical system it generates. In particular, time-independent flows $u$ such that the operator $u \cdot \nabla$ has sufficiently smooth eigenfunctions are not relaxation-enhancing. Here we extend results of [2] to time-periodic flows $u(x,t)$ and in particular show that there exist flows such that for each fixed time the flow is Hamiltonian, but the resulting time-dependent flow is relaxation-enhancing. Thus we confirm the physical intuition that time dependence of a flow may aid mixing. We also provide an extension of our results to the case of a nonlinear diffusion model. The proofs are based on a general criterion for the decay of a semigroup generated by an operator of the form $Γ+iAL(t)$ with a negative unbounded self-adjoint operator $Γ$, a time-periodic self-adjoint operator-valued function $L(t)$, and a parameter $A>>1$.
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