Diffusion in Fluid Flow: Dissipation Enhancement by Flows in 2D

We consider the advection-diffusion equation on ℝ2, with u a periodic incompressible flow and A ≫ 1 its amplitude. We provide a sharp characterization of all u that optimally enhance dissipation in the sense that for any initial datum φ0 ∈ L p (ℝ2), p < ∞, and any τ > 0, Our characterization is expressed in terms of simple geometric and spectral conditions on the flow. Moreover, if the above convergence holds, it is uniform for φ0 in the unit ball in L p (ℝ2), and ‖·‖∞ can be replaced by any ‖·‖ q , with q > p. Extensions to higher dimensions and applications to reaction-advection-diffusion equations are also considered.

[1]  Yuri Kifer,et al.  On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles , 1980 .

[2]  Andrej Zlatos,et al.  Pulsating front speed-up and quenching of reaction by fast advection , 2007, 0704.1164.

[3]  Quenching of combustion by shear flows , 2004, math/0405397.

[4]  R. Shterenberg,et al.  Relaxation Enhancement by Time-Periodic Flows , 2007, 0706.4411.

[5]  A. Fannjiang,et al.  Dissipation time and decay of correlations , 2003, math/0311209.

[6]  J. Norris.  Long-Time Behaviour of Heat Flow: Global Estimates and Exact Asymptotics , 1997 .

[7]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[8]  P. Constantin,et al.  Diffusion and mixing in fluid flow , 2005 .

[9]  Chii-Ruey Hwang,et al.  The behavior of the spectral gap under growing drift , 2009 .

[10]  J. Nash.  Continuity of Solutions of Parabolic and Elliptic Equations , 1958 .

[11]  A. Fannjiang,et al.  Noise Induced Dissipation in Lebesgue-Measure Preserving Maps on d-Dimensional Torus , 2002, math/0209231.

[12]  N. Nadirashvili,et al.  Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena , 2005 .

[13]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[14]  A. Majda,et al.  SIMPLIFIED MODELS FOR TURBULENT DIFFUSION : THEORY, NUMERICAL MODELLING, AND PHYSICAL PHENOMENA , 1999 .

[15]  Quenching of reaction by cellular flows , 2005, math/0505654.

[16]  George Papanicolaou,et al.  Convection Enhanced Diffusion for Periodic Flows , 1994, SIAM J. Appl. Math..

[17]  Sergei Petrovich Novikov,et al.  On Dynamical Systems with an Integral Invariant on a Torus , 1991 .

[18]  M. Freidlin.  Reaction-Diffusion in Incompressible Fluid: Asymptotic Problems , 2002 .

[19]  Mark Freidlin,et al.  Random perturbations of Hamiltonian systems , 1994 .

[20]  Y. Kifer.  Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states , 1990 .

[21]  Peter Meier,et al.  On the critical exponent for reaction-diffusion equations , 1990 .

[22]  Yu Shu-Xiang,et al.  On dynamical systems with an integral invariant on the torus , 1984 .

[23]  M. Freidlin,et al.  Functional Integration and Partial Differential Equations. (AM-109), Volume 109 , 1985 .

[24]  S. Bates,et al.  Toward a precise smoothness hypothesis in Sard’s theorem , 1993 .

[25]  J. Roquejoffre.  Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders , 1997 .

[26]  M. Freidlin.  Functional Integration And Partial Differential Equations , 1985 .

[27]  Andrej Zlatoš.  Quenching and propagation of combustion without ignition temperature cutoff , 2004, math/0409172.

[28]  Quenching of flames by fluid advection , 2000, nlin/0006024.