Enhanced Dissipation for Two-Dimensional Hamiltonian Flows

Abstract

Let H∈C1∩W2,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\in C^1\\cap W^{2,p}$$\\end{document} be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field b=∇⊥H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b=\ abla ^\\perp H$$\\end{document}. We give sharp upper bounds on the enhanced dissipation rate of b in terms of the properties of the period T(h) of the closed orbit {H=h}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{H=h\\}$$\\end{document}. Specifically, if 0<ν≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\ u \\ll 1$$\\end{document} is the diffusion coefficient, the enhanced dissipation rate can be at most O(ν1/3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\ u ^{1/3})$$\\end{document} in general, the bound improves when H has isolated, non-degenerate elliptic points. Our result provides the better bound O(ν1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\ u ^{1/2})$$\\end{document} for the standard cellular flow given by Hc(x)=sinx1sinx2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{\ extsf{c}}(x)=\\sin x_1 \\sin x_2$$\\end{document}, for which we can also prove a new upper bound on its mixing rate and a lower bound on its enhanced dissipation rate. The proofs are based on the use of action-angle coordinates and on the existence of a good invariant domain for the regular Lagrangian flow generated by b.