Homogenization of two-dimensional linear flows with integral invariance

Tamir Tassa

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

We study the homogenization of two-dimensional linear transport equations, ut + a + (Combining right arrow above sign)(x + (Combining right arrow above sign)/ε) · ∇x + (Combining right arrow above sign) u = 0, where a + (Combining right arrow above sign) is a nonvanishing vector field with integral invariance on the torus T2. When the underlying flow on T2 is ergodic, we derive the efficient equation which is a linear transport equation with constant coefficients and quantify the pointwise convergence rate. This result unifies and illuminates the previously known results in the special cases of incompressible flows and shear flows. When the flow on T2 is nonergodic, the homogenized limit is an average, over T1, of solutions of linear transport equations with constant coefficients; the convergence here is in the weak sense of W-1,∞loc(ℝ1), and the sharp convergence rate is φ(ε). One of the main ingredients in our analysis is a classical theorem due to Kolmogorov, regarding flows with integral invariance on T2, to which we present here an elementary and constructive proof.

Original languageEnglish
Pages (from-to)1390-1405
Number of pages16
JournalSIAM Journal on Applied Mathematics
Volume57
Issue number5
DOIs
StatePublished - 1 Jan 1997
Externally publishedYes

Keywords

  • Dynamical systems
  • Ergodic theory
  • Homogenization
  • Linear transport equations

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