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Stochastic homogenization of diffusions in turbulence driven by non-local symmetric Lévy operators

Xin Chen, Jian Wang, Kun Yin

Abstract

We investigate the stochastic homogenization of a class of turbulent diffusions generated by non-local symmetric Lévy operators with divergence-free drift fields in ergodic random environments, where neither the drift fields nor their associated stream functions are assumed to be bounded. A pivotal step in our proof is the establishment of $W_{loc}^{1,q}$ estimates with $q\in (1,2)$ for the corresponding correctors, under mild prior regularity conditions imposed on the Lévy measure and the stream function.

Stochastic homogenization of diffusions in turbulence driven by non-local symmetric Lévy operators

Abstract

We investigate the stochastic homogenization of a class of turbulent diffusions generated by non-local symmetric Lévy operators with divergence-free drift fields in ergodic random environments, where neither the drift fields nor their associated stream functions are assumed to be bounded. A pivotal step in our proof is the establishment of estimates with for the corresponding correctors, under mild prior regularity conditions imposed on the Lévy measure and the stream function.
Paper Structure (10 sections, 8 theorems, 234 equations)

This paper contains 10 sections, 8 theorems, 234 equations.

Table of Contents

  1. Introduction and main result
  2. Background
  3. Framework and main result
  4. The corrector and homogenized coefficients
  5. The existence of the corrector
  6. Homogenized coefficients
  7. The scaled resolvent equation and properties of the corrector
  8. The existence of solution to the scaled resolvent equation \ref{['e1-6']}
  9. Properties of the corrector $\phi$
  10. Proof of Theorem \ref{['t1-2']}

Key Result

Theorem 1.4

Suppose that Assumptions a1-2--, a1-2 and a1-3 are satisfied. Then, for any $\lambda>0$, $h\in \mathscr{G}_\lambda$, a.s. $\omega\in \Omega$ and every $R\geqslant 1$, $p\geqslant 1$, Here $u^{\varepsilon}(x;\omega)$ and $\bar{u}$ are solutions to the equations e1-6 and t1-2-2, respectively.

Theorems & Definitions (19)

  • Theorem 1.4
  • Remark 1.5
  • Example 1.6
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • ...and 9 more