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Localized Enhanced Dissipation: A Hypocoercivity Approach

Siming He

Abstract

In this paper, we consider the passive scalar solutions in shear flows with critical points. With a detailed hypocoercivity functional, we develop streamline-wise enhanced dissipation estimates.

Localized Enhanced Dissipation: A Hypocoercivity Approach

Abstract

In this paper, we consider the passive scalar solutions in shear flows with critical points. With a detailed hypocoercivity functional, we develop streamline-wise enhanced dissipation estimates.
Paper Structure (3 sections, 8 theorems, 62 equations, 2 figures)

This paper contains 3 sections, 8 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. Technical Lemmas

Key Result

Theorem 1.1

Consider the smooth solution $f$ to the equation EQ:hypo_PS. Further assume that the shear flow profile $U$ only has finitely many nondegenerate critical points and the diffusion coefficient $\nu\in(0,1]$. Then the functional $\Phi$Hypo_F has the following estimate Here, $\delta\in(0,1)$ is a constant that depends only on $U.$

Figures (2)

  • Figure 1: Critical Layer
  • Figure 2: The weight $\log W$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: $\alpha$-estimate
  • Lemma 2.4: $\beta$-estimate
  • Lemma 2.5: $\gamma$-estimate
  • proof : Proof of Theorem \ref{['thm:ED']}
  • proof : Proof of Lemma \ref{['lem:nondegenerate al']}
  • ...and 6 more