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Accelerated relaxation enhancing flows cause total dissipation

Keefer Rowan

Abstract

We show that by "accelerating" relaxation enhancing flows, one can construct a flow that is smooth on $[0,1) \times \mathbb{T}^d$ but highly singular at $t=1$ so that for any positive diffusivity, the advection-diffusion equation associated to the accelerated flow totally dissipates solutions, taking arbitrary initial data to the constant function at $t=1$.

Accelerated relaxation enhancing flows cause total dissipation

Abstract

We show that by "accelerating" relaxation enhancing flows, one can construct a flow that is smooth on but highly singular at so that for any positive diffusivity, the advection-diffusion equation associated to the accelerated flow totally dissipates solutions, taking arbitrary initial data to the constant function at .
Paper Structure (3 sections, 2 theorems, 31 equations)

This paper contains 3 sections, 2 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Discussion
  3. Proofs

Key Result

Theorem 1.6

Suppose that $u \in L^\infty_{loc}([0,\infty) \times \mathbb{T}^d)$ such that $\nabla \cdot u =0$ is relaxation enhancing. Then there exists some acceleration of $u$ that is totally diffusive on $[0,1]$ for all diffusivities $\nu>0$, that is there exists some smooth diffeomorphism $\sigma : [0,1) \t

Theorems & Definitions (8)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • proof : Proofs of Theorems \ref{['thm.main-result']} and \ref{['thm.bound']}