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Enhanced Dissipation and Global Well-Posedness for a Three-Dimensional Flame Propagation Model with Couette Flow

Yoshiyuki Kagei, Lijuan Wang

Abstract

We study a three-dimensional gravity-induced flame front model under a Couette flow. By exploiting the enhanced dissipation induced by the Couette flow, we prove global-in-time well-posedness of the Cauchy problem in $\mathbb{R}^3$ and derive decay estimates for the solution and its spatial derivatives in $L^p$ norms for all $p \ge 1$. The analysis is based on a Green's function approach for the associated variable-coefficient linearized operator. Since an explicit representation of the Green's function is unavailable, we first establish decay estimates in the spectral domain and then transfer them to physical space. These results show that enhanced dissipation induced by the Couette flow is the key mechanism leading to global existence in the whole space, in the large initial data regime.

Enhanced Dissipation and Global Well-Posedness for a Three-Dimensional Flame Propagation Model with Couette Flow

Abstract

We study a three-dimensional gravity-induced flame front model under a Couette flow. By exploiting the enhanced dissipation induced by the Couette flow, we prove global-in-time well-posedness of the Cauchy problem in and derive decay estimates for the solution and its spatial derivatives in norms for all . The analysis is based on a Green's function approach for the associated variable-coefficient linearized operator. Since an explicit representation of the Green's function is unavailable, we first establish decay estimates in the spectral domain and then transfer them to physical space. These results show that enhanced dissipation induced by the Couette flow is the key mechanism leading to global existence in the whole space, in the large initial data regime.

Paper Structure

This paper contains 9 sections, 15 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Green's Function and Computational Lemmas
  3. Green's Function
  4. Lp Estimates for the Green's Function ($p \ge 2$)
  5. Lp Estimates for the Green's Function $(1\le p<2)$
  6. Global Existence of Solutions
  7. Local Existence of the Solution
  8. Global Existence of the Solution
  9. Decay Estimates of Solution

Key Result

Theorem 1.1

Let $\phi_0(x,y,z) \in W^{4,\infty}(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)$ be the initial data, and let $\varepsilon>0$ be a fixed constant, with the assumption that $\kappa > \frac{1}{16} + \varepsilon.$ Then there exists a positive constant $A_0 = A_0(\phi_0)$ such that, for any $A \ge A_0$, the Cau Moreover, the solution satisfies the following decay estimates for all derivatives: where $D^k = \

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 16 more