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

TL;DR

This paper proves global well-posedness and decay estimates for a 3D flame propagation model with Couette flow, highlighting enhanced dissipation as the key mechanism for large initial data.

Contribution

It establishes global existence and decay for a 3D flame model with Couette flow using Green's function and spectral analysis, a novel approach in this context.

Findings

Enhanced dissipation leads to global existence in 3D.

Decay estimates are derived for solutions and derivatives in $L^p$ norms.

Green's function approach is successfully applied without explicit formulas.

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.