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Existence theory for a bushfire equation

Serena Dipierro, Enrico Valdinoci, Glen Wheeler, Valentina-Mira Wheeler

Abstract

In a recent paper, we have introduced a new model to describe front propagation in bushfires. This model describes temperature diffusion in view of an ignition process induced by an interaction kernel, the effect of the environmental wind and that of the fire wind. This has led to the introduction of a new partial differential equation of evolutionary type, with nonlinear terms both of integral kind and involving the gradient of the solution. This equation is, in a sense, ``hybrid'', since it encodes both analytical and geometric features of the front propagation. This new characteristic makes the equation particularly interesting also from the mathematical point of view, often falling outside the territory already covered by standard methods. In this paper, we start the mathematical treatment of this equation by establishing the short time and global existence theory for this equation.

Existence theory for a bushfire equation

Abstract

In a recent paper, we have introduced a new model to describe front propagation in bushfires. This model describes temperature diffusion in view of an ignition process induced by an interaction kernel, the effect of the environmental wind and that of the fire wind. This has led to the introduction of a new partial differential equation of evolutionary type, with nonlinear terms both of integral kind and involving the gradient of the solution. This equation is, in a sense, ``hybrid'', since it encodes both analytical and geometric features of the front propagation. This new characteristic makes the equation particularly interesting also from the mathematical point of view, often falling outside the territory already covered by standard methods. In this paper, we start the mathematical treatment of this equation by establishing the short time and global existence theory for this equation.
Paper Structure (8 sections, 15 theorems, 135 equations)

This paper contains 8 sections, 15 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['MAIN:THEOREM']}
  3. Toolbox
  4. The fixed-point argument
  5. Proof of Theorem \ref{['MAIN:THEOREM:LIN']}
  6. Proof of Theorem \ref{['MAIN:THEOREM:GLO']}
  7. Toolbox
  8. The iteration argument

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain of $\mathds{R}^n$ with $C^2$ boundary. Let also $K\in L^2(\Omega\times\Omega)$, $\Theta\in L^\infty(\Omega\times (0,1))$, $\omega\in L^\infty(\Omega\times (0,1))$, $\beta\in W^{1,\infty}(\mathds{R})$, and $\gamma\in(1,2]$. Then, there exists $T_\star\in(0,1]$, depend possessesFor simplicity, the notion of solution adopted in this paper is the weak one, see e.g. MR2

Theorems & Definitions (30)

  • Theorem 1.1: Short time existence for a bushfire equation
  • Theorem 1.2: Short time existence for a bushfire equation with convective terms with linear scaling
  • Theorem 1.3: Global in time existence for a bushfire equation
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 20 more