标题： 阻火蔓延的螺旋策略的存在性 显示英文标题

标题： Existence of spiral strategies for blocking fire spreading

摘要： 在本文中，我们研究通过构建形状类似螺旋的墙$\zeta$来阻断火灾的问题。当只有一个消防员以有限的建造速度$\sigma$构建墙时，这被认为是最佳策略：满足此建造速度限制的障碍物称为可接受的。我们在这种情况下证明了Bressan火灾猜想的精确版本，即当可接受的障碍物是类似螺旋的曲线时：存在一个类似螺旋的障碍物将火灾限制在$\mathbb R^2$的有界区域中，当且仅当障碍物的建造速度$\sigma$严格大于临界速度$\bar \sigma = 2.614...$。对于$\sigma > \bar \sigma$的封闭螺旋障碍物的存在性已经为人所知 [Bressan A. 等, 2008, Klein R. 等, 2019]，而我们则专注于负面情况，即如果$\sigma \leq \bar \sigma$，则没有可接受的螺旋障碍物能阻止火灾。这些结果的证明依赖于：1）螺旋障碍物的精确定义及其表示；2）作为滞后微分方程（RDE）的饱和螺旋障碍物的分析，其思想源于 [Klein R. 等, 2019]；3）将猜想重新表述为一个给定泛函的最小化问题；4）最优闭合螺旋的构造；5）对可接受螺旋的不同可微路径的分析，在该路径上泛函是可微的，并且特别地，当从最优螺旋移动到其他任何螺旋时，泛函是增加的（同伦论证）。由于解决方案的复杂性，为了证明泛函是增加的所需量的评估是数值进行的。 显示英文摘要

摘要： In this paper we address the problem for blocking fire by constructing a wall $\zeta$ whose shape is spiral-like. This is supposed to be the best strategy when a single firefighter is constructing the wall with a finite construction speed $\sigma$: the barriers which satisfy this bound on the construction speed are called admissible. We prove a sharp version of Bressan's Fire Conjecture in this case, i.e. when admissible barriers are spiral-like curves: namely, there exists a spiral-like barrier confining the fire in a bounded region of $\mathbb R^2$ if and only if the speed of construction of the barrier $\sigma$ is strictly larger than a critical speed $\bar \sigma = 2.614...$. The existence of confining spiral barriers for $\sigma > \bar \sigma$ is already known [Bressan A. et al., 2008, Klein R. et al., 2019], while we concentrate on the negative side, i.e. if $\sigma \leq \bar \sigma$ no admissible spiral blocks the fire. The proof of these results relies on: 1) the precise definition of spiral barrier and its representation; 2) the analysis of saturated spiral barriers as a Retarded Differential Equation (RDE) in the spirit of [Klein R. et al., 2019]; 3) the equivalent reformulation of the conjecture as a minimum problem for a prescribed functional; 4) the construction of the optimal closing spiral; 5) the analysis of a differentiable path of admissible spirals along which the functional is differentiable, and in particular increasing when moving from the optimal spiral to any other one (homotopy argument). Due to the complexity of the solution, the evaluation of the quantities needed to prove that the functional is increasing is performed numerically.