Abstract

In this talk we will introduce Bressan’s Fire Conjecture: it is concerned with the model of wild fire spreading in a region of the plane and the possibility to block it using barriers constructed in real time. The fire starts spreading at time $t=0$ from the unit ball $B_1(0)$ in every direction with speed $1$, while the length of the barrier constructed within the time $t$ has to be lower than $\sigma t$, where $\sigma$ is a positive constant (construction speed). If $\sigma \leq 1$ Bressan proved that no barrier can block the spreading of the fire, while if $\sigma > 2$ there exists always a strategy that confine the fire. In 2007 Bressan conjectured that if $\sigma \leq 2$ then no barrier can block the fire.

In this talk we will prove Bressan’s Fire Conjecture in the case barriers are spirals. Spirals are thought to be the best strategies a firefighter can do in order to confine the fire for $\sigma \leq 2$. We will introduce the new concept of family of generalized barriers and we will prove that, if there exists such a family satisfying a diverging condition, then no spiral can confine the fire.

This is a joint work with Stefano Bianchini.

Mathematics in Conversation