Some results on mixing flows and on the blocking fire problem

This thesis is developed following two lines of research in analysis and applied mathe matics: the study of mixing phenomena arising in a non-smooth setting and dynamic blocking problems of fire propagation. The first part of this thesis is devoted to the study of mixing from the point of view of Ergodic Theory and the one of Fluid dynamics. The major result achieved here is the construction of infinitely many Exponential Mixers, that is divergence-free vector fields that have strong mixing properties. Moreover we give an example of a vector field which is weakly mixing but not strongly mixing. The second part is devoted to the study of the Fire Problem first proposed by Bressan in 2007 [14]. Here, we study the properties of Optimal Blocking Strategies, proving Bressan’s fire conjecture for spiral-like strategies.