ETD

• Isotropic stochastic flows
• Kraichnan model
• Mixing
• Scale function
• Two-point motion

An interesting problem in physics and mathematics is the understanding of the evolution of a passive scalar, such as dust on a fluid's surface. In particular, it is relevant to study the dispersion of the passive scalar, a phenomenon known as mixing.

A rigorous way to formulate the mixing problem is through a negative homogeneous Sobolev norm of the passive scalar, called the mixing norm. Lower values of these norms denote more mixed states.

In this thesis, we study the mixing phenomenon in the Kraichnan model, where the velocity field is the time derivative of a divergence-free, isotropic Wiener process that is smooth in space, where we can use classical stochastic analysis tools.

In the first part of the thesis, after studying stochastic flows and isotropic covariance functions, we focus on the study of the two-point motion equation, showing that, for isotropic flows, the two-point distance is a closed one-dimensional diffusion.

Finally, the main contribution of the thesis is a polynomial decay rate for the expected mixing norm. The proof is based on the connection between the mixing norm and the two-point motion and relies, as a key tool, on the closed one-dimensional evolution of the two-point distance in the Kraichnan model. While the asymptotic behaviour of the two-point distance near zero and at infinity suggests, respectively, exponential and polynomial decay, the rigorous argument uses the scale function to control the evolution of the two-point motion on the full space.