The concept of transport is fundamental and has great influence in a wide range of fields across science. This dissertation provides three topics possessing the character of transport phenomena from the perspective of partial differential equations. The three parts include:

(1) Commutator method for averaging lemma: A new commutator method is introduced to prove a new type of averaging lemmas, the regularizing effect for the velocity average of solutions for kinetic equations. This novel approach shows a new range of assumptions that are sufficient for the velocity average to be in $L^2([0,T],H^{1/2}_x)$ and improves the regularity result for the measure-valued solutions of scalar conservation laws in space one-dimensional case.

(2) Unmixing property of incompressible flows on 2d tori: The local Hamiltonian structure of a 2d torus is utilized to show that the unmixing property of incompressible flows can be preserved under a sup-norm perturbation on stream functions. With this perturbation result, a quantitative statement was provided by considering vector fields in the form of a random Fourier series. This statement offers an interesting observation for the unmixing property from the perspective of Fourier analysis.

(3) Memory effect on animal migration: The goal of this work is to obtain a better understanding of the memory effect on the animals' migration patterns under periodic environments. A memory model and a corresponding memory-driven dynamic were constructed. Through simulations, it is discovered that in order to have periodic movement, the individual must be able to gather and carry sufficient information from both short-term memory and long-term memory, and possess the ability to discriminate which information is more important with appropriate time scales. Furthermore, our mathematical model is general and can be used to test the memory effect under different circumstances. Several interesting examples are demonstrated.