First, we'll develop the basic mathematical theory of Fluid Dynamics beginning with the main equations that govern fluid motion, including questions concerning existence, uniqueness, and regularity of solutions and then introduce the basic ideas and tools of computer-assisted proofs through a collection of specific examples. We will explore how computation can be used to identify interesting approximate solutions and how rigorous estimates can then prove the existence of exact solutions nearby. More broadly, the course will show how numerical insight and mathematical analysis can work together to address challenging questions in PDEs.