In this article, we examine L 2 well-posedness and stabilization property of the dispersion-generalized Benjamin-Ono equation with periodic boundary conditions. The main ingredient of our proof is a development of dissipation-normalized Bourgain space, which gains smoothing properties simultaneously from dissipation and dispersion within the equation. We will establish a bilinear estimate for the derivative nonlinearity using this space and prove the linear observability inequality leading to small-data stabilization.

I would like to thank my committee-Prof. Thomas Sideris and Prof. Hector Ceniceros, for being true role models and mentors, and for taking the time to read my dissertation and make helpful suggestions that improve the presentation of this work. Everything I have learned about PDEs has been inspired by these three professors. In addition, I would like to thank Prof. Ken Millett and Medina Price-you have told me many things I need to hear before knowing I needed to hear them. Finally, a heartfelt thanks to my family. Words cannot express how grateful I am to my mother, father and sister for all of the sacrifices that you have made on my behalf. I would also like to thank my friends and colleagues who have offered encouraging words. At the end, I would like to thank my supportive and loving husband, especially for the care you offer our little ones. v

Nonlocal operators that have appeared in a variety of physical models satisfy identities and enjoy a range of properties similar to their classical counterparts. In this paper, we obtain Helmholtz-Hodge type decompositions for two-point vector fields in three components that have zero nonlocal curls, zero nonlocal divergence, and a third component which is (nonlocally) curl-free and divergence-free. The results obtained incorporate different nonlocal boundary conditions, thus being applicable in a variety of settings. Keywords Nonlocal operators • Nonlocal calculus • Helmholtz-Hodge decompositions Mathematics Subject Classification (2010) 35R09 • 45A05 • 45P05 • 35J05 • 74B99 This project was developed during the week-long workshop Women in Mathematics of Materials held at University of Michigan in May 2018 with support from the

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in H s (T), s > 2. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.2010 Mathematics Subject Classification. Primary: 35Q53, 93B05, 93D15.

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