Smoothing properties of certain dispersive nonlinear partial differential equations

This thesis is primarily concerned with the smoothing properties of dispersive equations and systems. Smoothing in this context means that the nonlinear part of the solution flow is of higher regularity than the initial data. We establish this property, and some of its consequences, for several equations. The first part of the thesis studies a periodic coupled Korteweg-de Vries (KdV) system. This system, known as the Majda-Biello system, displays an interesting dependancy on the coupling coefficient α linking the two KdV equations. Our main result is that the nonlinear part of the evolution resides in a smoother space for almost every choice of α. The smoothing index depends on number-theoretic properties of α, which control the behavior of the resonant sets. We then consider the forced and damped version of the system and obtain similar smoothing estimates. These estimates are used to show the existence of a global attractor in the energy space. We also use a modified energy functional to show that when the damping is large, the attractor is trivial. The next chapter studies the Zakharov and related Klein-Gordon-Schrödinger (KGS) systems on Euclidean spaces. Again, the main result is that the nonlinear part of the solution is smoother than the initial data. The proof relies on a new bilinear Bourgain-space estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. As an application, we give a simplified proof of the existence of global attractors for the KGS flow in the energy space for dimensions two and three. We also use smoothing in conjunction with a high-low decomposition to show global well-posedness of the KGS evolution on R4 below the energy space for sufficiently small initial data. In the final portion of the thesis we consider well-posedness and regularity properties of the “good” Boussinesq equation on the half line. We obtain local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. We also establish a smoothing result, obtaining up to half derivative smoothing of the nonlinear term. The results are sharp within the framework of the restricted norm method that we use and match known results on the full line. ii Acknowledgements I am most grateful to my advisor, Assoc. Professor Nikolaos Tzirakis for his invaluable instruction and encouragement. This project would not have begun, much less finished, without his patient guidance. I am grateful to Professors Jared Bronski, M. Burak Erdoğan, and Marius Junge for their service on my thesis committee, as well as for their instruction in previous years. I would like to thank the Mathematics Department at the University of Illinois for the opportunity to study here, and the many excellent faculty and staff members who made doing so a pleasure. I am also grateful to the faculty of the University of North Florida Mathematics Department, who showed me there’s math after calculus, and encouraged me to study it. I am grateful to the National Physical Science Consortium for its support of my graduate education. I appreciate its generosity, and the opportunities it gave me to explore applied mathematics. Finally, I wish to thank my family for their unfailing love and encouragement.