## Research and Publications

- F. Linares, G. Ponce, D. L. Smith,
*On the regularity of solutions to a class of nonlinear dispersive equations*, submitted for publication October 2015. arXiv:1510.02512 - J. Segata, D. L. Smith,
*Propagation of regularity and persistence of decay for fifth order dispersive models*. Journal of Dynamics and Differential Equations, arXiv:1502.01796

My dissertation research focuses on smoothing properties of solutions to dispersive nonlinear partial differential equations. For the Korteweg-de Vries (KdV) equation, it was recently shown that appropriate regularity present in the initial data on the positive, or right, half-line travels to the left with infinite speed. In collaboration with J. Segata, we extended this propagation of regularity phenomenon to equations in the KdV hierarchy. However, propagation of regularity does not depend on the equation being completely integrable or even having constant coefficients. In a joint work with F. Linares and G. Ponce, we demonstrated the result for a family of quasilinear KdV-type equations. More generally, smoothing effects play a crucial role in the well-posedness and control theory of nonlinear dispersive equations, topics which I intend to pursue in future research. I have also identified potential research projects related to my dissertation work which are suitable for undergraduates.

I will speak about the quasilinear results during the upcoming Joint Mathematics Meetings in Seattle.