Speaker: Brian Reyes Velez
University of Notre Dame
Will give a PhD Defense entitled:
"Evolution of the radius of spatial analyticity for the dispersion modified Degasperis-Procesi equation"
Abstract: The local well-posedness of the Cauchy problem for the dispersion modified \textbf{b}-equation with data in Sobolev spaces $H^s(\mathbb{R})$ and analytic Gevrey spaces $G^{\delta, s}(\mathbb{R})$ is proved for any $s>1/4$. However, for =3$, which is the modified Degasperis-Procesi equation, a sharper result is established. In this case, the equation behaves as a nonlocal perturbation of the Kordeweg-de Vries (KdV) equation and well-posedness is shown for $s>-3/4$. Furthermore, for =3$ this equation possesses a twisted-$L^2$ conservation law. This yields an almost conservation law in the analytic Gevrey spaces $G^{\delta, 0}$. Using this almost conservation law global solutions are established and a lower bound, given by $c/t^{\frac 43+}$, for their radius of spatial analyticity is proved. Key ingredients in the proof of this result are the Paley-Wiener Theorem and bilinear estimates for the nonlinearity of the modified Degasperis-Procesi equation.
Date: 03-21-2024
Time: 12:00 pm
Location: 258 Hurley Bldg
Zoom Link: https://notredame.zoom.us/j/