Global boundary stabilization and regularization of Burgers´ equation

We consider several results from our earlier work (1998) concerning stabilization and regularization of Burgers´ equation. We consider the viscous Burgers equation under previously proposed nonlinear boundary conditions which guarantee global asymptotic stabilization and semiglobal exponential stabilization in the H¹ sense. We show global existence and uniqueness of classical solutions with initial data which are assumed to be only in L². To do this, we establish a priori estimates of up to four spatial and two temporal derivatives, and then employ the Banach fixed point theorem to the integral representation with a heat kernel. Our result is global in time and allows arbitrary size of initial data. It strengthens results by Byrnes, Gilliam, and Shubov (1998), Ly, Mease, and Titi (1997), and Ito and Yan (1995). We include a numerical result which illustrates the performance of the boundary controller