Department of Mathematics

In this talk, we consider the fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus, with or without an advection term. We will show that in the absence of advection, there exists initial data which will make the solution blow up in finite time; while in the advective case, if the imposed advection is sufficiently mixing, the global existence of the solutions can be achieved. Finally, we will make some further remarks on the general framework on how advection can guarantee the global existence of certain non-linear equations. This talk is based on several joint work with Yu Feng, Xiaoqian Xu and Yeyu Zhang.