8th Croatian Mathematical Congress

Improved equilibration rates to self-similarity for a thin-film equation

2 Jul 2024, 17:05

20m

D9 (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek)

Talk PDE: Partial Differential Equations and Applications Partial Differential Equations and Applications

Speaker

Mario Bukal (University of Zagreb Faculty of Electrical Engineering and Computing)

Description

We consider long-time behavior of solutions to the thin-film equation $ \partial_th = -\partial_x(h \partial_x^3h)$ on the real line with initial datum of finite second moment. The equation describes the interface dynamics of a thin fluid neck of thickness $2h(x,t)$ in the Hele-Shaw cell. Upon rescaling the equation in such a way that the second moment is constant in time, precise rate of convergence to the steady state is given in terms of corresponding relative Rényi entropy. For intermediate times, this improves the rate of convergence given in terms of the usual relative entropy and thus, improves the rate of convergence in the $L^1$-norm. The result is based on ideas developed by Carrillo and Toscani (Nonlinearity 27 (2014), 3159-3177) for second-order nonlinear diffusion equations and utilizes the concavity property of the Rényi entropy power.