Stability of the Bunsen Flame Profiles in the Kuramoto–Sivashinsky Equation

Abstract

The stability of the conical stationary solutions of the Kuramoto–Sivashinsky equation $u_t + \Delta ^2 u + \Delta u + |\nabla u|^2 = c^2 $ in one and two space dimensions is studied. It is shown that these solutions are unstable in the whole space. Next the problem is studied in the one-dimensional (1D) case in a bounded interval $|x| \leq l$ and in the 2D case in a disc $0 \leq r < l$ with natural boundary conditions. It is proved that for a large slope c the above stationary solutions are stable. In the 1D case part of the proof is computer assisted.