Global estimates and shocks for the noiseless conserved Kardar - Parisi - Zhang equation

We present a series of results concerning the equation , which describes non-equilibrium surface dynamics in the presence of volume conservation. We investigate whether a singularity in the solution can be formed in finite time, as was reported recently in the literature. For we derive several integrals of motion, which allow us to prove that u always remains bounded, even if a singularity is formed. For we prove that the norm of the solution decays at least exponentially, on a time scale of the order of , where L is the system size. Moreover, we show that the solution remains bounded for all times if the norm of the derivative is sufficiently small initially. Based on these results and on the numerical integration of the equation we conjecture that no singularities occur for . For we predict wedge-like singularities to appear at local maxima, in which changes sign, but u as well as remain continuous. We show that a self-similar approach to such singularities is possible and present some numerical evidence for this scenario. We are able to continue the simulations beyond the formation of singularities. The simulations show the formation of non-moving shocks (in ), which decay algebraically in time.