On the existence of solutions for amorphous molecular beam epitaxy
This article deals with the following surface growth equation arising in molecular beam epitaxy \[ \partial_{t}h(t,x)=-\Delta^2h(t,x)-\mu\Delta h(t,x)-\Delta|\nabla h(t,x)|^2+\xi(t,x), \] where \(h:[0,T]\times[0,L]^{d}\rightarrow R\), \(d=1,2\), is subject to periodic boundary conditions and the initial condition \(h(0,\cdot)=h_0(\cdot)\). The stochastic term \(\xi\) is a Gaussian space-time white noise. In the case \(d=1\) the global existence of a martingale solution to the considered problem is proved. First, an approximating sequence of stochastic ordinary differential equations is solved and its boundedness in certain norms is shown. The results of Prokhorov and Skorokhod enable the authors to pass to a converging subsequence and the limit turns out to be a martingale solution of the considered problem.
scientific article; zbMATH DE number 1915789