Global Smooth Solutions of the Three-dimensional Modified Phase Field Crystal Equation

Abstract. The Modified Phase Field Crystal (MPFC) equation, a generalized damped wave equation for which the usual Phase Field Crystal (PFC) equation is a special case, is analyzed in detail in three dimensions. A time-discrete numerical scheme, based on a convex splitting for the functional energy, is utilized to construct an approximate solution, which is then shown to converge to a solution of the MPFC equation as the time step approaches zero. In detail, a uniform-in-time bound of the pseudo energy for the numerical solution is obtained owing to the structure of the convex-splitting scheme. As an immediate result, we obtain a bound of the Ls (0, T ;H 2 per) norm of the numerical solution. More detailed energy estimates, which are obtained by taking the inner product of the numerical scheme with (−∆)(φ − φ), give bounds for the numerical solution and it first and second temporal backward differences in the Ls (0, T ;H m+3 per ), L ∞ s (0, T ;H m per) and Ls (0, T ;H m−3 per ) norms, respectively. These estimates of the numerical solutions in turn result in a global weak solution (with m = 0) and a unique global strong solution (m = 3) upon passage to the limit as the time step size approaches zero. A global smooth solution can also be established by taking arbitrarily large values of m.