INFINITE SPEED OF SUPPORT PROPAGATION FOR THE DERRIDA-LEBOWITZ-SPEER-SPOHN EQUATION AND QUANTUM DRIFT-DIFFUSION MODELS

Abstract. We show that weak solutions of the Derrida-Lebowitz-Speer-Spohn equation display infinite speed of support propagation. We apply our method to the case of the quantum drift-diffusion equation which augments the DLSS equation with a drift term and possibly a second-order diffusion term. The proof is accomplished using weighted entropy estimates, Hardy's inequality and a family of singular weight functions to derive a differential inequality; the differential inequality shows exponential growth of the weighted entropy, with the growth constant blowing up very fast as the singularity of the weight becomes sharper. To the best of our knowledge, this is the first example of a nonnegativity-preserving higher-order parabolic equation displaying infinite speed of support propagation

A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on $R^d$ are studied. These equations constitute gradient flows for the perturbed information functionals $F[u] = 1/(2\alpha) \int | D (u^\alpha) |^2 dx + \lambda/2 \int |x|^2 u dx$ with respect to the $L^2$-Wasserstein metric. The value of $\alpha$ ranges from $\alpha=1/2$, corresponding to a simplified quantum drift diffusion model, to $\alpha=1$, corresponding to a thin film type equation.Comment: 33 pages, no figure

High order collocation software for the numerical solution of fourth order parabolic PDEs

viii, 114 leaves : ill. ; 29 cm.Includes abstract.Includes bibliographical references (leaves 109-114).BACOL is an efficient software package for solving systems of second order parabolic PDEs in one space dimension. A significant feature of the package is that it employs adaptive error control in both time and space. A second order PDE depends on the solution, u, and its first and second derivatives, ux and uxx. However, many applications lead to mathematical models which involve fourth order PDEs. Fourth order PDEs depend on u, ux, uxxx, and uxxxx. One contribution of the thesis is that it provides a survey of applications in which fourth order PDEs arise. The thesis focuses on how to extend BACOL so that it can handle fourth order PDEs. We have explored a somewhat novel approach that involves converting the fourth order PDE to a coupled system which contains one second order PDE and one second order PDE (in space). A careful investigation of the BACOL package is carried out in order to extend it so that it can treat this coupled PDE/ODE system directly; the new software is called BACOL42. For comparison purposes we have also considered an approximate form of the converted system that can be solved using the original BACOL software. Numerical results are provided to demonstrate the effectiveness of BACOL42. The thesis also provides a numerical study of two other PDE solvers, pdepe and MOVCOL4, that can be applied to solve fourth order PDEs