Nonlinear evolution equations of fourth- and higher-order in spatial derivatives emerge in various models of mathematical physics. This thesis is devoted to the study of nonlinear higher-order diffusion equations, which arise in the quantum modeling of semiconductor and plasma physics, and describe the evolution of densities of charged particles in a quantum fluid.
These equations appear as quantum corrections to the classical models of the transport of charged particles.
Primary questions in the mathematical analysis of nonlinear higher-order equations are the existence and uniqueness of solutions, long-time behaviour and positivity of solutions, growth of the support and speed of propagation. In order to obtain the answers, many approaches rely on certain a priori estimates, called entropy production inequalities. These estimates are results of mathematical dissipation of some nonlinear functionals (entropies) along solutions of the equation under consideration, but often they also reflect the underlying physical laws, namely that of conservation of mass and energy, or the dissipation of the physical entropy. As a consequence, they provide necessary uniform bounds for solutions in corresponding Sobolev (semi-)norms.
The first part of the thesis considers an algebraic approach for proving entropy production inequalities for radially symmetric solutions to a class of higher-order diffusion equations in multiple space dimensions.
The approach is an extension of the previously developed method for nonlinear evolution equations of even order in one space dimension. Key idea is to translate the problem of proving the integral inequlities into a decision problem about nonnegativity of corresponding polynomials. A benefit of this procedure is that the latter problem is always solvable in an algorithmic way. In application of the method, novel entropy production inequalities are derived for the thin-film equation, the fourth-order Derrida-Lebowitz-Speer-Spohn equation, and the sixth-order quantum diffusion equation.
In the second part, the initial-value problem for the sixth-order quantum diffusion equation with periodic boundary conditions is studied.
The concept of weak nonnegative solutions for this equation is introduced and it is proved that the equation admits the global-in-time solutions in two and three space dimensions. Moreover, these solutions are smooth and classical whenever the particle density is strictly positive and particular energy functional is uniformly bounded. In addition, the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate is observed. The analysis strongly uses a special entropy production inequality, which is a direct consequence of the dissipation property of the physical entropy.
Finally, the third part is devoted to novel approximations of the fourth-order quantum diffusion equation, also known as the Derrida-Lebowitz-Speer-Spohn equation. Two different approaches are discussed, which have the common goal of preserving some qualitative properties of solutions on a (semi-)discrete level. First, the semi-discrete two-step backward difference (BDF-2) method of a reformulated equation yields the discrete entropy stability property and second-order convergence of the method in a specific case. Next, a particular variational structure of the equation is used to introduce the discrete variational derivative method in the onedimensional case. The method preserves the mass and the dissipation property of the corresponding energy (Fisher information) functional on a discrete level. Furthermore, the method is extended to the temporally more accurate multistep discrete variational derivative methods, which possess generalized discrete dissipation properties.