Entropies for radially symmetric higher-order nonlinear diffusion equations

Abstract

Abstract. A previously developed algebraic approach to proving entropy production inequalities is extended to deal with radially symmetric solutions for a class of higher-order diffusion equations in multiple space dimensions. In application of the method, novel a priori estimates are derived for the thin-film equation, the fourth-order Derrida-Lebowitz-Speer-Spohn equation, and a sixth-order quantum diffusion equation.Key words. Higher-order diffusion equations, thin-film equation, quantum diffusion model, polynomial decision problem, quantifier elimination.AMS subject classifications. 35B45, 35G25, 35K55, 76A20, 82C70. IntroductionIn the last two decades, there has been a growing interest in the mathematical analysis of fourth and higher-order nonlinear diffusion equations. Such equations arise, for instance, in lubrication theory and as models for the electron transport in semiconductors; below, we will briefly review several specific examples and their origins in physics. Rigorous results about the existence of solutions and their qualitative behavior are typically much harder to obtain than in the context of the well-studied secondorder diffusion equations. One of the principal difficulties is the non-applicability of comparison principles. To substitute for this loss, one has to rely on suitable a priori estimates.In [11], the last two authors have proposed a systematic approach to the derivation of a priori estimates for certain classes of nonlinear evolution equations of even order. This procedure allows one to determine Lyapunov functionals, which we call entropies in the following, and to derive integral bounds from their dissipation, called entropy production inequalities. The developed method has been successfully applied to several equations in one space dimension. The main idea is to translate the procedure of integration by parts -which is the core element in most derivations of a priori estimates -into an algebraic problem about the positivity of polynomials. Roughly speaking, to each evolution equation, a polynomial in the spatial derivatives of the solution is associated, and integration by parts allows one to modify the coefficients of this polynomial. If a suitable change of coefficients can be found that makes the resulting polynomial nonnegative, then this corresponds (formally) to a proof of an a priori estimate on the solutions. The key point is that such polynomial decision

A multidimensional nonlinear sixth-order quantum diffusion equation

“…It is well known for the fourth-order equation (3), that weak solutions may not be unique [13]. We expect the same phenomenon to occur for (1).…”

“…are spaces of functions that are 1-periodic in each spatial coordinate direction. The derivation of the sixth-order equation 1in [3] was performed on R d and hence, it does not include the derivation of physically relevant boundary conditions. In this work, we have chosen periodic boundary conditions to simplify the analysis.…”

“…In particular, integration by parts plays a pivotal role in our derivation of a priori estimates, and the boundary integrals vanish for periodic functions. We note that in [3], radially symmetric solutions for (1) satisfying no-flux-type boundary conditions have been considered instead.…”

“…Forh > 0, however, the quantum exponential is a complicated, genuinely nonlocal operator. An asymptotic expansion in terms ofh has been performed in [3,Appendix], leading to the following local approximation of A in terms of n:…”

“…An asymptotic expansion of the nonlocal model in terms of the scaled Planck constanth 2 leads to a family of parabolic equations for the particle densities n(t; x). The first member of this family is the classical heat equation ∂ t n = n. The second one is the fourth-order Derrida-Lebowitz-Speer-Spohn (DLSS) equation, see (3) below, which is analyzed in [9,13]. This paper is concerned with the third family member, obtained from an expansion to orderh 4 (see [3,Appendix]), which reads as ∂ t n = div n∇ 1 2 ∂ 2 ij log n 2 + 1 n ∂ 2 ij n∂ 2 ij log n .…”