New dissipated energies for the thin fluid film equation

Laugesen, Richard S.

doi:10.3934/cpaa.2005.4.613

Cited by 20 publications

(39 citation statements)

References 12 publications

(31 reference statements)

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“…The same result holds for periodic boundary conditions [11]. This bound turns out to be sharp, at least in the one-dimensional case [17]. Moreover, the entropy production Q α in (1.4) can be made explicit: a valid choice is Q α [U ] = Ω |(U (α+β)/2 ) xx | 2 dx with a suitable c > 0 if 3/2 < α + β < 3; see [11].…”

Section: Introductionsupporting

confidence: 58%

“…This correspondence -which is summarized in Lemma 2.1 below -constitutes an extension of the ideas previously developed for entropy estimates in one spatial dimension by the last two authors [11]; see also [17] for an alternative approach. The proof of the main theorems are then obtained by solution of the associated algebraic problems.…”

Section: Decision Problem and Shift Polynomialsmentioning

confidence: 98%

“…The principal idea for our definition of u ε in (5.1) is borrowed from Laugesen's construction of a "trial function" in one space dimension [17]. Our definition and also the proof of I(u ε ) → −∞ are more straightforward, since we work under the assumption of strict homogeneity (2.1); the proof in [17] has been designed for a slightly more general situation.…”

Section: Absence Of Entropiesmentioning

confidence: 99%

“…Similarly as in [11,17], it is possible to prove that certain functionals E α cannot be entropies. Below, we generalize Theorem 19 in [11] to the multidimensional, radially symmetric situation.…”

Section: Absence Of Entropiesmentioning

confidence: 99%

“…Our definition and also the proof of I(u ε ) → −∞ are more straightforward, since we work under the assumption of strict homogeneity (2.1); the proof in [17] has been designed for a slightly more general situation. The functions u ε are chosen as suitable ε-regularizations of the radially symmetric power functionũ(x) = r σ .…”

Section: Absence Of Entropiesmentioning

confidence: 99%

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Entropies for radially symmetric higher-order nonlinear diffusion equations

Bukal

Jüngel

Matthes

2011

Communications in Mathematical Sciences

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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

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Section: Introductionsupporting

confidence: 58%

Section: Decision Problem and Shift Polynomialsmentioning

confidence: 98%