A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Matthes, Daniel; McCann, Robert J.; Savaré, Giuseppe

doi:10.1080/03605300903296256

Cited by 139 publications

(253 citation statements)

References 37 publications

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“…For n = 1, this was proven rigorously in [18] and refined results are contained for instance in [15,16,30,45]. In particular in [30], one of the authors of this paper was able to prove that (in suitably chosen moving coordinates) arbitrarily many derivatives of h converge to the source-type self-similar solution, which has the explicit form H(Z) = C 1 (C 2 − Z 2 ) 2 for |Z| ≤ C 2 with C 1 , C 2 > 0, first found by Smyth and Hill in [53].…”

Section: Introductionmentioning

confidence: 99%

A dynamical systems approach for the contact-line singularity in thin-film flows

2016

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Abstract. We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system. In particular, we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles. Key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points. The results are extended to radially-symmetric source-type solutions in higher dimensions. Furthermore, we give dynamical systems proofs for the existence and uniqueness of self-similar droplet solutions in the nonzero dynamic contact-angle case.

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Section: Introductionmentioning

confidence: 99%

A dynamical systems approach for the contact-line singularity in thin-film flows

2016

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“…However, this leads, even in simple situations, to huge polynomial expressions, and the corresponding algebraic problem is too complex to be solved directly, even with the aid of computer algebra systems. The method has been successfully adapted to deal with certain multidimensional equations of second order [15,19] and fourth order [12,20], but the systematic extension of the scheme to the general multidimensional case is still under development. In this paper, we propose a further adaption that works generally for radially symmetric solutions to higherorder nonlinear equations of a certain homogeneity, and we prove its practicability by applying our scheme to the Equations (1.1)-(1.3) listed below.…”

Section: Introductionmentioning

confidence: 99%

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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“…For instance, it is possible to prove L ∞ bounds on ρ if ρ 0 ∈ L ∞ and V ∈ C 2 (see Section 7.4.1 in [84]), which is enough to transform the H 1 estimates on m−1/2 into estimates on m . Another possibility is the use of estimates obtained via a technique called flow interchange (where the optimality of a density in the JKO scheme is tested against its evolution via the gradient flow of another functional, see [66]), which allows to modify the exponent of in (4.18).…”

Section: How To Prove Convergence Of the Jko Schemementioning

confidence: 99%

{Euclidean, metric, and Wasserstein} gradient flows: an overview

2017

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This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan-Kinderlehrer-Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

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A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Cited by 139 publications

References 37 publications

A dynamical systems approach for the contact-line singularity in thin-film flows

A dynamical systems approach for the contact-line singularity in thin-film flows

Entropies for radially symmetric higher-order nonlinear diffusion equations

{Euclidean, metric, and Wasserstein} gradient flows: an overview

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