Felix Otto

THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION

We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show that the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.

The variational formulation of the Fokker-Planck equation

The Fokker--Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It ...

A compactness result in the gradient theory of phase transitions

in the plane. As ° ! 0, this functional favours jrAj = 1 and penalizes singularities where jrrAj concentrates. Our main result is a compactness theorem: if fE ° (A ° )g° #0 is uniformly bounded, then frA ° g° #0 is compact in L . Thus, in the limit ° ! 0, A solves the eikonal equation jrAj= 1 almost everywhere. Our analysis uses entropy relations’ and the div-curl lemma,’ adopting Tartar’ s approach to the interaction of linear di® erential equations and nonlinear algebraic relations.

EULERIAN CALCULUS FOR THE CONTRACTION IN THE WASSERSTEIN DISTANCE

We consider the porous medium equation on a compact Riemannian manifold and give a new proof of the contraction of its semigroup in the Wasserstein distance. This proof is based on the insight that the porous medium equation does not increase the size of infinitesimal perturbations along gradient flow trajectories and on an Eulerian formulation for the Wasserstein distance using smooth curves. Our approach avoids the existence result for optimal transport maps on Riemannian manifolds.

Uniform energy distribution for an isoperimetric problem with long-range interactions

We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and long-range interactions. The short-range interaction is attractive and comes from an interfacial energy, and the long-range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers. A natural conjecture is that in all space dimensions, minimizers are essentially periodic on an intrinsic scale. However, proving any periodicity result turns out to be a formidable task in dimensions larger than one. In this paper, we address a weaker statement concerning the distribution of energy for minimizers. We prove in any space dimension that each component of the energy (interfacial and nonlocal) of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale. Moreover, we also prove an L 1 bound on the optimal potential associated with the long-range interactions. This bound allows for an interesting interpretation: Note that the average volume fraction of the optimal pattern in a subsystem of size R fluctuates around the system average m. The bound on the potential yields a rate of decay of these fluctuations as R tends to +1. This rate of decay is stronger than the one for a random checkerboard pattern. In this sense, the optimal pattern has less large-scale variations of the average volume fraction than a pattern with a finite correlation

Coarsening dynamics of the convective Cahn-Hilliard equation

Abstract We characterize the coarsening dynamics associated with a convective Cahn-Hilliard equation (cCH) in one space dimension. First, we derive a sharp-interface theory through a matched asymptotic analysis. Two types of phase boundaries (kink and anti-kink) arise, due to the presence of convection, and their motions are governed to leading order by a nearest-neighbors interaction coarsening dynamical system ( CDS ). Theoretical predictions on CDS include: • The characteristic length L M for coarsening exhibits the temporal power law scaling t1/2; provided L M is appropriately small with respect to the Peclet length scale L P . • Binary coalescence of phase boundaries is impossible. • Ternary coalescence only occurs through the kink-ternary interaction; two kinks meet an anti-kink resulting in a kink. Direct numerical simulations performed on both CDS and cCH confirm each of these predictions. A linear stability analysis of CDS identifies a pinching mechanism as the dominant instability, which in turn leads to kink-ternaries. We propose a self-similar period-doubling pinch ansatz as a model for the coarsening process, from which an analytical coarsening law for the characteristic length scale L M emerges. It predicts both the scaling constant c of the t1/2 regime, i.e. L M =ct 1/2 , as well as the crossover to logarithmically slow coarsening as L M crosses L P . Our analytical coarsening law stands in good qualitative agreement with large-scale numerical simulations that have been performed on cCH.

Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh-Bénard convection

For the infinite-Prandtl-number limit of the Boussinesq equations, the enhancement of vertical heat transport in Rayleigh–Benard convection, the Nusselt number

Two–dimensional modelling of soft ferromagnetic films

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Quantitative results on the corrector equation in stochastic homogenization

, is bounded above in terms of the Rayleigh number

Evolution of microstructure in unstable porous media flow: A relaxational approach

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