Christian Seis

Upper Bounds on Coarsening Rates in Demixing Binary Viscous Liquids

We consider the coarsening process of a binary viscous liquid after a temperature quench. In a first, diffusion-dominated coarsening regime (“evaporation-recondensation process”), the typical length scale

Rayleigh–Bénard convection: Improved bounds on the Nusselt number

\ell

Maximal mixing by incompressible fluid flows

increases according to the power law

A quantitative theory for the continuity equation

\ell\sim t^{1/3}

The Spectrum of a Family of Fourth-Order Nonlinear Diffusions Near the Global Attractor

, where t is the time. Siggia [Phys. Rev. A, 20 (1979), pp. 595–605] argued that in a second regime, coarsening should be mediated by viscous flow of the mixture. This leads to a crossover in the coarsening rates to the power law

Convergence Rates for Upwind Schemes with Rough Coefficients

\ell\sim t

On the coarsening rates for attachment-limited kinetics

. We consider a simple sharp-interface model which just allows for flow-mediated coarsening. For this model, we prove rigorously that coarsening cannot proceed faster than

Laminar Boundary Layers in Convective Heat Transport

\ell\sim t

On the Vortex Filament Conjecture for Euler Flows

. The analysis follows closely a method proposed in [R. V. Kohn and F. Otto, Comm. Math. Phys., 229 (2002), pp. 375–395], which is based on the gradient flow structure of the evolution. The analysis makes use of a Monge–Kantorowicz–Rubinstein transportation distance with logarithmic cost function as a proxy for the intrinsic ...

The thin-film equation close to self-similarity

We consider Rayleigh–Benard convection as modelled by the Boussinesq equations in the infinite-Prandtl-number limit. We are interested in the scaling of the average upward heat transport, the Nusselt number Nu, in terms of the non-dimensionalized temperature forcing, the Rayleigh number Ra. Experiments, asymptotics and heuristics suggest that Nu ∼ Ra1/3. This work is mostly inspired by two earlier rigorous work on upper bounds of Nu in terms of Ra. (1) The work of Constantin and Doering establishing Nu ≲ Ra1/3ln 2/3Ra with help of a (logarithmically failing) maximal regularity estimate in L∞ on the level of the Stokes equation. (2) The work of Doering, Reznikoff and the first author establishing Nu ≲ Ra1/3ln 1/3Ra with help of the background field method. The paper contains two results. (1) The background field method can be slightly modified to yield Nu ≲ Ra1/3ln 1/15Ra. (2) The estimates behind the background field method can be combined with the maximal regularity in L∞ to yield Nu ≲ Ra1/3ln 1/3ln Ra —...