# Christian Seis

ResearchThe area of my research is applied analysis with a main emphasis on nonlinear partial differential equations. The focus lies on mathematical models which originate from real problems in physics, materials science or engineering.

In the following, I give an overview on three specific areas I have worked on:

- Coarsening, mixing and transport equations
- Rayleigh-Bénard convection
- Long-time asymptotics for slow diffusions

### Coarsening, mixing and transport equations

Coarsening phenomena in physics and material
science have been the object of extensive theoretical, experimental, and numerical
research. Coarsening is observed in two-phase (or multi-phase) systems
far from equilibrium, where thermodynamics favors the separation of the
different phases and drives thus the formation of microstructure.
Typically, during the evolution, the average size of the one-phase regions
grows in time according to simple power laws. Together with coworkers,
I investigated these *coarsening rates* in two
physically relevant model.

Together with Yann Brenier, Felix Otto, and Dejan Slepčev, I focused on the evolution of binary viscous liquids. It turns out that in early stages of the evolution, the relevant material transport is diffusion while in later stages, material transport is dominated by the hydrodynamic bulk flow. Each stage is characterized by an individual coarsening rate.

- Felix Otto, Christian Seis, and Dejan Slepčev. Crossover in the coarsening rates
in demixing binary viscous liquids.
*Comm. Math. Sci.*, 11(2):441-464, 2013.

[Article, Preprint] - Yann Brenier, Felix Otto, and Christian Seis. Upper bounds on coarsening rates in
demixing binary viscous liquids.
*SIAM J. Math. Anal.*, 43(1):114-134, 2011.

[Article, Preprint]

In a joint work with Luca Mugnai, I studied the coarsening rates for attachment-limited kinetics, which is mathematically modeled by volume-preserving mean-curvature flow. Existence of global solutions for that model were obtained in colaboration with Luca Mugnai and Emanuele Spadaro.

- Luca Mugnai and Christian Seis. On the coarsening rates for attachment-limited kinetics.
*SIAM J. Math. Anal.*, 45(1):324-344, 2013.

[Article, Preprint] - Luca Mugnai, Christian Seis, and Emanuele Spadaro. Global solutions to the
volume-preserving mean-curvature flow.
*Calc. Var. and PDE.*55 (2016), no. 1, 55:18.

[Article, Preprint]

The efficiency of mixing of two (or more) fluid components and the search for optimal stirring strategies is of great importance for engineering applications. Modelling stirring by an imposed flow field, the question we are facing is "How efficient can a binary fluid be mixed by an incompressible flow?". A quantitative answer to this question necessitates to specify, on the one hand, the underlying measure for the degree of mixedness. On the other hand, a constraint on the velocity field has to be imposed which reflects the engineer's general set-up for the experiment. In a typical application, this is achieved by fixing the budget for the power (or viscous dissipation). I proved that mixing (measured in terms of negative Sobolev norms or logarithmic Kantorovich-Rubinstein distances) cannot proceed faster than exponentially in time. In particular, perfect mixing cannot be achieved in finite time.

- Christian Seis. Maximal mixing by incompressible fluid flows.
*Nonlinearity,*26(12):3279-3289, 2013.

[Article, Preprint]

Here are snapshots from a simulation of optimal mixing with power constraints:

Bounds on mixing rates can be considered as a quantification of the DiPerna-Lions theory for transport or continuity equations. Indeed, perfect mixing in finite time is equivalent to non-uniqueness of the associated Cauchy problem. Uniqueness is guaranteed if the transporting velocity field has spatial Sobolev or BV regularity. It is thus tempting to develop a well-posedness theory that reflects this regularity condition. In

- Christian Seis. A quantitative theory for the continuity equation.
*Ann. Inst. H. Poincaré Anal. Non Linéaire (in press).*

[Article, Preprint] - Christian Seis. Optimal stability estimates for continuity equations.
*Accepted for publication in Proc. Roy. Soc. Edinburgh Sect. A.*

[Preprint]

- Gianluca Crippa, Camilla Nobili, Christian Seis and Stefano Spirito.
Eulerian and Lagrangian solutions to the continuity and Euler equations with $L^1$ vorticity.

[Preprint]

- André Schlichting and Christian Seis. Convergence rates for upwind schemes with rough coefficients.
*SIAM J. Numer. Anal.*55(2), 812–840, 2017.

[Article, Preprint] - André Schlichting and Christian Seis. Analysis of the implicit upwind finite
volume scheme with rough coefficients.
*Submitted, arXiv:1702.02534.*

[Preprint]

### Rayleigh-Bénard convection

Rayleigh-Bénard convection is the flow of an incompressible Newtonian fluid in a container that is heated from below and cooled from above. It is one of the classical models of fluid dynamics. I am interested in a regime where the heat transport is mediated both by conduction and by a chaotic buoyancy-driven fluid flow. In this situation one observes a clear spatial separation of the relevant heat transfer mechanism: Thin boundary layers in the vicinity of the horizontal container boundaries in which heat is essentially conducted and a large bulk that is characterized by the convective heat flow. Here's a snapshot from a numerical simulation by Martin Zimmermann:

Together with Felix Otto, I investigated the behavior of the average upward heat flux, the so-called Nusselt number, as a function of the temperature forcing, traditionally measured by the Rayleigh number. Despite the complexity of the observed flow pattern, it is conjectured that the Nusselt number obeys a simple scaling law. We proved an upper bound on the Nusselt number that matches the expected scaling up to a double logarithm.

- Felix Otto and Christian Seis. Rayleigh-Bénard convection: Improved bounds on
the Nusselt number.
*J. Math. Phys.,*, 52(8):83702-83726, 2011.

[Article, Preprint]

In a second work on Rayleigh-Bénard convection, I focused on the conductive boundary layers. It is in these boundary layers where the main temperature drop between the hot bottom and the cold top plate happens. Therefore, the understanding of these layers yields further information on the efficiency of heat transport in the Rayleigh-Bénard experiment. I derived new bounds on the temperature field in the boundary layers and proved that the temperature profile is essentially linear.

- Christian Seis. Laminar boundary layers in convective heat transport.
*Comm. Math. Phys.*324(3):995-1031, 2013.

[Article, Preprint]

The techniques developed in the derivation of the Nusselt number bounds yield a general method that allows the study of a family of classical problems in fluid dynamics. More precisely, bounds on the average dissipation rate can be computed for channel flows, shear flows, more general Rayleigh-Bénard systems or porous medium convection.

- Christian Seis. Scaling bounds on dissipation in turbulent flows.
*J. Fluid Mech.*777: 591-603, 2015.

[Article, Preprint]

### Long-time asymptotics for slow diffusions

The porous medium equation is a nonlinear diffusion equation that (among other applications) describes the flow of an isentropic gas through a porous medium. The evolving quantity is the density of a gas, and as such, it is nonnegative. The equation is degenerate parabolic, that means, it is parabolic only on the support of the denisty function. As a consequence, solutions preserve their compact support (in contrast, for instance, to solutions to the heat equation) and thus the porous medium equation turns into a free boundary problem. It is well-known that the long-time behavior of solutions is governed by the self-similar Barenblatt solution. In a recent work, I constructed invariant manifolds for the porous medium equation close to the global attractor. These manifolds measure to what extend the porous medium dynamics can be described by a system of ODEs. Invariant manifolds can be used to study the higher-order asymptotics. Once an order of decay is specified, there exists a finite-dimensional invariant manifold such that all solutions that are sufficiently close to the self-similar Barenblatt solutions approach this invariant manifold with at least that rate. Thanks to the discreteness of the spectrum all eigenmodes are in principle accessible.

- Christian Seis. Long-time asymptotics for the porous medium equation: The spectrum
of the linearized operator.
*J. Differential Equations,*256(3):1191-1223, 2014.

[Article, Preprint] - Christian Seis. Invariant manifolds for the porous medium equation.
*Submitted, arXiv:1505.06657.*

[Preprint]

In joint work with Robert McCann, I studied the long-time behavior of solutions to a family of nonlinear fourth-order equation, which include the thin film equation (with linear mobility) and the quantum drift-diffusion equation. The thin film equation models the evolution of a thin viscous film on a solid substrate, while the quantum drift-diffusion equation describes the evolution of a density of electrons in a semiconductor crystal. Depending on the particular nonlinearity parameter, this family comes as a fourth-order analogue of the porous medium equation or of the heat equation and it shares some of their features. In particular, in the long-time limit, solutions converge to a self-similar Barenblatt-type solution or a Gaussian. We compute the complete spectrum of the linearized equation.

- Robert J. McCann and Christian Seis. The spectrum of a familiy of fourth-order
diffusions near the global attractor.
*Comm. Partial Differential Equations*40(2): 191-218, 2015.

[Article, Preprint]