Research WG Seis Applied Mathematics Münster

All of my publications can be found on Google Scholar, where you’ll usually find links to both the published version and a free preprint version. Below, I've listed my publications sorted by topic, namely:

V Navarro-Fernández, C Seis. Exponential mixing by random cellular flows. Preprint arXiv:2502.17273, 2025. [arXiv]

We study a passive scalar equation on the two-dimensional torus, where the advecting velocity field is given by a cellular flow with a randomly moving center. We prove that the passive scalar undergoes mixing at a deterministic exponential rate, independent of any underlying diffusivity. Furthermore, we show that the velocity field enhances dissipation and we establish sharp decay rates that, for large times, are deterministic and remain uniform in the diffusivity constant. Our approach is purely Eulerian and relies on a suitable modification of Villani's hypocoercivity method.

C Seis. Bounds on the rate of enhanced dissipation. Communications in Mathematical Physics 399 (3), 2071-2081, 2023. [OA]

We are concerned with flow enhanced mixing of passive scalars in the presence of diffusion. Under the assumption that the velocity gradient is suitably integrable, we provide upper bounds on the exponential rates of enhanced dissipation. Recent constructions indicate the optimality of our results.

V Navarro-Fernández, A Schlichting, C Seis. Optimal stability estimates and a new uniqueness result for advection-diffusion equations. Pure and Applied Analysis 4 (3), 571-596, 2022. [arXiv]

We present two main contributions. First, we provide optimal stability estimates for advection-diffusion equations in a setting in which the velocity field is Sobolev regular in the spatial variable. This estimate is formulated with the help of Kantorovich–Rubinstein distances with logarithmic cost functions. Second, we extend the stability estimates to the advection-diffusion equations with velocity fields whose gradients are singular integrals of L¹ functions entailing a new well-posedness result.

A Schlichting, C Seis. The Scharfetter–Gummel scheme for aggregation–diffusion equations. IMA Journal of Numerical Analysis 42 (3), 2361-2402, 2022. [arXiv]

In this paper we propose a finite-volume scheme for aggregation–diffusion equations based on a Scharfetter–Gummel approximation of the quadratic, nonlocal flux term. This scheme is analyzed concerning well posedness and convergence towards solutions to the continuous problem. Also, it is proven that the numerical scheme has several structure-preserving features. More specifically, it is shown that the discrete solutions satisfy a free-energy dissipation relation analogous to the continuous model. Consequently, the numerical solutions converge in the large time limit to stationary solutions, for which we provide a thermodynamic characterization. Numerical experiments complement the study.

D Meyer, C Seis. Propagation of regularity for transport equations. A Littlewood-Paley approach. Indiana Univ. Math Journal (in press). [arXiv]

It is known that linear advection equations with Sobolev velocity fields have very poor regularity properties: Solutions propagate only derivatives of logarithmic order, which can be measured in terms of suitable Gagliardo seminorms. We propose a new approach to the study of regularity that is based on Littlewood-Paley theory, thus measuring regularity in terms of Besov norms. We recover the results that are available in the literature and extend these optimally to the diffusive setting. As a consequence, we derive sharp bounds on rates of convergence in the zero-diffusivity limit.

C Seis. On the Littlewood–Paley spectrum for passive scalar transport equations, Journal of Nonlinear Science 30 (2), 645-656, 2020. [arXiv]

We derive time-averaged L¹ estimates on Littlewood–Paley decompositions for linear advection–diffusion equations. For wave numbers close to the dissipative cutoff, these estimates are consistent with Batchelor’s predictions on the variance spectrum in passive scalar turbulent mixing.

A Schlichting, C Seis. Analysis of the implicit upwind finite volume scheme with rough coefficients, Numerische Mathematik 139, 155-186, 2018. [arXiv]

We study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation in the low regularity DiPerna–Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at 1/2. The numerical error is estimated in terms of logarithmic Kantorovich–Rubinstein distances and provides thus a bound on the rate of weak convergence.

C Seis. A quantitative theory for the continuity equation. Annales de l'Institut Henri Poincaré C, Analyse non linéaire 34 (7), 1837-1850, 2017. [OA]

In this work, we provide stability estimates for the continuity equation with Sobolev vector fields. The results are inferred from contraction estimates for certain logarithmic Kantorovich–Rubinstein distances. As a by-product, we obtain a new proof of uniqueness in the DiPerna–Lions setting. The novelty in the proof lies in the fact that it is not based on the theory of renormalized solutions.

A Schlichting, C Seis. Convergence rates for upwind schemes with rough coefficients. SIAM Journal on Numerical Analysis 55 (2), 812-840, 2017. [arXiv]

This paper is concerned with the numerical analysis of the explicit upwind finite volume scheme for numerically solving continuity equations. We are interested in the case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable. We estimate the error between approximate solutions constructed by the upwind scheme and distributional solutions of the continuous problem in a Kantorovich--Rubinstein distance, which was recently used for stability estimates for the continuity equation [C. Seis, Ann. Inst. H. Poincaré Anal. Non Linéaire, https://doi.org/10.1016/j.anihpc.2017.01.001]. Restricted to Cartesian meshes, our estimate shows that the rate of weak convergence is at least 1/2 in the mesh size. The proof relies on a probabilistic interpretation of the upwind scheme [F. Delarue and F. Lagoutière, Arch. Ration. Mech. Anal., 199 (2011), pp. 229--268]. We complement the weak convergence result with an example that illustrates that for rough initial data no rates can be expected in strong norms. The same example suggests that the weak 1/2 rate is optimal.

C Seis. Optimal stability estimates for continuity equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 148 (6) 1279 - 1296, 2018. [arXiv]

This review paper is concerned with the stability analysis of the continuity equation in the DiPerna–Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently, but optimality was not discussed. We revisit the results from our 2017 paper, compare the new estimates with previously known estimates for Lagrangian flows and demonstrate how these can be applied to produce optimal bounds in applications from physics, engineering and numerical analysis.

C Seis. Scaling bounds on dissipation in turbulent flows. Journal of Fluid Mechanics 777, 591 - 603, 2015. [arXiv]

We propose a new rigorous method for estimating statistical quantities in fluid dynamics such as the (average) energy dissipation rate directly from the equations of motion. The method is tested on shear flow, channel flow, Rayleigh–Bénard convection and porous medium convection.

C Seis. Laminar boundary layers in convective heat transport. Communications in Mathematical Physics 324, 995-1031, 2013. [arXiv]

We study Rayleigh–Bénard convection in the high-Rayleigh-number regime and infinite-Prandtl-number limit, i.e., we consider a fluid in a container that is exposed to strong heating of the bottom and cooling of the top plate in the absence of inertia effects. While the dynamics in the bulk are characterized by a chaotic heat flow, close to the horizontal walls, the fluid is essentially motionless.

We derive local bounds on the temperature field in the boundary layers and prove that the temperature profile is essentially linear. The results depend only logarithmically on the system parameters. An important tool in our analysis is a new Hardy-type estimate for the convecting velocity field, which yields control of the fluid motion in the layer. The bounds on the temperature field are derived via local maximal regularity estimates for convection-diffusion equations.

C Seis. Maximal mixing by incompressible fluid flows. Nonlinearity 26 (12), 3279, 2013. [arXiv]

We consider a model for mixing binary viscous fluids under an incompressible flow. We prove the impossibility of perfect mixing in finite time for flows with finite viscous dissipation. As measures of mixedness we consider a Monge–Kantorovich–Rubinstein transportation distance and, more classically, the H^-1 norm. We derive rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time. The rate of the exponential decay is uniform in the initial data.

L Mugnai, C Seis. On the coarsening rates for attachment-limited kinetics. SIAM Journal on Mathematical Analysis 45 (1), 324-344, 2013.

We study the coarsening rates for attachment-limited kinetics which is modeled by volume-preserving mean-curvature flow. Attachment-limited kinetics is observed during solidification processes, in which the system is divided into two domains of the two pure phases, more precisely, islands of a solid phase surrounded by an undercooled liquid phase, and the relaxation process is due to material redistribution form high to low interfacial curvature regions. The interfacial area between the phases decreases in time while the volume of each phase is preserved. Consequently, the domain morphology coarsens. Experiments, heuristics, and numerics suggest that the typical domain size ℓ of the solid islands grows according to the power law ℓ∼t^½, when t denotes time. In this paper, we focus on the two-dimensional case. We prove a weak one-sided version of this coarsening rate, namely, we prove that ℓ≲t^½ in time average. The bound on the coarsening rate is uniform in the sense that we do not impose any explicit assumptions on the initial configuration. However, during the evolution, we have to suppose that collisions of different domains are rare events. In this regard, our result is relevant and new for “relatively sparse” configurations. Our analytical approach is based on a method introduced by Kohn and Otto [Comm. Math. Phys., 229 (2002), pp. 375--395], which relies on the gradient flow structure of the dynamics.

F Otto, C Seis, D Slepčev. Crossover of the coarsening rates in demixing of binary viscous liquids. Comm. Math. Sci. 11 (2), 441-464, 2013.

We consider a model for phase separation in binary viscous liquids that allows for material transport due to cross-diffusion of unlike particles and convection by the hydrodynamic bulk flow. Typically, during the evolution, the average size of domains of the pure phases increases with time—a phenomenon called coarsening. Siggia [24] predicts that at an initial stage, coarsening proceeds mainly by diffusion, which leads to the well-known evaporation-recondensation growth law l∼ t^1/3, when l denotes the average domains size and t denotes time. Furthermore, he argued that at a later stage, convection by the bulk flow becomes the dominant transport mechanism, leading to a crossover in the coarsening rates to l∼ t. Siggia’s predictions have been confirmed by experiments and numerical simulations.

In this work, we prove the crossover in the coarsening rates in terms of time-averaged lower bounds on the energy, which scales like an inverse length. We use a method proposed by Kohn and the first author [15], which exploits the gradient flow structure of the dynamics. Our adaption uses techniques from optimal transportation. Our main ingredient is a dissipation inequality. It measures how the optimal transportation distance changes under the effects of convective and diffusive transport.

F Otto, C Seis. Rayleigh–Bénard convection: improved bounds on the Nusselt number. Journal of mathematical physics 52 (8), 2011.

We consider Rayleigh–Bénard 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∼ Ra^1/3. 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≲ Ra^1/3ln^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≲ Ra^1/3 ln^1/3Ra with help of the background field method. The paper contains two results. The background field method can be slightly modified to yield Nu≲ Ra^1/3 ln^1/15 Ra. The estimates behind the background field method can be combined with the maximal regularity in L∞ to yield Nu ≲ Ra^1/3ln^1/3 ln Ra — an estimate that is only a double logarithm away from the supposedly optimal scaling.

Y Brenier, F Otto, C Seis. Upper bounds on coarsening rates in demixing binary viscous liquids. SIAM journal on mathematical analysis 43 (1), 114-134, 2011.

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 ℓ increases according to the power law ℓ∼t^1/3, 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 ℓ∼t. 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 ℓ∼t. 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 distance, which is not known explicitly.

Theory of Euler equations

N De Nitti, D Meyer, C Seis. Optimal regularity for the 2D Euler equations in the Yudovich class. Journal de Mathématiques Pures et Appliquées 191, 103631, 2024.

We analyze the optimal regularity that is exactly propagated by a transport equation driven by a velocity field with a BMO gradient. As an application, we study the 2D Eulerequations in case the initial vorticity is bounded. The sharpness of our result for the Euler equations follows from a variation of Bahouri and Chemin's vortex patch example.

C Seis, E Wiedemann, J Woźnicki. Strong Convergence of Vorticities in the 2D Viscosity Limit on a Bounded Domain. Preprint arXiv:2406.05860, 2024.

In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a purely interior framework, Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary but finite time under the assumption that on each compactly contained subset of the domain, the enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the p-enstrophies, p>2. The key novel idea is the analysis of the evolution of the weak convergence defect.

C Nobili, C Seis. Renormalization and energy conservation for axisymmetric fluid flows. Mathematische Annalen 382 (1), 1-36, 2022.

We study vanishing viscosity solutions to the axisymmetric Euler equations without swirl with (relative) vorticity in L^p with p>1. We show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions. Moreover, we show that the kinetic energy is preserved provided that p>3/2 and the vorticity is nonnegative and has finite second moments.

HJ Nussenzveig Lopes, C Seis, E Wiedemann. On the vanishing viscosity limit for 2D incompressible flows with unbounded vorticity, Nonlinearity 34 (5), 3112, 2021.

We show strong convergence of the vorticities in the vanishing viscosity limit for the incompressible Navier–Stokes equations on the two-dimensional torus, assuming only that the initial vorticity of the limiting Euler equations is in L^p for some p> 1. This substantially extends a recent result of Constantin, Drivas and Elgindi, who proved strong convergence in the case p=∞. Our proof, which relies on the classical renormalisation theory of DiPerna–Lions, is surprisingly simple.

C Seis. A note on the vanishing viscosity limit in the Yudovich class. Canadian Mathematical Bulletin 64 (1), 112-122, 2021.

We consider the inviscid limit for the two-dimensional Navier–Stokes equations in the class of integrable and bounded vorticity fields. It is expected that the difference between the Navier–Stokes and Euler velocity fields vanishes in bound, which slightly improves upon earlier results by Chemin.

G Crippa, C Nobili, C Seis, S Spirito. Eulerian and Lagrangian Solutions to the Continuity and Euler Equations with Vorticity. SIAM Journal on Mathematical Analysis 49 (5), 3973-3998, 2017.

In the first part of this paper we establish a uniqueness result for continuity equations with a velocity field whose derivative can be represented by a singular integral operator of an L1 function, extending the Lagrangian theory in [F. Bouchut and G. Crippa, J. Hyperbolic Differ. Equ., 10 (2013), pp. 235--282]. The proof is based on a combination of a stability estimate via optimal transport techniques developed in [C. Seis, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear] and some tools from harmonic analysis introduced in [F. Bouchut and G. Crippa, J. Hyperbolic Differ. Equ., 10 (2013), pp. 235--282]. In the second part of the paper, we address a question that arose in [M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes, Arch. Ration. Mech. Anal., 179 (2006), pp. 353--387], namely, whether 2 dimensional Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data have low integrability. We show that this is the case even when the initial vorticity is only in L1, extending the proof for the Lp case in [G. Crippa and S. Spirito, Comm. Math. Phys., 339 (2015), pp. 191--198].

Vortex dynamics in fluids

D Meyer, L Niebel, and C Seis. Steady bubbles and drops in inviscid fluids. Preprint arXiv:2503.05503.

We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform vorticity distribution in this inner phase and they have a vortex sheet on its surface. Our construction relies on a perturbative approach around an explicit spherical solution, given by Hill's vortex enclosed by a spherical vortex sheet. The construction is sensitive to the Weber numbers describing the flow. At critical Weber numbers, we perform a bifurcation analysis utilizing the Crandall-Rabinowitz theorem in Sobolev spaces on the 2-sphere. Away from these critical numbers, our construction relies on the implicit function theorem. Our results imply that the model containing surface tension is richer than the ordinary one-phase Euler equations, in the sense that for the latter, Hill's spherical vortex is unique (modulo translations) among all axisymmetric simply connected uniform vortices of a given circulation.

D Meyer, C Seis. Steady Ring-Shaped Vortex Sheets. Peprint arXiv:2409.08220, 2024.

In this work, we construct traveling wave solutions to the two-phase Euler equations, featuring a vortex sheet at the interface between the two phases. The inner phase exhibits a uniform vorticity distribution and may represent a vacuum, forming what is known as a hollow vortex. These traveling waves take the form of ring-shaped vortices with a small cross-sectional radius, referred to as thin rings. Our construction is based on the implicit function theorem, which also guarantees local uniqueness of the solutions. Additionally, we derive asymptotics for the speed of the ring, generalizing the well-known Kelvin--Hicks formula to cases that include surface tension.

S Ceci, C Seis. On the dynamics of vortices in viscous 2D flows. Mathematische Annalen 388 (2), 1937-1967, 2024.

We study the 2D Navier–Stokes solution starting from an initial vorticity mildly concentrated near N distinct points in the plane. We prove quantitative estimates on the propagation of concentration near a system of interacting point vortices introduced by Helmholtz and Kirchhoff. Our work extends the previous results in the literature in three ways: The initial vorticity is concentrated in a weak (Wasserstein) sense, it is merely integrable for some p>2, and the estimates we derive are uniform with respect to the viscosity.

S Ceci, C Seis. On the dynamics of point vortices for the two-dimensional Euler equation with L^p vorticity, Philosophical Transactions of the Royal Society A 380 (2226), 20210046, 2022.

We study the evolution of solutions to the two-dimensional Euler equations whose vorticity is sharply concentrated in the Wasserstein sense around a finite number of points. Under the assumption that the vorticity is merely L^p integrable for some p>2, we show that the evolving vortex regions remain concentrated around points, and these points are close to solutions to the Helmholtz–Kirchhoff point vortex system.

S Ceci, C Seis. Vortex dynamics for 2D Euler flows with unbounded vorticity, Revista Matematica Iberoamericana 37 (5), 1969-1990, 2021.

It is well known that the dynamics of vortices in an ideal incompressible two-dimensional fluid contained in a bounded not necessarily simply connected smooth domain is described by the Kirchhoff–Routh point vortex system. In this paper, we revisit the classical problem of how well solutions to the Euler equations approximate these vortex dynamics, and extend previous rigorous results to the case where the vorticity field is unbounded. More precisely, we establish estimates for the 2-Wasserstein distance between the vorticity and the empirical measure associated with the point vortex dynamics. In particular, we derive an estimate on the order of weak convergence of the Euler solutions to the solutions of the point vortex system.

RL Jerrard, C Seis. On the vortex filament conjecture for Euler flows. Archive for Rational Mechanics and Analysis 224, 135-172, 2017.

In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in R³, we prove that the curve evolves to leading order by binormal curvature flow. Our approach combines new estimates on the distance of the corresponding Hamiltonian-Poisson structures with stability estimates recently developed in Jerrard and Smets (J Eur Math Soc (JEMS) 17(6):1487–1515, 2015).

Nonlinear diffusions

C Seis, D Winkler. Fine large-time asymptotics for the axisymmetric Navier–Stokes equations. Journal of Evolution Equatons 24 (3), 72, 2024.

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier–Stokes equation in R³. We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard L²-based entropy methods.

C Seis, D Winkler. Invariant manifolds for the thin film equation. Archive for Rational Mechanics and Analysis 248 (2), 27, 2024.

The large-time behavior of solutions to the thin film equation with linear mobility in the complete wetting regime on R^N is examined. We investigate the higher order asymptotics of solutions converging towards self-similar Smyth–Hill solutions under certain symmetry assumptions on the initial data. The analysis is based on a construction of finite-dimensional invariant manifolds that solutions approximate to an arbitrarily prescribed order.

C Seis, D Winkler. Stability of traveling waves for doubly nonlinear equations. Preprint arXiv:2401.10597. 2024

In this note, we investigate a doubly nonlinear diffusion equation in the slow diffusion regime. We prove stability of the pressure of solutions that are close to traveling wave solutions in a homogeneous Lipschitz sense. We derive regularity estimates for arbitrary derivatives of the solution's pressure by extending existing results for the porous medium equation (Ref. 15).

B Choi, C Seis. Finite-dimensional leading order dynamics for the fast diffusion equation near extinction. Discrete and Continuous Dynamical Systems 44 (9), 2697-2712, 2024.

The fast diffusion equation is analyzed on a bounded domain with Dirichlet boundary conditions, for which solutions are known to extinct in finite time. We construct invariant manifolds that provide a finite-dimensional approximation near the vanishing solution to any prescribed convergence rate.

B Choi, RJ McCann, C Seis. Asymptotics near extinction for nonlinear fast diffusion on a bounded domain, Archive for Rational Mechanics and Analysis 247 (2), 16, 2023.

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap; this refines and confirms a 1980 conjecture of Berryman and Holland. We also improve on a result of Bonforte and Figalli by providing a new and simpler approach which is able to accommodate the presence of zero modes, such as those that occur when the vanishing profile fails to be isolated (and possibly belongs to a continuum of such profiles).

C Seis, D Winkler. A well-posedness result for a system of cross-diffusion equations, Journal of Evolution Equations 21, 2471-2489, 2021.

This work’s major intention is the investigation of the well-posedness of certain cross-diffusion equations in the class of bounded functions. More precisely, we show existence, uniqueness and stability of bounded weak solutions under a smallness assumption on the intial data. As an application, we provide a new well-posedness theory for a diffusion-dominant cross-diffusion system that originates from a hopping model with size exclusions. Our approach is based on a fixed point argument in a function space that is induced by suitable Carleson-type measures.

C Seis. The thin-film equation close to self-similarity, Analysis & PDE 11 (5), 1303-1342, 2018.

We study well-posedness and regularity of the multidimensional thin-film equation with linear mobility in a neighborhood of the self-similar Smyth–Hill solutions. To be more specific, we perform a von Mises change of dependent and independent variables that transforms the thin-film free boundary problem into a parabolic equation on the unit ball. We show that the transformed equation is well-posed and that solutions are smooth and even analytic in time and angular direction. The latter gives the analyticity of level sets of the original equation, and thus, in particular, of the free boundary.

C Seis. Invariant manifolds for the porous medium equation. Preprint arXiv:1505.06657, 2015.

In this paper, we investigate the speed of convergence and higher-order asymptotics of solutions to the porous medium equation posed in R^N. Applying a nonlinear change of variables, we rewrite the equation as a diffusion on a fixed domain with quadratic nonlinearity. The degeneracy is cured by viewing the dynamics on a hypocycloidic manifold. It is in this framework that we can prove a differentiable dependency of solutions on the initial data, and thus, dynamical systems methods are applicable. Our main result is the construction of invariant manifolds in the phase space of solutions which are tangent at the origin to the eigenspaces of the linearized equation. We show how these invariant manifolds can be used to extract information on the higher-order long-time asymptotic expansions of solutions to the porous medium equation.

RJ McCann, C Seis. The spectrum of a family of fourth-order nonlinear diffusions near the global attractor. Communications in Partial Differential Equations 40 (2), 191-218, 2015.

The thin film and quantum drift diffusion equations belong to a fourth-order family of evolution equations proposed in to be analogous to the (second-order) porous medium family. They are 2-Wasserstein (=d₂) gradient flows of the generalized Fisher information I(v) just as the porous medium family was shown to be the d₂ gradient flow of the generalized entropy E(v) by Otto. The identity aI(v) = bE(v) + |∇ d 2 E(v)|²/2 implies a Hess d₂ I(v *) = Hess d₂ E(v *)(b + Hess d₂ E(v *)) formally, when the equation is rescaled and linearized around the resulting self-similar critical profile v *. We couple this relation with the diagonalization of Hess d₂ E(v *) for the porous medium flow computed in [46]. This yields information about the leading- and higher-order asymptotics of the equation on R^N which—outside of special cases—was inaccessible previously.

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

We compute the complete spectrum of the displacement Hessian operator, which is obtained from the confined porous medium equation by linearization around its stationary attractor, the Barenblatt profile. On a formal level, the operator is conjugate to the Hessian of the entropy via similarity transformation. We show that the displacement Hessian can be understood as a self-adjoint operator and find that its spectrum is purely discrete. The knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor. More precisely, the inspection of the eigenfunctions allows to identify symmetries in RN with flows whose rates of convergence are faster than the uniform, translation-governed bound. The present work complements the analogous study of Denzler & McCann for the fast-diffusion equation.

Miscellaneous in Calculus of Variations

L Beck, E Cinti, C Seis. Optimal regularity of isoperimetric sets with Hölder densities. Calculus of Variations and Partial Differential Equations 62 (214), 1-20, 2023.

We establish a regularity result for optimal sets of the isoperimetric problem with double density under mild (α-)Hölder regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach in any dimension the regularity class C^{1‚α/2–α}. This class is indeed the optimal one for local minimizers of variational functionals with an integrand that depends α-Hölder continuous on the minimizer itself, and as such can (the boundary of) the isoperimetric set be locally written (with additional constraint).

L Mugnai, C Seis, E Spadaro. Global solutions to the volume-preserving mean-curvature flow. Calculus of Variations and Partial Differential Equations 55, 1-23, 2016.

In this paper, we construct global distributional solutions to the volume-preserving mean-curvature flow using a variant of the time-discrete gradient flow approach proposed independently by Almgren et al. (SIAM J Control Optim 31(2):387–438, 1993) and Luckhaus and Sturzenhecker (Calc Var Partial Differ Equ 3(2):253–271, 1995).

J Korman, RJ McCann, C Seis. Dual potentials for capacity constrained optimal transport. Calculus of Variations and Partial Differential Equations 54, 573-584, 2015.

Optimal transportation with capacity constraints, a variant of the well-known optimal transportation problem, is concerned with transporting one probability density f in L¹(R^N) onto another one g in L¹(R^N) so as to optimize a cost function c in L¹(R^n+m) while respecting the capacity constraints.

J Korman, RJ McCann, C Seis. An elementary approach to linear programming duality with application to capacity constrained transport. J. Convex Anal. 3, 797-808, 2015.

A new approach to linear programming duality is proposed which relies on quadratic penalization, so that the relation between solutions to the penalized primal and dual problems becomes affine. This yields a new proof of Levin's duality theorem for capacity-constrained optimal transport as an infinite-dimensional application.