Nonlinear Partial Differential Equations in Freiburg - Schedule

In 1996, Professors Rajagopal and Ruzicka introduced a new model for electrorheological fluids. In this talk, I will reflect on how this model sparked the beginning of a rewarding scientific journey shared between Michael Ruzicka and myself. I will also present some of the subsequent advancements of those results.

We consider the graphical mean curvature flow of maps $f : \mathbb{R}^m\to\mathbb{R}^n$, $m\ge 2$, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken. In the case of uniformly area decreasing maps $f : \mathbb{R}^m\to\mathbb{R}^2$, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander. This is joint work with Andreas Savas-Halilaj.

In systems of equations describing the motion of viscous fluids, solutions exhibit various interesting spatiotemporal pattern dynamics. In this talk, we will consider the bifurcation and stability problem of spatially periodic vortex patterns in a rotating fluid system. We will present results on the bifurcation of stationary periodic patterns and the stability and instability of bifurcating periodic patterns when the Mach number is small.

Given a closed riemannian manfiold of dimension $3$ $(M^3, [h])$, when will we fill in an asymptotically hyperbolic Einstein manifold of dimension 4 $(X^4, g_+)$ such that $r^2 g_+|_M= h$ on the boundary $M=\partial X$ for some defining function $r$ on $X^4$? This problem is motivated by the correspondance AdS/CFT in quantum gravity proposed by Maldacena in 1998 et comes also from the study of the structure of asymptotically hyperbolic Einstein manifolds.
In this talk, I discuss some existence and uniqueness issue of asymptotically hyperbolic Einstein manifolds in dimension 4. It is based on the recent works with Alice Chang.

In this talk, I will give an overview of results on linking the regularity of the solutions to the 3D Navier-Stokes equations and the geometry of the velocity/vorticity fields. In particular, starting from the celebrated criterion of Constantin and Fefferman (Indiana Univ. Math. J., 1993) about the vorticity direction, I will present some variations and improvement along various paths.

Anisotropic integrands are used to model surface energies of inhomogeneous materials, where the energy density depends on the direction of the surface. Under suitable assumptions, minimisers of the associated functionals are known to be so-called Wulff shapes, those being certain convex bodies which induce a Minkowski norm on Euclidean space. The first variation being the anisotropic mean curvature, the anisotropic Alexandrov-type theorem says that a surface with constant anisotropic mean curvature must be the corresponding Wulff shape. There are various related rigidity results for other anisotropic curvature functionals and in this talk we discuss their stability: In case the curvature condition is “almost” satisfied, is the surface “close” to the Wulff shape? This is joint work with Xuwen Zhang (University of Freiburg).

Homogeneous fractional Orlicz--Sobolev spaces extend classical fractional Sobolev spaces governed by the Gagliardo--Slobodetskii seminorm, which were introduced in late 1950's and which are defined in terms of a non-integer smoothness parameter. They provide a natural framework for solutions to nonlocal elliptic problems associated with non-polynomial nonlinearities. This is achieved by replacing the power type integrability with an integrability condition expressed in terms of a Young function. Fractional Orlicz--Sobolev spaces were introduced in 2019 by Fernandez-Bonder and Salort, and their functional properties have been intensively investigated ever since, either out of the pure mathematical curiosity, or in connection with one or more of its many applications.
Like for any type of Sobolev spaces, relations to other function spaces constitute a fundamental issue in the theory of fractional Orlicz--Sobolev spaces, as they provide a crucial tool for transferring regularity from the data to solution in related differential equations. Properties of functions in these spaces are governed by the smoothness parameter and the Young function.
We shall focus on various types of regularity of functions belonging to fractional Orlicz--Sobolev spaces in both the so-called subcritical and supercritical regimes. In the former regime we shall mainly concentrate on sharp embeddings into spaces defined in terms of global integrability properties of functions, called rearrangement-invariant spaces, while, in the latter regime, we shall call for finer properties of functions such as criteria for continuity, optimal moduli of continuity, or a control of mean oscillation expressed by a membership into spaces of generalized Campanato type.
The aptness of the notion of fractional Orlicz--Sobolev spaces which we adopt is supported by the fact that, unlikely in the classical case, setting the smoothness parameter as an integer exactly matches their counterparts for integer-order Orlicz--Sobolev spaces. Interestingly, customary techniques that have proved appropriate for classical fractional Sobolev spaces, such as characterizations of Hölder spaces in terms of Campanato spaces, Littlewood--Paley decompositions and Hardy-type inequalities fail to yield optimal conclusions for fractional Orlicz--Sobolev spaces. This discrepancy forces us to adopt novel approaches.
In the talk we shall give a survey of recent results on fractional Orlicz--Sobolev spaces obtained jointly with Angela Alberico, Andrea Cianchi and Lenka Slavíková.

The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this talk, I will present recent work that resolves this conjecture: The exterior mass $m$ of an asymptotically flat support surface $S \subset \mathbb{R}$ with nonnegative mean curvature and outermost free boundary minimal surface $D$ is bounded in terms of \[ m \geq \sqrt{\frac{\vert D \vert}{\pi}} . \] If equality holds, then the unbounded component of $S \setminus \partial D$ is a half-catenoid. To prove this result, we develop the theory of a weak foliation of the region above $S$ by minimal capillary surfaces supported on $S$ that emerges from $D$ and admits a nondecreasing quantity associated with its leaves.

Let us consider the dilation operator, defined in the usual way by \[ D_rf(t) = f(rt), \qquad t>0, \] where $r>0$ is the dilation parameter and $f:[0,\infty)\to\mathbb R$ is a Lebesgue-measurable function. With $p\in[1,\infty]$, a space $(X,\|\cdot\|_X)$, consisting of functions defined on $[0,\infty)$, is called $p$-homogeneous if the function norm $\|\cdot\|_X$ satisfies \[ \|D_rf\|_X = r^{-\frac1p}\|f\|_X \] for every $f\in X$ and $r>0$. We will focus on the homogeneity property within the class of rearrangement-invariant spaces, i.e., function spaces $X$ where the norm $\|f\|_X$ depends only on the measure of the level sets of $f$. More precisely, given a measurable $f:[0,\infty)\to\mathbb R$, its nonincreasing rearrangement is given by \[ f^*(t)=\inf\{s>0;\ \lambda(\{x\in [0,\infty):|f(x)|>s\})\le t\},\quad t>0, \] where $\lambda$ stands for the Lebesgue measure. A Banach function space $(X,\|\cdot\|_X)$ is then called rearrangement-invariant (r.i.) if $\|f\|_X=\|g\|_X$ whenever $f^*=g^*$.
In the talk, various properties of homogeneous r.i. spaces will be discussed. We will see that a typical example of a such space is the Lorentz $L^{p,q}$ space. However, the homogeneity property is not restricted only to this class of spaces, which will be shown by constructing other examples, for instance by using certain interpolation and extrapolation techniques.

In this talk, we show that localised, weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed $n$-dimensional Kähler Ricci flow always hold. These integral estimates improve and extend the integral curvature estimates shown in an earlier paper by the speaker. If $M^4$ is closed and four dimensional, and the spatial $L^p$ norm of the scalar curvature is uniformly bounded for some $p>2$, for $t\in [0,T),$ $T< \infty$, then we show:

A recent topic in the area of numerics of PDEs is their approximation in higher space dimensions. While different strategies are available by now that are supported by motivating heuristics, it is that most of them lack a rigorous convergence theory. In the talk, I propose a new discretization approach for the Fokker-Planck equation, and prove its convergence. The method combines tools from stochastics and mathematical statistics, as does the related convergence theory. This is joint work with Max Jensen (UC London) and Fabian Merle (U Tübingen).

In this talk, we will focus on a stationary fractional nonlinear equation with exponential non-linearity defined on the whole real line in presence of a singular term. We will show the non-degeneracy of its solutions. This particular equation appears as a limit problem to physical models for the description of galvanic corrosion phenomena for simple electrochemical systems. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a Lipschitz bounded domain. We conclude by proving the simplicity of the corresponding eigenvalue. The argument used to prove our main result can also be applied to prove that the second eigenvalue of Steklov’s problem on the ellipse is simple, as long as the ellipse is not a circle. The talk is based on a work done in collaboration with G. Mancini and A. Pistoia.

Experimental and numerical results can not yet settle whether, between horizontal coaxial cylinders, if the curvature is large, the first transition for convection is an exchange of stability or rather an Hopf bifurcation. We directly show that if the curvature tends to infinity, no periodic linear perturbation exists when the Rayleigh number is equal to the critical one for non-linear stability.

In this talk, we introduce our recent developments, joint with Professor Guofang Wang, on capillary constant mean curvature (CMC) hypersurfaces, including a resolution to a question of Ros and Sternberg-Zumbrun on classification of stable capillary CMC hypersurfaces in a Euclidean ball via Minkowski-type formula, and a new proof of Alexandrov-Wente’s theorem on embedded capillary CMC hypersurfaces in a half-space via Heintze-Karcher-type inequality.

In this talk, a fully-discrete approximation of an unsteady electro-rheological fluid flow model employing an implicit Euler step in time and a discretely inf-sup-stable finite element approximation in space is examined for its well-posedness, stability, and (weak) convergence under minimal regularity assumptions on the data. Furthermore, numerical experiments are presented to complement the theoretical findings.

In this talk, we provide characterizations on the Gromov-Hausdorff stability to a flat torus from a closed Riemannian manifold with Ricci and integral scalar curvature bounds, in terms of harmonic maps and harmonic forms. Applications include a new topological stability result to a flat torus. This is a joint work with C. Ketterer, I. Mondello, R. Perales and C. Rigoni.

Since 1992 I have been thinking about Fluid flow. My first publication was about the Boussinesq approximation of power-law materials. My co-authors chose it in order to apply well established tools to a slightly nonstandard problem - the flow of magma. One of them was Michael Růžička. A few years later we started to build convection models together with the right frame for their validation with Yoshiyuki Kagei and Michael Růžička. When collecting ideas for my habilitation I decided to let power-law materials convect again and presented model derivation and solvability theory in one volume. After that from 2006 on Arianna Passerini invited Michael Růžička and myself to study flow in annuli. Which seemed simple enough but keeps us interested and busy up to today. I invite the audience to revisit the Boussinesq approximation and the study of convection in different domains.

In this talk we study the relation between two geometric quantities for smooth closed $2d$-surfaces $\Sigma$ - the Willmore bending energy $W(\Sigma)$ and the minimal length of a closed geodesic $\ell(\Sigma)$. It turns out that for surfaces of Willmore energy less than $6\pi$ (with normalized area), $\ell(\Sigma)$ is bounded below in terms of $W(\Sigma)$. The threshold of $6\pi$ is optimal for such a result - we will see that surfaces above this threshold can indeed have geodesic bottlenecks. Our inequality can be proved very easily if one assumes that the shortest closed geodesic has no self-intersections. The discussion of this assumption leads to intriguing insights. This is joint work with Fabian Rupp (Vienna) and Christian Scharrer (Bonn).

We consider the free boundary problem of the incompressible Navier-Stokes system in the scaling critical Besov space. By translation into the Lagrange coordinate, the system can be considered on the fixed region without the convection term, while all the spatial derivatives are changed into the covariant derivatives involving higher order nonlinearity and the system becomes a quasi-linear system. To control such terms, we introduce endpoint maximal regularity of the solution and show the global existence of the free boundary problem for the Naiver-Stokes system near the half Euclidean space. This talk is based on a joint work with Senjo Shimizu (Kyoto University).

Firey introduced the Gauss curvature flow in 1974 to model evolution of tumbling stones. Andrews proved the convergence to a round point in dimension two in 1999. The same result holds for in high dimensions, the flow converges to a soliton (Guan-Ni) and the soliton is sphere (Brendle-Choi-Daskalopoulos). The convergence relies on the estimates of entropy type quantity and related entropy points. This approach can be adapted to deal with variations of Gauss curvature type flows: inhomogeneous and anisotropy Gauss curvature flows, and application to the $L^p$-Minkowski type problems. We will also discuss a type of anisotropic flow arising from the $L^p$ Christoffel-Minkowski problem.