Nonlinear Partial Differential Equations in Freiburg
Wednesday, 2.10.24, 09:00-10:00, Hörsaal II, Albertstr. 23b
Homepage von Analysis-Gruppe:\nhttp://home.mathematik.uni-freiburg.de/analysis/
A Journey with Michael Ruzicka
Wednesday, 2.10.24, 09:45-10:45, Hörsaal II, Albertstr. 23b
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.
Codimension two mean curvature flow of entire graphs
Wednesday, 2.10.24, 11:00-12:00, Hörsaal II, Albertstr. 23b
We consider the graphical mean curvature flow of maps \(f:\bmathbb{R}^m\bto\bmathbb{R}^n\), \(m\bge 2\), 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 \(f:\bmathbb{R}^m\bto\bmathbb{R}^2\), 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.
On the stability of bifurcating periodic patterns of compressible viscous fluid equations
Wednesday, 2.10.24, 11:45-12:45, Hörsaal II, Albertstr. 23b
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.
Conformal fill in by Poincaré-Einstein metrics in dimension 4
Wednesday, 2.10.24, 14:00-15:00, Hörsaal II, Albertstr. 23b
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 \(M=\bpartial X\) 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.\nIn 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.
Geometrical criteria and "regular" solutions to the Navier-Stokes equations
Wednesday, 2.10.24, 14:45-15:45, Hörsaal II, Albertstr. 23b
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.
Stability for anisotropic curvature functionals
Wednesday, 2.10.24, 16:00-17:00, Hörsaal II, Albertstr. 23b
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).
Regularity of functions in fractional Orlicz--Sobolev spaces
Wednesday, 2.10.24, 16:45-17:45, Hörsaal II, Albertstr. 23b
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.\b\b\nLike 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.\b\b\nWe 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.\b\b\nThe 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.\b\b\nIn 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 Penrose inequality in extrinsic geometry
Wednesday, 2.10.24, 17:30-18:30, Hörsaal II, Albertstr. 23b
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 \(S\bsubset \bmathbb{R}\) with nonnegative mean curvature and outermost free boundary minimal surface D is bounded in terms of\n\[\nm\bge \bsqrt{\bfrac{|D|}{\bpi}}.\n\]\nIf equality holds, then the unbounded component of \(S\bsetminus\bpartial D\) 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.
Homogeneous rearrangement-invariant function spaces
Thursday, 3.10.24, 09:00-10:00, Hörsaal II, Albertstr. 23b
Let us consider the dilation operator, defined in the usual way by\n\[\nD_r f(t)=f(rt),\bquad t>0,\]\nwhere \(r>0\) is the dilation parameter and \(f:[0,\binfty)\bto \bmathbb{R}\) is a Lebesgue-measurable function. With \(p\bin [1,\binfty]\), a space \((X,\b|\bcdot\b|_X)\), consisting of functions defined on \([0,\binfty)\), is called \(p\)-homogeneous if the function norm \(\b|\bcdot\b|_X\) satisfies\n\[\n\b|D_r f \b|_X=r^{−\bfrac1p}\b|f\b|_X\]\nfor every \(f\bin X\) 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 \(\b|f\b|_X\) depends only on the measure of the level sets of \(f\). More precisely, given a measurable \(f:[0,\binfty)\bto R\), its nonincreasing rearrangement is given by\n\[\nf^∗(t)=inf{s>0; \blambda({x\bin [0,\binfty):|f(x)|>s})\ble t},\bquad t>0,\]\nwhere \(\blambda\) stands for the Lebesgue measure. A Banach function space \((X,\b|\bcdot\b|_X)\) is then called rearrangement-invariant (r.i.) if \(\b|f\b|_X=\b|g\b|_X\) whenever \(f^∗=g^∗\). \n\nIn 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.
Ricci flow with \(L^p\) bounded scalar curvature
Thursday, 3.10.24, 09:45-10:45, Hörsaal II, Albertstr. 23b
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 \(t\bin [0,T)\), \(T<\binfty\), then we show:\n\na) a uniform bound on the spatial \(L^2\) norm of the Riemannian curvature tensor for \(t\bin [0,T)\),\n\nb) uniform non-expanding and non-inflating estimates for \(t\bin [0,T)\),\n\nc) convergence to an orbifold as \(t\bto T\),\n\nd) existence of an extension of the flow to times \(t\bin [0,T+\bsigma)\) for some \(\bsigma >0\) using the orbifold Ricci flow.\n\nThis is joint work with Jiawei Liu.
A new convergent method to solve the Fokker-Planck equation in higher dimensions
Thursday, 3.10.24, 11:00-12:00, Hörsaal II, Albertstr. 23b
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).
Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation
Thursday, 3.10.24, 11:45-12:45, Hörsaal II, Albertstr. 23b
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.
About the Onset of the Hopf Bifurcation for Convective Flows in Horizontal Annuli
Thursday, 3.10.24, 14:00-15:00, Hörsaal II, Albertstr. 23b
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.
Rigidity on capillary constant mean curvature hypersurfaces
Thursday, 3.10.24, 14:45-15:45, Hörsaal II, Albertstr. 23b
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.
Fully-discrete finite element approximation of an unsteady electro-rheological fluid flow model: convergence and error analysis
Thursday, 3.10.24, 16:00-17:00, Hörsaal II, Albertstr. 23b
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.
Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds
Thursday, 3.10.24, 16:45-17:45, Hörsaal II, Albertstr. 23b
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.
Magma and aircraft hulls: Convection is fascinating
Thursday, 3.10.24, 17:30-18:30, Hörsaal II, Albertstr. 23b
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.
Short closed geodesics and the Willmore energy
Friday, 4.10.24, 09:00-10:00, Hörsaal II, Albertstr. 23b
In this talk we study the relation between two geometric quantities for smooth closed \(2d\)-surfaces \(\bSigma\) -- the Willmore bending energy \(W(\bSigma)\) and the minimal length of a closed geodesic \(\bell(\bSigma)\). It turns out that for surfaces of Willmore energy less than \(6\bpi\) (with normalized area), \(\bell(\bSigma)\) is bounded below in terms of \(W(\bSigma)\). The threshold of \(6\bpi\) 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).
Endpoint maximal regularity and free boundary problem for the Navier-Stokes system in scaling invariant case
Friday, 4.10.24, 09:45-10:45, Hörsaal II, Albertstr. 23b
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).
Gauss curvature flow: it's variations and applications
Friday, 4.10.24, 11:00-12:00, Hörsaal II, Albertstr. 23b
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 Lp-Minkowski type problems. We will also discuss a type of anisotropic flow arising from the \(L^p\) Christoffel-Minkowski problem.
The Frobenius relation in string topology
Monday, 21.10.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
String topology is the study of algebraic operations on the homology of the free loop space of a closed manifold. Two prominent operations are the Chas-Sullivan product and the Goresky-Hingston coproduct. It is an important question what structure these two operations form together. We show that under a transversality condition a Frobenius-type relation for the product and the coproduct holds. As an application this yields the behaviour of the coproduct on product manifolds. This talk is based on joint work with Nathalie Wahl.\n
Mathematics and Arts
Tuesday, 22.10.24, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
Our civilization has been developed based on modern knowledge and tradition, or oriental and occidental, but there is an important third source which has often been neglected: interdisciplinary and intercultural inference and exchange. In this talk, Professor Cai will focus on the interplay between science and humanity and, in particular, the similarities between mathematics, arts, and daily life. Prof. Cai will demonstrate it through classical and modern paintings together with photographs that he has taken all around the world. \n\nDr. Cai Tianxin is a mathematician, poet and essayist, an outstanding professor of Zhejiang University, China. He is the author of over 30 books of literature, academic and popular science works. His work has been translated and published into more than 20 languages, including 7 in English. He is a winner of the 2013 Naji Naaman Poetry Award (Beirut) and the 2019 Kadark Literature Award (Darka), the 2017 National Award of Science and Technology (Beijing) for his book Mathematical Legends and the 2024 National Award for Science Popularization, he is the only mathematician who won the prize awarded every 4 years.\n
Verbindung zwischen schulischer und akademischer Mathematik – Lehramtsaufgaben als Mittel zur Adressierung der doppelten Diskontinuität?
Tuesday, 22.10.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Lehramtsstudierende zeigen häufig Schwierigkeiten, Verbindungen zwischen der akademischen\nMathematik, die sie an der Hochschule lernen, und der Schulmathematik, die sie später unterrichten\nsollen, herzustellen. Viele Hochschulen versuchen diesem bereits von Felix Klein (1908)\nbeschriebenen Problem der doppelten Diskontinuität durch den Einsatz professionsspezifischer\nÜbungsaufgaben zu begegnen. Diese „Lehramtsaufgaben“ adressieren explizit Verbindungen\nzwischen Schul- und Hochschulmathematik und sollen dazu beitragen, dass Studierende\nHochschulmathematik als relevant für die Schulmathematik wahrnehmen. In dem Vortrag wird\nanhand einer Fragebogenstudie erörtert, inwiefern Lehramtsaufgaben einer wahrgenommenen\ndoppelten Diskontinuität entgegenwirken können. Ergänzend werden Ergebnisse einer\nInterviewstudie vorgestellt zu der Frage, was genau Lehramtsaufgaben für Studierende relevant\nmacht, sodass zusammenfassend diskutiert werden kann, welche Ansatzpunkte sich zur\nWeiterentwicklung professionsspezifischer Lerngelegenheiten zur Adressierung der doppelten\nDiskontinuität ableiten lassen.
Semi-algebraic differential forms
Friday, 25.10.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Modifying a construction by Hardt, Lambrechts, Turchin and Volić, we will present (\(\bmathbb{Q}\)-)semi-algebraic differential forms and explain connections to period numbers.
A comparison principle based on couplings of partial integro-differential operators
Friday, 25.10.24, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we present a new perspective on the comparison principle for viscosity solutions of Hamilton-Jacobi (HJ), HJ-Bellman, and HJ-Isaacs equations. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test-function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with Lévy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to both continuous and discrete operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to derive well-posedness results for partial integro-differential operators. In the context of spatially dependent Lévy operators, we show that the comparison principle is implied by a Wasserstein-contractivity property on the Lévy jump measures.\n\nJoint work with Serena Della Corte (TU Delft), Richard Kraaij (TU Delft) and Max Nendel (University of Bielefeld)
Self-adjoint codimension 2 boundary conditions for Dirac operators
Monday, 28.10.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Joint work with Nadine Große.\n\nLet \(N\) be an oriented compact submanifold in an oriented complete Riemannian manifold \(M\). We assume that \(M\bsetminus N\) is spin and carries a unitary line bundle \(L\). We study the associated twisted Dirac operator, a priori defined on smooth section with compact support in the interior of \(M\bsetminus N\). We are interested in self-adjoint extensions of this operator.\n\nIf \(N\) has codimension~\(1\), then this is the well-studied subject of classical\nboundary values for Dirac operators. If \(N\) has codimension at least \(3\), or if \(N\) has codimension \(2\) and if \(L\) has trivial monodromy around \(N\), then we obtain a unique self-adjoint extension which coincides with the classical self-adjoint Dirac operator on \(M\). The submanifold \(N\) is thus ``invisible''.\n\nThe main topic of this talk is thus the case of codimension~\(2\) with non-trivial monodromy. We will classify all selfadjoint extensions.\n\nThis work is motivated by work of Portman, Sok and Solovej, who treated the special case of \(M=S^3\) with a link, a case important in mathematical physics.\nWe thank Boris Botvinnik and Nikolai Saveliev for stimulating discussions about this topic.
On a Complete Riemannian Metric on the Space of Closed Embedded Curves
Tuesday, 29.10.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
On a Complete Riemannian Metric on the Space of Closed Embedded Curves\njoint work with Elias Döhrer and Henrik Schumacher (Chemnitz University\nof Technology / Univ. of Georgia)\n\nIn pursuit of choosing optimal paths in the manifold of closed embedded\nspace curves we introduce a Riemannian metric which is inspired by a\nself-contact avoiding functional, namely the tangent-point potential.\nThe latter blows up if an embedding degenerates which yields infinite\nbarriers between different isotopy classes.\n\nFor finite-dimensional Riemannian manifolds the Hopf—Rinow theorem\nstates that the Heine—Borel property (bounded sets are relatively\ncompact), geodesic completeness (long-time existence of geodesic\nshooting), and metric completeness of the geodesic distance are\nequivalent. Moreover, it states that existence of length-minimizing\ngeodesics follows from each of these statements. Albeit the Hopf—Rinow\ntheorem does not hold true in this generality for infinite-dimensional\nRiemannian manifolds, we can prove all its four assertions for a\nsuitably chosen Riemannian metric on the space of closed embedded\ncurves.\n