Abstract

TBA

Abstract

We present a complete picture of coercive Korn-type inequalities for generalised incompatible fields, and optimally extend and unify several previously known inequalities that are crucial to the existence theory for a variety of models in continuum mechanics.

Abstract

We study the classical inverse problem to determine the shape of a three-dimensional scattering obstacle from measurements of scattered waves or their far-field patterns. Previous research on this subject has mostly assumed the object to be star-shaped and imposed a Sobolev penalty on the radial function or has defined the penalty term in some other ad-hoc manner which is not invariant under coordinate transformations.

For the case of curves in $\mathbb{R}^2$, Julian Eckardt suggests in his PhD thesis to use the bending energy as regularisation functional and proposes Tikhonov regularization and regularized Newton methods on a shape manifold. The case of surfaces in $\mathbb{R}^3$ is considerably more demanding. First, a suitable space (manifold) of shapes is not obvious. The second problem is to find a

stabilizing functional for generalised Tikhonov regularisation which on the

one hand should be bending-sensitive and on the other hand prevent the surface from self-intersections during the reconstruction.

The tangent-point energy is a parametrization-invariant and repulsive surface energy that is constructed as the double integral over a power of the tangent point radius with respect to two points on the surface, i.e. the smallest radius of a sphere being tangent to the first point and intersecting the other. The finiteness of this energy also provides $C^{1,\alpha}$ Hölder regularity of the surfaces. Using this energy as the stabilising functional, we choose general surfaces of Sobolev-Slobodeckij reguality, which are naturally connected to this energy.

The proposed approach works for surfaces of arbitrary (known) topology.

In numerical examples we demonstrate that the flexibility of our approach in handling the reconstruction of rather general shapes.

Authors: Jannik Rönsch, Henrik Schumacher, Max Wardetzky, and Thorsten Hohage

Abstract

TBA

Abstract

The transient and long time behavior of Langevin equations has received increased attention over the last years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic Fokker-Planck operator which is a consequence of the partially deterministic nature of a second order stochastic differential equation. Similar challenges arise within the study of the continuous version of (stochastic) Cucker-Smale type flocking models where the resulting hypocoercive PDE additionally becomes nonlinear. Introducing controls on the finite-dimensional level naturally lead to abstract infinite-dimensional bilinear control problems with an unbounded but admissible control operator. By means of an artificial diffusion approach, solutions to a class of hypoercives PDEs as well as to associated optimal control problems are analyzed under smallness assumptions on the initial data.

Abstract

In this minicourse, we will delve into various aspects of topological obstructions within the framework of Sobolev spaces. To illustrate fundamental principles, we will initially explore a Sobolev adaptation of the Brouwer Fixed Point theorem. This exploration will naturally lead us to considerations regarding the definition of degree for Sobolev maps between manifolds. Subsequently, we will examine Sobolev maps with restricted rank, alongside several examples illustrating topological obstructions encountered in the approximation or extension of Sobolev maps. It is assumed that participants are acquainted with the theory of Sobolev spaces in Euclidean contexts. Any topological concept will be carefully defined throughout the course.

https://en.wikipedia.org/wiki/Hopf_fibration#/media/File:Hopf_Fibration.png

Abstract

In this minicourse, we will delve into various aspects of topological obstructions within the framework of Sobolev spaces. To illustrate fundamental principles, we will initially explore a Sobolev adaptation of the Brouwer Fixed Point theorem. This exploration will naturally lead us to considerations regarding the definition of degree for Sobolev maps between manifolds. Subsequently, we will examine Sobolev maps with restricted rank, alongside several examples illustrating topological obstructions encountered in the approximation or extension of Sobolev maps. It is assumed that participants are acquainted with the theory of Sobolev spaces in Euclidean contexts. Any topological concept will be carefully defined throughout the course.

https://en.wikipedia.org/wiki/Hopf_fibration#/media/File:Hopf_Fibration.png

Abstract

In this minicourse, we will delve into various aspects of topological obstructions within the framework of Sobolev spaces. To illustrate fundamental principles, we will initially explore a Sobolev adaptation of the Brouwer Fixed Point theorem. This exploration will naturally lead us to considerations regarding the definition of degree for Sobolev maps between manifolds. Subsequently, we will examine Sobolev maps with restricted rank, alongside several examples illustrating topological obstructions encountered in the approximation or extension of Sobolev maps. It is assumed that participants are acquainted with the theory of Sobolev spaces in Euclidean contexts. Any topological concept will be carefully defined throughout the course.

https://en.wikipedia.org/wiki/Hopf_fibration#/media/File:Hopf_Fibration.png

Abstract

In 1995 Toponogov authored the following conjecture: “Every smooth strictly convex and complete surface of the type of a plane has an umbilic point, possibly at infinity”. In our talk, we will outline a proof, in collaboration with Brendan Guilfoyle, namely that (i) the Fredholm index of an Riemann Hilbert boundary problem for holomorphic discs associated to a putative counterexample is negative. Thereby, (ii) no solutions may exist for a generic perturbation of the boundary condition (iii) however, the geometrization by a neutral metric gives rise to barriers for the continuity method to prove existence of a holomorphic disc.

Abstract

Introduced to model rarefied gas, the Boltzmann equation is a flexible tool to model particles (or, more generally, entities) evolving via binary interaction. In this seminar, we review the basic elements of this model, the entropy dissipation theorem, and discuss its approximation via particle systems.

Abstract

We consider here a 2d incompressible fluid in a periodic channel, whose density is advected by pure transport, and whose velocity is given by the Stokes equation with gravity source term. Dirichlet boundary conditions are taken for the velocity field on the bottom and top of the channel, and periodic conditions in the horizontal variable. We prove that the affine stratified density profile is stable under small perturbations in Sobolev spaces and show convergence of the density to another limiting stratified density profile for large time with an explicit algebraic decay rate. Moreover, we are able to precisely identify the limiting profile as the decreasing vertical rearrangement of the initial density. Finally, we show that boundary layers are formed for large times in the vicinity of the upper and lower boundaries. These boundary layers, which had not been identified in previous works, are given by a self-similar Ansatz and driven by a linear mechanism. This allows us to precisely characterize the long-time behavior beyond the constant limiting profile and enlighten the optimal decay rate.

This is a joint work with Anne-Laure Dalibard and Julien Guillod (Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris, France), part of my PhD thesis.

Abstract

We study the discretization of a homogenized and dimension reduced model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić in 2014. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proven for a multi-affine finite element discretization of the involved three-dimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the two-dimensional isometry-constrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper.

The doctoral researchers of the Research Training Group (EDDy) are pleased to announce the next Math Chat (everyone is invited).

Math Chats are informal gatherings hosted by PhD students for bachelor’s and master’s students who are interested in finding out more about what research is like, how to find a thesis advisor, which seminars & opportunities are available through EDDy at the RWTH, etc.

Abstract

The shallow water equations are a frequently used approximation for modelling currents in rivers, lakes or oceans. But what exactly are the shallow water equations? In this Basic Notion Seminar we will derive the shallow water equations, analyse their properties and briefly discuss their numerical implementation.

Abstract

In this talk I will consider a variational problem which appears in models of bilayer membranes. After introducing and deriving the model I will establish the existence of volume-constrained minimizers where the energy functional consists of two competing terms: a surface energy term penalizing transitions between sets and a nonlocal energy involving the Wasserstein distance between equal volume sets. In the second part of the talk I will consider the maximization of the minimum Wasserstein distance between two given sets, and show that this maximum is obtained by a micella. These results are drawn from joint works with Almut Burchard, Davide Carazzato, Michael Novack, and Raghavendra Venkatraman.

Abstract

I plan to present a survey talk on regularity of harmonic and n-harmonic maps into compact Riemannian manifolds, putting the problem in a historical perspective and discussing known results, from the papers of R. Schoen and K. Uhlenbeck which appeared 40 years ago, through the work of F. Hélein on harmonic maps on planar domains, to the recent results of A. Schikorra and my Polish collaborators, M. Miśkiewicz and B. Petraszczuk.

I shall also present a new example by Petraszczuk who proved that a specific mildly nonlinear elliptic system in the plane (with a quadratic nonlinearity in the gradient) - considered already by J. Frehse in 1973 has the following property: given an arbitrary compact set K in the disc, there exists a solution which is discontinuous precisely on K, and smooth elsewhere. A few related open questions will be stated at the end.

Abstract

Nonlinear partial differential equations appear naturally in many biological or chemical systems. E.g., activator-inhibitor systems play a role in morphogenesis and may generate different patterns. Noisy random fluctuations are ubiquitous in the real world. The randomness leads to various new phenomena and may have a non-trivial impact on the behaviour of the solution. The presence of the stochastic term (or noise) in the model often leads to qualitatively new types of behaviour, which helps to understand the real processes and is also often more realistic. Due to the interplay of noise and nonlinearity, noise-induced transitions, stochastic resonance, metastability, or noise-induced chaos may appear. Noise in stochastic Turing patterns expands the range of parameters in which Turing patterns appears.

The topic of the talk is a nonlinear partial differential equation disturbed by stochastic noise. Here, we will present recent results about the existence of martingale solutions using a stochastic version of a Tychanoff-Schauder type Theorem. In particular, we will introduce the stochastic Klausmeier system, a system that is not monotone, nor does it satisfy a maximum principle. So, the existence of a solution can only be shown using compactness arguments.

In the talk, we first introduce stochastic (partial) differential equations, and then we will present the Klausmeier system. Secondly, we will introduce the notion of martingale solutions and present our main result. Finally, we will outline the proof of our main result, i.e., the proof of the existence of martingale solutions.

Abstract

The aim of this lecture is to revise two basic results on the structure of a $C^2$ function near a critical point. The first result is the classical Morse Lemma; it asserts that, in a special coordinate system, a function near a nondegenerate critical point coincides with the quadratic form given by its Hessian up to an additive constant. The nondegeneracy assumption here means that the Hessian has a trivial kernel. The second result is the Gromoll-Meyer splitting lemma; it handles the general critical point. Both results hold in infinite-dimensions. We believe these results would be useful for those whose research touches on the topic of gradient descent.

Abstract

The Dirac equation was the first quantum wave equation that combined in a consistent way the principles of quantum mechanics and special relativity. It describes spin-$1/2$ massive particles such as electrons. Classically, the Dirac equation is written in terms of Dirac spinors, which are functions of time and space taking values in $\mathbb{C}^4$. In this talk we will present a reformulation of the Dirac equation in a more compact form using the space algebra $\mathcal{Cl}(3)$, which is a Clifford algebra that augments the vector calculus in $\mathbb{R}^3$ with a multiplicative structure. We consider a nonlinear extension of the Dirac equation introduced by French physicist Daviau. Unlike the well-known cubic Dirac equations, which is also known as Soler model and which has attracted considerable interest in the mathematical and physical community, this new model is not well-studied. Unlike the cubic model, it has the same homogeneity as the linear Dirac equation, but the nonlinearity is only Lipschitz continuous. This nonlinear Dirac equation admits a clean and symmetric split into the left and right-handed spinor components. We will discuss a global existence result for a regularized equation and derive a hydrodynamics formulation analogous to the Madelung transform for the Schrödinger equation.

Abstract

This work is devoted to the homogenization of perfect incompressible fluid flows described by the 2D Euler equations in several heterogeneous settings. Starting from the vorticity formulation, the question is naturally split into the homogenization of the div-curl problem defining the fluid velocity and the homogenization of the transport equation for the vorticity. Yet, due to heterogeneities, the fluid velocity typically has large small-scale oscillations, on top of its large-scale variations. In such a multiscale setting, the homogenization of the corresponding transport equation for the vorticity becomes a highly delicate question and a well-posed limit equation might not exist. As we shall see, this difficulty is related to the possibility of a localization phenomenon: part of the vorticity may in principle become trapped due to heterogeneities. We illustrate the topic by focussing on two model problems: the homogenization of the 2D Euler equations with impermeable inclusions and the homogenization of the 2D lake equations. In both cases, we shall see how vorticity localization can be ruled out in some situations.

Abstract

We study equilibration rates for nonlocal Fokker-Planck equations arising in swarm manufacturing. The PDEs of interest are characterised by a time-dependent nonlocal diffusion coefficient and a nonlocal drift, modeling the relaxation of a large swarms of agents, feeling each other in terms of their distance, towards the steady profile characterized by a uniform spreading over a domain. The result follows by combining entropy methods for quantifying the decay of the solution towards its quasi-stationary distribution, with the properties of the quasi-stationary profile.

Bibliography: [1] F. Auricchio, G. Toscani, M. Zanella. Trends to equilibrium for a nonlocal Fokker-Planck equation. Applied Mathematics Letters, 145: 108746, 2023. [2] F. Auricchio, G. Toscani, M. Zanella. Fokker-Planck modeling of many-agent systems in swarm manufacturing: asymptotic analysis and numerical results. Communications in Mathematical Sciences, 21(6):1655-1677, 2023.