Spring 2018

Time/place: Tuesday 19th of June 2018, 13:15-14:00, room 734
Speaker: Ilya Chevyrev (Oxford)
Title: An isomorphism between branched and geometric rough paths
Abstract: In this talk, I will present an isomorphism between branched and geometric rough paths. The construction relies on an isomorphism between the Grossman–Larson and tensor algebra. This gives a canonical version of the Itô–Stratonovich conversion formula established by Hairer–Kelly. I will also discuss applications of the isomorphism to study signatures of branched rough paths. Joint work with H. Boedihardjo.
Time/place: Friday 15th of June 2018, 13:15-14:00, room 656
Speaker: Nathaël Alibaud (Besançon)
Title: Optimal and dual stability results for \(L^1\) viscosity and \(L^\infty\) entropy solutions
Abstract: We revisit stability results for two central notions of weak solutions for nonlinear PDEs: entropy and viscosity solutions originally introduced for scalar conservation laws and Hamilton-Jacobi equations. Here, we consider two second order model equations, the Hamilton-Jacobi-Bellman (HJB) equation \[ \partial_t \varphi=\sup_\xi \{b(\xi) \cdot D \varphi+\mathrm{tr}(a(\xi) D^2\varphi)\}, \] and the anisotropic degenerate parabolic equation \[ \partial_t u+\mathrm{div} F(u)=\mathrm{div} (A(u) D u). \] The viscosity solutions of the first equation and the entropy solutions of the second satisfy contraction principles in \(L^\infty\) and \(L^1\) respectively. Our aim is to get similar results for viscosity solutions in \(L^1\) and entropy solutions in \(L^\infty\). For the first equation, we identify the smallest Banach topology which is stronger than \(L^1\) for which we have stability. We then construct a norm such that a quasicontraction principle holds. For the second equation, we propose a new weighted \(L^1\) contraction principle allowing for pure \(L^\infty\) solutions. Our main contribution is to show that the solutions of the HJB equation can be used as weights and that this choice is optimal. Interestingly, this reveals a new type of duality between entropy and viscosity solutions.
Time/place: Tuesday 05th of June 2018, 13:15-14:00, room 734
Speaker: W. Steven Gray (Old Dominion University)
Title: A Combinatorial Hopf Algebra Underlying Poincaré's Center-Focus Problem
Abstract: Poincaré's classical center-focus problem asks for sufficient conditions under which a planar polynomial system of non-autonomous differential equations has a center or focus at the origin. It is well understood that Abel's equation naturally describes the center-focus problem in a convenient coordinate system. The present work builds on Devlin's approach to finding polynomial solutions of the Abel equation using the shuffle product on combinatorial words. In particular, it is shown that the primary algebraic structure underlying the problem is a certain Faà di Bruno type Hopf algebra induced by the composition of iterated integrals. For example, there is a direct relationship between Devlin's canonical polynomials and the graded components of the antipode of this Hopf algebra. The link is made primarily using operations similar to coderivations to describe new recursions for the Hopf algebra structure maps. Devlin also generalized his approach beyond the classical Abel equation, and a central aim of the research presented here is to extend the Hopf algebraic setting accordingly.
(Based on joint work with K. Ebrahimi-Fard.)
Time/place: Tuesday 08th of May 2018, 13:15-14:00, room 734
Speaker: José G. Llorente (Universitat Autònoma de Barcelona)
Title: On the asymptotic mean value property for p-harmonic functions in the plane
Abstract: The classical mean value property characterizes harmonic functions and provides the fundamental link between Linear Potential Theory and Probability.
A classical result tracing back to Privalov, Blaschke and Zaremba says that a continuous function \(u\) in a domain \(\mathbb{R}^n\) is harmonic if \(u\) satisfi es the so called asymptotic mean value property, that is: \[ u(x) = -\hspace-10.5pt \int_{B(x,r)}\hspace{-0.2cm}u + o(r^2)\] at each \(x\in \Omega\).
In the last ten years an increasing interest has developed to figure out which stochastic processes are related to other nonlinear differential operators, the key being the identification of the corresponding mean value properties. It turns out that if \(1 < p < \infty\) then solutions of the \(p\)-laplacian \(div (\nabla u | \nabla u |^{p-2}) = 0\) (so called \(p\)-harmonic functions) are closely related to the following (asymptotic) nonlinear mean value property \[ u(x) = \frac{p-2}{ p+n } \cdot \frac{1}{2}\Big ( \sup_{B(x,r)}u + \inf_{B(x,r)}u \Big ) + \frac{2+n}{p+n}\, -\hspace-11.5pt \int_{B(x,r)} \hspace{-0.2cm}u(y)dy + o(r^2 ) \, \, \, \, (*)\] It is known that continuous functions satisfying (*) are \(p\)-harmonic in any dimension but the converse is open if \(n>2\). More information is available if \(n=2\) due to the special representation of the complex gradient of a \(p\)-harmonic function. In this case, recent results of Lindqvist -Manfredi and Arroyo-Llorente have established that planar \(p\)-harmonic functions do satisfy the asymptotic MVP (*). In the talk we will give an overview of the connection between (*) and \(p\)-harmonic functions and then we will move into the peculiarities of the case \(n=2\).
Time/place: Monday 30th of April 2018, 14:15-15:00, room S21
Speaker: Alberto Bressan (Penn State)
Title: On the optimal shape of tree roots and branches
Abstract: Living organisms come in an immense variety of shapes, such as roots, branches, leaves, and flowers in plants, or bones in animals. In many cases it is expected that, through natural selection, these organisms have evolved into a best possible shape. From a mathematical perspective, it is thus of interest to study functionals whose minimizers correspond to some of the many shapes found in the biological world.
As a step in this direction, we consider two functionals that may be used to describe the optimal configurations of roots and branches in a tree. The first one, which we call the sunlight functional models the total amount of sunlight captured by the leaves of a tree. The second one, which we call the harvest functional, models the amount of nutrients collected by the roots. The above functionals will be combined with a ramified transportation cost for transporting nutrients from the roots to the base of the trunk, or from the base of the trunk to the leaves.
The talk will address the semicontinuity of these functionals, and the existence and properties of optimal solutions, in a space of measures.
Time/place: Tuesday 24th of April 2018, 13:15-14:00, room S21
Speaker: Juan Luis Vázquez (Madrid)
Title: Elliptic and parabolic equations with fractional operators
Abstract: After a brief presentation of the topic, we will discuss some recent results of the author and collaborators on the theory of linear and nonlinear elliptic and parabolic equations, especially when posed on bounded domains. New problems will be mentioned.
Time/place: Friday 23rd of March 2018, 13:15-14:00, room 734
Speaker: Danyu Yang (NTNU)
Title: Integration of rough paths
Abstract: The theory of rough path provides a mathematical tool to model the evolution of controlled systems driven by highly oscillating signals, and the continuity theorem explains the convergence of controlled systems. We present a simple approach to the integration of rough paths, and interpret it as a non-abelian counterpart of the classical Young integration.​
Time/place: Tuesday 13th of March 2018, 13:15-14:00, room 656
Speaker: Christina Lienstromberg (Hannover)
Title: Travelling Waves in Dilatant Non-Newtonian Thin Films with Surfactant
Abstract: We prove the existence of a travelling wave solution for a gravity-driven thin film of a viscous and incompressible dilatant/shear-thickening fluid coated with an insoluble surfactant. The governing system of second order partial differential equations for the film’s height \(h\) and the surfactant’s concentration \(\gamma\) is derived by means of lubrication theory applied to the non-Newtonian Navier–Stokes system.
Time/place: Friday 09th of March 2018, 13:15-14:00, room 734
Speaker: Alexander Schmeding (TU Berlin)
Title: Euler-Arnold theory for SPDEs?!
Abstract: In 1966 V. Arnold demonstrated that Euler's equations for an ideal fluid can be understood as the geodesic equation on the group of volume preserving diffeomorphisms with respect to a suitable Riemannian metric. Subsequently this bridge between PDEs on finite dimensional manifolds and ODEs on infinite-dimensional manifolds has been used to study the so called Euler-Arnold equations (e.g. Ebin/Marsden 1970).
In this talk we will give a short tour to this theory and its key ideas. Our aim is to discuss an extensions of these techniques to certain SPDEs which have recently been considered in Fluid dynamics (Crisan, Flandoli, Holm 2017).
This is joint with K. Modin (Chalmers, Gothenburg) and M. Maurelli (WIAS Berlin).
Time/place: Friday 02nd of March 2018, 13:15-14:00, room 734
Speaker: Yvain Bruned (Imperial College London)
Title: Renormalisation of singular SPDEs
Abstract: In this talk, we will present some recent developments on the resolution of singular SPDEs by the theory of Regularity Structures introduced by Martin Hairer. After having exposed the new renormalisation tools, we will compute some examples of the renormalised equation.
Time/place: Tuesday 27nd of February 2018, 13:15-14:00, room 656
Speaker: Vincent Duchêne (Rennes)
Title: A full dispersion model for the propagation of gravity waves in the shallow water regime
Abstract: We will motivate a model for the propagation of gravity water waves, which can be seen as a modification of the so-called Green-Naghdi system, where nonlocal operators (Fourier multipliers) have been inserted. We will then discuss some basic properties shared by a class of systems including the original Green-Naghdi system and the modified one, namely the well-posedness of the Cauchy problem, and the existence of solitary waves.
Time/place: Thursday 22nd of February 2018, 11:15-12:00, room 734
Speaker: Charles Curry (NTNU)
Title: Rough differential equations on homogeneous spaces
Abstract: Lyons’ theory of rough paths was introduced to handle ordinary differential equations driven by control signals of low regularity (Hölder continuity < 1/2), with particular applications in stochastic calculus where the low regularity of Brownian or fractional Brownian paths presented an apparently insurmountable obstacle to the construction of a pathwise/deterministic theory. In a sense, the idea is to look closer at the Taylor expansion of the assumed solution, and enforce that sufficiently many terms are given a meaning to allow a coherent construction of the series. Few mathematicians are more familiar with Taylor expansions of ODEs than numerical analysts, and our understanding of these objects when constructing and justifying Lie group integrators on homogeneous spaces provides the blueprint for an analogous analytic theory of rough ODEs in this setting.
Over the course of this talk I aim to present an accessible introduction to rough differential equations in (finite-dimensional) vector spaces, explain why the theory is natural to those versed in the theory of numerical ODEs/SDEs, and outline some of our recent results in extending the setting to homogeneous spaces.
Joint work with Kurusch Ebrahimi-Fard (NTNU), Dominique Manchon (Univ. Clermont-Auvergne), Hans Munthe-Kaas (UiB)
Time/place: Friday 16th of February 2018, 13:15-14:00, room 734
Speaker: Félix del Teso (NTNU)
Title: Discretizations of fractional powers of the Laplacian in bounded domains
Abstract: The fractional Laplacian is a differential operator of non-integer order that has been extensively studied in the last few decades and is naturally defined on the whole \(\mathbb{R}^N\). As many other fractional order derivatives and integrals, this operator has been often used to model transport processes which generalize classical Brownian motion. However, many physical problems of interest are defined in bounded domains and the use of the fractional Laplacian as modeling tool in this context poses the challenge of providing a meaningful interpretation of the operator in these settings.

Following the heat semi-group formalism, we consider a family of operators which are boundary conditions dependent and discuss a suitable approach for their numerical discretizations by combining quadratures rules with finite element methods.

This approach will provide a flexible strategy for numerical computations of fractional powers of operators in bounded settings with different homogeneous boundary conditions in multi-dimensional (possibly irregular) domains. We also discuss the corresponding fractional Poisson problem and provide a natural way of defining different types of non-homogeneous boundary conditions.

This is joint work with Nicole Cusimano (BCAM), Luca Gerardo-Giorda (BCAM) and Gianni Pagnini (BCAM).
2018-08-15, Markus Grasmair