Titles and abstracts

Boris Andreianov

Title: On generalized second-order models of traffic

Abstract: We start by discussing bottleneck models based upon Lighthill-Whitham-Richards equation, with a focus on models able to reproduce capacity drop and self-(dis)organization behaviors at the bottleneck. Then a non-local second-order model with built-in self-organization dynamics ("orderliness") on the whole road is formulated; existence analysis and finite volume approximation are discussed (joint works with Abraham Sylla). To this end, we exploit and enrich the 1D renormalization theory developed by Evgeni Panov for the Keyfitz-Kranzer system. Next, we address the Aw-Rascle and Zhang model  and its generalizations  (including a local variant of the "orderliness" model) by viewing them as a couple of scalar discontinuous-flux equations.  [If time permits], on this occassion we will revisit general aspects of discontinuous-flux scalar conservation laws, interpreting the jump conditions as Interface Coupling Conditions and relating the notion(s) of solution admissibility to modeling assumptions and to the choice of numerical fluxes at interfaces. This viewpoint is also applicable to networks; but only order-preserving node coupling conditions can be inserted within our framework.

Stefano Bianchini

Title: Properties of mixing BV vector fields

Abstract: We consider the density properties of divergence-free vector fields \(b \in L^1([0,1],\mathrm{BV}([0,1]^2))\) which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow \(X_t\) is an ergodic/weakly mixing/strongly mixing measure preserving map when evaluated at = 1. Our main result is that there exists a \(G_\delta\)-set \(\mathcal U \subset L^1_{t,x}([0,1]^3)\) made of divergence free vector fields such that 

  • the map \(\Phi\) associating b with its RLF \(X_t\) can be extended as a continuous function to the \(G_\delta\)-set ​​​​\(\mathcal{U}\);
  • ergodic vector fields b are a residual \(G_\delta\)-set in \(\mathcal{U}\);
  • weakly mixing vector fields b are a residual \(G_\delta\)-set in \(\mathcal{U}\);
  • strongly mixing vector fields b are a first category set in \(\mathcal{U}\);
  • exponentially (fast) mixing vector fields are a dense subset of \(\mathcal{U}\).

The proof of these results is based on the density of BV vector fields such that \(X_{t=1}\) is a permutation of subsquares, and suitable perturbations of this flow to achieve the desired ergodic/mixing behavior. These approximation results have an interest of their own.

A discussion on the extension of these results to d ≥ 3 is also presented.

Alberto Bressan

Title: Remarks and open problems on hyperbolic conservation laws

Gui-Qiang Chen

Title: Cavitation and Concentration in Entropy Solutions of the Euler/Euler-Poisson Equations and Related Nonlinear PDEs

Abstract: We will discuss the intrinsic phenomena of cavitation/decavitation and concentration/deconcentration in entropy solutions of the compressible Euler/Euler-Poisson equations and related nonlinear PDEs, which are fundamental to understanding the well-posedness and solution behavior of nonlinear PDEs. We will start to discuss the formation process of cavitation and concentration in entropy solutions of the isentropic Euler equations with respect to the initial data and the vanishing pressure limit. Then we will analyze a longstanding fundamental problem in fluid dynamics:  Does the concentration occur generically so that the density develops into a Dirac measure at the origin generically in spherically symmetric entropy solutions of the multi-dimensional compressible Euler/Euler-Poisson equations? We will report our recent results and approaches developed for solving this longstanding open problem for the Euler/Euler-Poisson equations and related nonlinear PDEs and discuss its close connections with entropy methods and the theory of divergence-measure fields. Further related topics, perspectives, and open problems will also be addressed. 

Snorre Christiansen

Title: Finite element systems 

Abstract: The finite element systems framework is designed to supply some missing information in Ciarlet's definition of a finite element. Instead of phrasing things in terms of degrees of freedom, the framework emphasizes continuity conditions, expressed by restriction operators to subcells. Thus one has to identify spaces on subcells. The framework is well suited to studying complexes of finite elements, both de Rham complexes and elasticity complexes, especially their cohomological properties.

Rinaldo Colombo

Title: Space Dependent Conservation Laws and Hamilton–Jacobi Equations

Abstract: We propose a framework where the well posedness of both Conservation
Laws and Hamilton–Jacobi equations can be proved, in the scalar, 1D,
space dependent case. No convexity is required and growth conditions
are relaxed to a coercivity assumption, resulting in this framework
being alternative to the ones classically used for these equations. A
key role is played by the construction of sufficiently many stationary
entropy solutions (not necessarily in BV).

This work is in collaboration with Vincent Perrollaz and Abraham Sylla.

Espen Jakobsen

Title: Existence and uniqueness results for Mean Field Game systems with fractional, mixed and/or nonlinear diffusions

Abstract: Mean Field Games (MFGs) is currently a very active area of research. In these games of large/infinite number of agents, the Nash equilibria can sometimes be described by a coupled system of nonlinear PDEs:

(i) A backward in time Hamilton–Jacobi–Bellman equation for the decision making
of the generic agent, and

(ii) a forward in time Fokker–Planck equation for the distribution of agents.

In the presence of noise, both equations include diffusion terms, and in this talk we will
discuss existence, uniqueness (and regularity) of solutions in two different cases:

(a) The diffusion is nonlocal/fractional or even mixed local/nonlocal: The noise has long-distance interactions/fat tails, typically modelled by a Levy jump process. In this case the MFG system will be nonlocal. Nonlocal diffusions are sometimes called anomalous diffusions, and are common in e.g. Physics and Finance. 

(b) The diffusion is nonlinear: This means that in the underlying controlled SDE, not
only the drift is controlled but also the diffusion. In many applications controlled
diffusions occur naturally, especially in Finance.

Both results are among the first such extensions to the MFG setting, and include even
results for strongly degenerate diffusions (of low order). Ingredients of the proofs include various regularity, compactness and stability results, adjoint methods, heat kernel bounds, semigroup/(very) weak/viscosity solutions/strong formulations, and extensions and improvements of (“Lasry-Lions”) fixed point and uniqueness arguments for MFGs. Compared to much of the literature, we work in the whole space and not on the compact torus, which makes proofs more complicated. We avoid explicit moment assumptions.

Joint work with Olav Ersland, Indranil Chowdhury, Milosz Krupski, and Artur Rutkowski.

Christian Klingenberg

Title: On solutions of the multi-dimensional compressible Euler equations

Abstract: This talk will survey some results for the two- or three-space dimensional compressible Euler equations, results both in theory and numerics. We shall present

  • non-uniqueness results of weak entropy solutions using convex integration
  • solution concepts beyond weak solutions
  • convergence to the incompressible limit of the compressible Euler equations with gravity
  • the relationship between stationary preservation and asymptotic preserving numerical methods
  • introduce a new numerical method that holds promise to achieve this

This is joint work among others with Wasilij Barsukow, Eduard Feireisl, Simon Markfelder and Phil Roe.

Ujjwal Koley

Title: Convex integration solutions for transport equation forced by random noise

Abstract: We will consider a transport equation forced by random noise of three types. Via convex integration modified to probabilistic setting, we prove existence of a divergence-free vector field with spatial regularity in Sobolev space and corresponding solution to a transport equation with spatial regularity in Lebesgue space, and consequently non-uniqueness in law at the level of probabilistically strong solutions globally in time.

Christian Lubich

Title: Splitting for the semiclassical Schrödinger equation

Abstract: Convergence properties of splitting integrators applied to the semiclassically scaled time-dependent Schrödinger equation are presented in this talk. This includes error bounds for the approximation to the wave function in the \(L^2\) norm (which are not uniform with respect to the semiclassical small scaling parameter) and error bounds for the approximation to expectation values of observables (which are uniform in the small parameter). This talk is based on joint work with Caroline Lasser, Computing quantum dynamics in the semiclassical regime, Acta Numerica 2020.

Christian Rohde

Title: Hyperbolic Modelling and Homogenization of Two-Phase Flow using Diffuse Interface Ideas

Abstract: We will consider the multi-scale modeling and numerics for two-phase flow problems in unbounded and confined domains. Two-phase flow with topological changes can hardly be accessed by classical sharp-interface ideas, and in the past decades various diffuse-interface models have been suggested to overcome this problem. In  the lecture, we will  present a class of diffuse-interface models that build on the classical compressible Navier–Stokes–Korteweg model but can be recast in the form of a system of hyperbolic conservation laws.   We will show that this model can provide the basis for reliable computations of convection-dominated processes in porous media-like domains.  Moreover, we will show that the (hyperbolized) diffuse-interface ansatz is compatible with homogenization techniques.  Thus, we  conclude the lecture with a discussion of up-scaling compressible two-phase flow in a  porous medium.

Anders Szepessy

Title: Canonical mean-field molecular dynamics derived from quantum mechanics

Abstract: Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature  the ab intio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be  $\mathcal O(M^{-1})$, provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and $M$ is the ratio of nuclei and electron masses. For higher temperature several eigenvalues and paths corresponding to excited electron states are required to obtain $\mathcal O(M^{-1})$ accuracy.  In the talk I will present mean-field molecular dynamics that approximates canonical quantum correlation observables. The proof of accuracy uses path integrals. 

Eitan Tadmor

Title: Hydrodynamic alignment with pressure

Franziska Weber

Title: TBA

Published Sep. 5, 2022 11:35 AM - Last modified Sep. 12, 2022 11:19 AM