Schedule

Nonlinear parabolic equations involving nonlocal diffusion

We report on the progress done on the theory of evolution equations that combine the strongly nonlinear parabolic character with the presence of different fractional operators representing long-range interaction effects. The results deal with the topics of optimal existence, regularity, selfsimilarity and asymptotics.

Conservation laws with moving constraints arising in traffic modeling

We consider the Cauchy problem for strongly coupled PDE-ODE systems modelling the influence of controlled single vehicles or platoons on the surrounding road traffic. The models consist of a conservation law with discontinuous flux describing the main traffic evolution and ODEs accounting for the trajectories of the slower vehicles, which depend on the downstream traffic conditions. The moving constraint is operated by an inequality on the flux, which accounts for the bottleneck created on the road by the presence of the controlled vehicles. We present some analytical results and a finite volume scheme able to capture exactly the non-classical discontinuities that may arise at the constraint position. Optimal control problems for traffic management are also addressed numerically.

A Lipschitz metric for the Camassa–Holm equation

The Camassa—Holm equation $u_t+u u_x+p_x= 0$, $p - p_{xx}=u^2+ \frac12 u_x^2$ has received considerable attention since it was first studied by Camassa and Holm in 1993. Part of the interest stems from the fact that the solution develops singularities in finite time while keeping the $H^1$-norm finite. At wave breaking uniqueness is lost as the there are infinitely many ways to extend the solution beyond wave breaking. We study the so-called conservative solutions and show how to construct a Lipschitz metric comparing two conservative solutions. This is joint work with J. A. Carrillo (Oxford) and K. Grunert (NTNU).

Functional analysis for mixed-dimensional partial differential equations

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more d+1-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. This work is a collaboration with Wietse Boon and Jon Eivind Vatne.

Traveling waves for nonlinear wave equations

Local, classical traveling waves can serve as building blocks for global traveling waves by gluing them together. We will illustrate both for the Camassa-Holm equation and the nonlinear variational wave equation how this can be done by using ideas from scalar conservation laws.

Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities

We establish boundedness estimates for solutions to generalized porous medium equations of the form

$$

\partial_t u+(-\mathfrak{L})[u^m]=0\quad\quad\text{in $\mathbb{R}^N\times(0,T)$},

$$

where $m\geq1$ and $-\mathfrak{L}$ is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, L\'evy operators. Our quantitative bounds take the form of precise $L^1$--$L^\infty$-smoothing effects, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of $-\mathfrak{L}$.

In both the linear ($m=1$) and nonlinear ($m>1$) setting, we explore equivalences between smoothing effects and Gagliardo-Nirenberg-Sobolev inequalities. This is in turn equivalent to heat kernel estimates in the linear case, from which our needed Green function estimates can be deduced.

The presentation is based on a joint work with Matteo Bonforte (Universidad Autónoma de Madrid).

Conservation laws on networks

We review some theory of conservation laws posed on a graph, and look at some recent results on well-posedness of entropy solutions via the convergence of a finite volume method. This is joint work with Nils Henrik Risebro and Markus Musch (UiO).

Noise-driven bifurcations in a system of PDEs modelling grid cells

Grid cells are neurons involved in the internal navigational system of mammals. Due to their characteristic spatial hexagonal firing patterns, they are believed to play a pivotal role in spatial representation. In this talk I will present a system of nonlinear PDEs aiming to describe the network activity. In this model, the emergence of network patterns can be understood as noise-driven bifurcations. I will show you an analytical result about the apparition of these patterns which my collaborators, José Antonio Carrillo and Pierre Roux, and I have obtained. From a mathematician’s perspective, I will also discuss the possible biological implications of this.

Shallow water and the deep hazards of Haugesund

Predicting ocean waves in the nearshore region and understanding surfzone dynamics involves various systems of partial differential equations. These systems are usually obtained asymptotically from the Euler or Navier-Stokes equations under the assumption that the wavelength is much larger than the water depth. In this lecture, we give an overview over some classical and recent results on long-wave models such as the shallow-water and Boussinesq systems. We focus on the propagation of bores, the description of wave breaking, and the interaction of shock waves. We also report on predictions and observations of wave runup at the Norwegian coast.

Lie Butcher series in geometry and applications

B-series are Taylor series indexed by trees, originating in the seminal work of John Butcher

half a century ago, and with roots going 150 years back to Arthur Cayley. It has been the main tool for analysing structure preservation in numerical integration of differential equations, and has found applications in other areas such as rough path theory and renormalisation. The modern understanding is that B-series is a canonical expansion for mappings respecting Euclidean symmetries, based on pre-Lie algebras as the algebra of canonical connections on Euclidean spaces.

In recent years B-series have been generalised to special algebras of affine connections on a manifold. Goals of this research has so far been to understand all the special algebras arising in the cases of invariant connections, where the torsion and the curvature are parallel. This yields Lie-Butcher (LB)-series for flows on Lie groups and general Klein geometries.

The general goal of defining similar expansions expansions for more general connections, such as e.g. the Levi—Civita connection of Riemannian geometries, seemed for a long time to be outside reach, but is now finding a nice solution in terms of postLie algebras. This opens a lot of possibilities for computational approaches to Riemannian and other affine geometries, such as numerical integration algorithms, rough paths and Chen-type characterisations of flow compositions.

In the talk we will give a survey of classical and recent results, as well as interesting work in progress.

Wellposedness for the Korteweg-de Vries hierarchy

In this joint work with Friedrich Klaus and Baoping Liu we study wellposedness for all equations of the KdV hierarchy in $H^{-1}$. We use the seminal ideas of Killip and Visan, but with crucial modifications. A key point is the essential equivalence of the KdV hierarchy, the Gardner hierarchy and an additional seemingly new hierarchy.

Existence of Davey--Stewartson type solitary waves for the fully dispersive Kadomtsev--Petviashvilii equation

We prove existence of small-amplitude modulated solitary waves for the full-dispersion Kadomtsev--Petviashvilii (FDKP) equation with weak surface tension. The resulting waves are small-order perturbations of scaled, translated and frequency-shifted solutions of a Davey--Stewartson (DS) type equation. The construction is variational and relies upon a series of reductive steps which transform the FDKP functional to a perturbed scaling of the DS functional, for which least-energy ground states are found. We also establish a convergence result showing that scalings of FDKP solitary waves converge to ground states of the DS functional as the scaling parameter tends to zero.

Our method is robust and applies to nonlinear dispersive equations with the properties that (i) their dispersion relation has a global minimum (or maximum) at a non-zero wave number, and (ii) the associated formal weakly nonlinear analysis leads to a DS equation of elliptic-elliptic focussing type. We present full details for the FDKP equation.

This is joint work with Mark D. Groves and Dag Nilsson, Saarland University.

Mean field game system and the master equation associated with local and nonlocal diffusions in the whole space.

We study well-posedness and regularity for the mean field game (MFG) system in the whole Euclidean space. The system consists of a backward Hamilton–Jacobi equation coupled with a forward Fokker–Planck equation. Both equations are driven by a Lévy operator which is allowed to have both local and nonlocal components. In addition to that, we derive the master equation associated to the MFG system and we prove that it has a unique solution, in the spirit of the seminal book by Cardaliaguet­, Delarue, Lasry, and Lions. The talk is based on a joint work with Espen Jakobsen.

A rough path Euler equation

In this talk I will present a rough path Euler equation which arise from random (rough path) perturbation of the Lagrangian trajectories. To find the structure of this equation we will use techniques from geometric hydrodynamics as initiated by Vladimir Arnold. We will briefly touch upon well-posedness results for this equation, i.e. Beale-Kato-Majda type results for blow-up. Finally, I will discuss how to estimate parameters in the equations from observations of the Lagrangian trajectories.

Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths

In this talk we want to discuss a path-by-path uniqueness result in the sense of A. M. Davie for SDE's driven by a fractional Brownian motion with a Hurst parameter $H\in (0,1/2)$, where the drift vector field is allowed to be merely bounded and measurable.

SPDEs as singular SDEs and their stability.

In this talk I will discuss some recent results on stochastic fluid models. I will first talk about a general framework for singular SDEs such that a large class of models from fuid mechanics can be solved in a unified way. Nonlocal transport noise and regularization effect of large noise will be emphasized. Then I will discuss the stability of the solutions. After introducing the notion of stability of exiting times, I will discuss the examples showing that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data for certain stochastic transport type equations.

On analogue of Laplace operator on one dimensional quaternionic space

A Cauchy–Fueter operator on one-dimensional quaternionic space is analogous to the Cauchy-Riemann operator for a function of one complex variable. Continuing the analogy with the analysis of complex variables we consider $k$-Cauchy–Fueter operators that are the higher-order operators on the Cauchy–Fueter complex, which is a generalization of the Dolbeault complex in several complex variables. After the introduction we discuss the possible construction of solutions of $k$-Cauchy–Fueter equations, that are overdetermined, and their compatibility conditions are given by the $k$-Cauchy–Fueter complex.

$k$-Cauchy–Fueter operator is the Euclidean version of spin $k/2$ massless free field operator on the Minkowski space in physics, corresponding to the Dirac–Weyl equation for $k = 1$, Maxwell’s equation for $k = 2$, the Rarita–Schwinger equation for $k = 3$, the linearized Einstein’s equation for $k = 4$, etc.

This is joint work with D.C.Chang, Georgetown University, USA, and W.Wang, Zhejiang University, PR China.

Strong interaction of solitary waves for the fmKdV equation

The fractional modified Korteweg-de Vries equation :

\begin{align*}

\partial_t u + \partial_x \left( \vert D \vert^\alpha u + u^3 \right)=0,

\end{align*}

for $\alpha\in (1,2)$ enjoys the existence of solitary waves : those solutions keep their form along the time and move with a constant velocity in one direction. Since the existence of multi-solitary waves with different velocities has been established by A. Eychenne , we are interested in constructing solutions behaving at large time as a sum of two solitary waves with the same velocity. I will first introduce the equation and the asymptotic behaviour of solitary waves, and state the existence of solutions whose asymptotic behaviour is a sum of two strongly interacting solitary waves with almost the same velocity. This is a joint work with Arnaud Eychenne.