The third Norwegian meeting on PDEs will gather Norwegian experts on partial differential equations, as well as three international keynote speakers. The meeting will be held at the Blindern campus of the University of Oslo from Wednesday 5 June (at 13:00) to Friday 7 June 2024 (at lunch time).
There will be a poster session on Thursday afternoon.
Venue: Niels Henrik Abels Hus, 12th floor (Abels utsikt).
Program
Wednesday
13:00 – Opening
13:15 – Didier Pilod
Title: On the anisotropic fractional Schrödinger equation
Abstract: We study the initial value problem (IVP) associated to the semi-linear anisotropic fractional Schödinger equation. We deduce several properties of the anisotropic fractional elliptic operator modeling the dispersion relation and used them to establish the well-posedness of the problem. Also, we obtain some unique continuation results for the solutions of this IVP.
This talk is based on a joint work with Carlos Kenig, Gustavo Ponce and Luis Vega
13:50 – Andrey Piatninski
Title: Homogenization of Lévy-type operators
Abstract: The talk will focus on homogenization of nonlocal Lévy-type operators of the
form
\(A^\varepsilon u(x)=\int_{\mathbb R^d}\frac{\Lambda\big(\frac x\varepsilon,\frac y\varepsilon\big)\, \big(u(y)-u(x)\big)\,dy}{|x-y|^{d+\alpha}}, \quad 0<\alpha<2,\)
in \(L^2(\mathbb R^d)\) with a small parameter 𝜀 > 0. Both periodic and random stationary environments will be studied. It will be shown that under natural coerciveness and symmetry conditions the family \(A^\varepsilon\) admits homogenization (strong resolvent convergence) as 𝜀 → 0, and that the limit operator is also a Lévy-type operator with constant coefficient. We will also discuss the case of non-symmetric coefficients and the rate of convergence in the operator norm.
This presentation is based on joint works M. Kassmann, T. Suslina, V. Sloushch and E. Zhizhina.
14:25 – Sigmund Selberg
Tittel: A conservative stochastic Dirac-Klein-Gordon system
Abstract: I will talk about joint work with Evgueni Dinvay, where we consider a nonlinear dispersive stochastic system consisting of Dirac and Klein-Gordon equations coupled through the Yukawa interaction, and obeying the least action formalism. By truncation and an adaptiation of Bourgain’s Fourier restriction norm method, we prove local well-posedness of the Cauchy problem. We also establish pathwise almost sure conservation of the charge, and use this to prove global existence for a class of initial data with finite charge.
14:50 – Coffee break
15:20 – Susanne Solem
Title: Asymptotic stability of a stochastic neural field in the form of a PDE
Abstract: At the last NorPDE-meeting I talked about a PDE representing stochastic neural fields with the aim of better understanding the impact of noise on grid cells. This PDE representation allows a study of noise-induced behaviors in a deterministic framework. So far, the PDE has been rigorously derived from a stochastic particle system and (some of) its noise-driven spatial-pattern-forming bifurcations have been characterized. However, due to its nonlinear and non-local nature, combined with a tricky boundary condition, it is not obvious how to determine the stability of the stationary states for different noise strengths. In this talk, I will present some recent results in this direction.
15:55 – Oana Silvia Serena
Title: Singularly perturbed stochastic control systems and their associated Hamilton Jacobi Belmann equations. Asymptotic behavior and optimality conditions
Abstract: We aim at studying stochastic singularly perturbed control systems, their associated averaged systems and Hamilton Jacobi Belmann equations. The first part is presenting linear (primal and dual) formulations for classical control problems. Necessary and sufficient support criteria conditions for optimality are provided. Motivated by these remarks, we provide linearized formulations associated to the value function in the averaged dynamics setting. In the last part of the presentation, these formulations are used to infer criteria allowing to identify the optimal trajectory of averaged stochastic systems.
Note that when dealing with nonlinear controlled perturbed dynamics, it is very difficult to characterize the optimal trajectories using the classical methods. Indeed, these criteria involve Pontryagin's maximum principle which is difficult to study if one does not
fully understand the averaged dynamics.
We propose an alternative to these classical methods. Our approach consists in
embedding the controlled trajectories into a space of probability measures
satisfying a convenient constraint. This condition is given in terms of the
coefficient functions. The results allow to characterize the set of constraints
as the closed and convex hull of occupational measures associated to controls. Finally, we propose support criteria for the optimality of measures in this setting. We
emphasize that it does not require to effectively compute the averaged dynamics.
18:45 – Conference dinner
Vaaghals, located at Dronning Eufemias gate 8.
Thursday
9:30 – Benjamin Gess
Title: Optimal regularity for nonlocal porous media equations
Abstract: We prove optimal regularity for solutions to a class of nonlocal porous media equations in Sobolev spaces with respect to time and space. The proof builds on the kinetic form of solutions, velocity averaging techniques, and a careful micro-local analysis of pseudodifferential operators. This extends to the nonlocal case the previous results [Tadmor, Tao; CPAM, 2007], [Gess, JEMS; 2021], [Gess, Sauer, Tadmor, Anal. PDE; 2020].
10:30 – Espen R. Jakobsen
Title: On Mean Field Games with nonlocal and nonlinear diffusions.
Abstract: Mean Field Games (MFGs) are limits for N-player games as the number of players N tends to infinity. In the limit a Nash equilibrium is characterized by a coupled system of nonlinaer PDEs – the MFG system - a backward Bellman equation for the optimal strategy of a generic player and a forward Fokker-Planck equation for the distribution of players. The mathematical theory goes back to 2006 and work of Lasry-Lions and Cains- Haung-Malhame, and important questions addressed by the literature include well-posedness, approximations/numerical methods, and the convergence problem – rigorously proving the limit as N tends to infinity. The latter problem involves the so-called Master equation, a PDE posed on the set of probability measures, whos characteristic equations are precisely the above mentioned MFG system. In most of the results in the literature, the diffusion is local/Gaussian and linear/uncontrolled.
In this talk I will discuss recent results on MFGs with (i) nonlocal and (ii) nonlinear diffusions. Case (i) corresponds to a MFGs where players are affected by independent non-Gaussian/Levy induvidual noises, leading to nondegenerate PDEs with linear nonlocal diffusion terms. Results on well-posedness, numerical approximations, and the corresponding Master equation will be addressed. In case (ii), the indepdent individual noises are controlled by the players, and the PDEs become fully nonlinear. We will address well-posedness results for the MFG system in this case.
The talk is based on joint work with former phd’s and postdoc’s, O. Ersland (NTNU), I. Chowdhury (IIT Kanpur), M. Krupski (U Wroclaw), and A. Rutkowski (TU Wroclaw).
11:05 – Coffee break
11:20 – Håkon Hoel
Title: Multiindex Monte Carlo for semilinear stochastic partial differential equations
Abstract: This talk presents an exponential-integrator mulitiindex Monte Carlo method (MIMC) for weak approximations of mild solutions of semilinear stochastic partial differential equations (SPDE). We show that multiindex coupled solutions of the SPDE are stable and satisfy multiplicative error estimates, and describe how these properties can be utilized to obtain a tractable MIMC method. Numerical examples demonstrate that MIMC outperforms alternative methods, such as multilevel Monte Carlo, in settings with low-regularity.
11:55 – Peter Pang
Title: Dissipative solutions of the stochastic Camassa--Holm equation
Abstract: The Camassa–Holm equation is a bi-Hamiltonian, nonlocal, nonlinear equation that exhibits non-uniqueness after a particular mode of blow-up known as wave-breaking. In this talk I will outline a general schema for establishing existence of solutions in nonlinear SPDEs in the context of the Camassa–Holm equation with gradient noise. This talk is aimed at a general PDEs audience, but I shall touch on interesting particulars of this equation and of the inclusion of gradient-type noise. This is primarily based on joint work with Helge Holden (NTNU), Kenneth Karlsen (Oslo), and Luca Galimberti (KCL) (2024, JDE).
12:30 – Lunch
14:00 – Ola I. H. Mæhlen
Title: The particle paths of hyperbolic conservation laws
Abstract: Scalar hyperbolic conservation laws are traditionally viewed as transport equations, and the viewpoint leads to the study of generalized characteristics. Here, we instead view these PDEs as continuity equations with an implicitly defined velocity field; this motivates the analysis of the associated particle paths (solutions of the corresponding ODE). We show that the particle paths of a weak solution of the PDE are well-defined (the ODEs well-posed) if, and only if, the weak solution is the entropy solution. This contrast the ODEs corresponding to characteristics of a solution, which can be ill-posed even for the entropy solution.
This is joint work with Ulrik Skre Fjordholm (UiO) and Magnus Christie Ørke (UiO)
14:35 – Sondre T. Galtung
Title: Equivalence of entropy solutions and gradient flows for the 1D pressureless Euler–Poisson system
Abstract: The Euler–Poisson (EP) system may serve as a model for, e.g., self-gravitating matter or plasma. Here we show how two apparently distinct notions of solution for the one-dimensional EP system happen to be the same. The first notion is based on a differential inclusion, coming from a gradient flow, with a minimal selection principle, while the second is based on a conservation law. It turns out that the minimal selection principle coincides with the Oleinik E condition for entropy solutions of the conservation law, and so the two notions yield the same distributional solutions of the EP system. The ideas are illustrated with several examples.
This is joint work with José A. Carrillo (University of Oxford).
15:10 – Coffee break
15:30 – Kjetil O. Lye
Title: Learning partial differential equations with differentiable simulators
Abstract: In this talk we investigate the feasibility of learning partial differential equations (PDE) from data. We first focus on our newly developed framework for learning pseudo-Hamiltonian Neural Networks (PHNN) for PDEs from observations of the solution variable, and compare this approach to a baseline approach using convolutional neural networks for learning the spatial derivatives. We furthermore look at the possibility of learning parts of a PDE, specifically source terms, with limited observation data. For this last part we highlight the importance of using automatic differentiation within simulators to enable this scenario.
17:00 – Poster session
Friday
9:30 – Massimiliano Berti
Title: Long time dynamics of water waves
Abstract: The study of the time evolution of water waves is a classical question, and one of the major problems in fluid dynamics. It is a boundary value problem where the dynamics of the free interface is governed by an infinite dimensional Hamiltonian and reversible system of quasi-linear equations. In the last years several important analytical results have been obtained concerning the long time dynamics of space periodic waves. In this talk I will survey this field focusing in particular on the existence of time quasi-periodic traveling Stokes waves, which can be regarded as the nonlinear superposition of multiple periodic Stokes waves with rationally independent speeds, interacting each other and retaining forever a quasi-periodic structure.
10:30 – Fredrik Hildrum
Title: Existence and properties in water-wave models
Abstract: Surface water waves travelling in one direction are generally described by the Euler equations or more generally of equations of the form
\(u_t + Lu_x + n(u)_x = 0,\)
where u(t, x) denotes the wave profile, L is a linear nonlocal dispersive operator and n(u) describes nonlinear effects. In addition to the study of time-dependent solutions covering well-posedness, wave breaking and long-term behaviour, particular interest have been given to the existence, symmetry and regularity of steady waves, both in the periodic and solitary-wave settings. One now has several constructions, ranging from small-amplitude, smooth steady solutions via optimisation to singular highest waves by means of global bifurcation.
I will survey some recent results emanating from the dispersive group in Trondheim.
11:05 – Coffee break
11:20 – Jan M. Nordbotten
Title: Momentum-balancing discretizations of linearized Cosserat materials and elasticity
Abstract: The Cosserat formulation of linearized elasticity contains an explicit constitutive law for the asymmetry of the stress tensor. This leads to a new field variable, the couple stress. Moreover, the constitutive law for the couple stress naturally deviates from the constitutive law of the linear stress by the length-scale squared, thus the Cosserat formulation can be seen as a two-scale formulation of elasticity. With this perspective, displacement is decomposed in (large scale) strain and (small scale) deviations in rotation, and the stress can be similarly interpreted. The above modeling considerations motivate the common description of materials modeled by the Cosserat equations as "micro-polar".
In this talk, we highlight that the Cosserat equations are attractive from the perspective of developing discretization methods. Of particular relevance is to develop robust methods, both in the classical sense of incompressible materials, but also more importantly in the sense of a vanishing Cosserat length scale. This latter requirement allows for a seamless transition between Cosserat and linear elastic models of the same physical object, which is attractive if a Cosserat formulation is only of relevance for certain regions of the domain.
We construct explicitly two families of mixed-finite element methods, as well as a finite volume method, all developed within the same framework as outlined above.
This is joint work with Wietse Boon, Omar Duran, and Eirik Keilegavlen.
11:55 – Kundan Kumar
Title: Robust splitting methods for coupled flow and elasticity in a fractured porous medium
Abstract: We consider splitting methods for a coupled wave and parabolic PDEs. The setting is a poromechanics model describing the coupling of flow and geomechanics in a fractured porous medium. The ingredients of the model include: linear elasticity equation for porous matrix; flow equation both in the porous matrix (3D) and along the fracture surface (2D) – leading to a mixed dimensional PDE model; transmission conditions across the fracture and matrix interface including the description of normal and frictional contact forces – leading to variational inequality. The resulting model is a variational inequality coupled to a system of parabolic PDEs. We develop iterative methods that decompose the problem in their individual constituents of flow and mechanical problems. Design of convergent and robust iterative schemes require careful considerations as poorly designed schemes may lead to unstable schemes. We propose a fully discrete iterative scheme for solving this model. We show that the split scheme is a contraction. The splitting methods allow flexibility in terms of time stepping for each of these equations - leading to development of multirate methods - and optimal choice of solvers and pre-conditioners.
12:30 – Lunch
Speakers
International keynote speakers
- Massimiliano Berti (SISSA, Italy)
- Benjamin Gess (University of Bielefeld and Max Planck institute Leipzig, Germany)
Confirmed national speakers
- Sondre Galtung (NTNU Trondheim)
- Markus Grasmair (NTNU Trondheim)
- Fredrik Hildrum (NTNU Trondheim)
- Håkon A. Hoel (University of Oslo)
- Espen R. Jakobsen (NTNU Trondheim)
- Kundan Kumar (University of Bergen)
- Kjetil O. Lye (SINTEF Oslo)
- Ola Mæhlen (University of Oslo)
- Jan M. Nordbotten (University of Bergen)
- Peter H. C. Pang (University of Oslo)
- Andrey Piatnitski (UiT – The Arctic University of Norway)
- Didier Pilod (University of Bergen)
- Sigmund Selberg (University of Bergen)
- Oana Silvia Serea (Western Norway University of Applied Sciences)
- Susanne Solem (Norwegian University of Life Sciences)
Funding
The conference is supported by the Trond Mohn Foundation.
Organising committee
Ulrik Fjordholm, Håkon Hoel, Snorre Christiansen, Kenneth Karlsen, and Nils Henrik Risebro.
Scientific committee
Ulrik Fjordholm, Espen Jakobsen, Helge Holden, Kenneth Karlsen and Didier Pilod.