Nonlocal Hydrodynamic Models of Interacting Agents
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
Agents in an interacting system typically organise their dynamics based on the behaviour of their neighbours. This is often observed in the animal kingdom - herds of mammals, schools of fish, flocks of birds - but also in a variety of economic, engineering, and social settings. Usually, due to huge number of agents in the system, detailed, microscopic description of such behaviour is impossible to analyse and to simulate numerically, and thus its applicability is rather limited.
Instead, in this project we propose to focus on the macroscopic models described by systems of Partial Differential Equations (PDEs). These equations allow us to capture interactions between multiple agents/species/phases in an elegant reformulation involving a small number of nonlocal hydrodynamic conservation laws. It often leads to completely unexplored classes of systems, but also opens the possibility of using a wide range of tools and techniques from the modern theory of PDEs to provide essential insight into the dynamics of large complex systems of interacting agents.
Solving the fundamental problems from the objectives of this proposal will provide a profound mathematical understanding of PDEs emerging in the modelling of collective behaviour. It will allow us to characterise the qualitative properties of the macroscopic models, and to asses whether they are fit to describe the achieving of consensus or emergence of complex patterns observable in nature. We are especially excited to gain this knowledge for the formally derived macroscopic models, as this could provide the arguments for their validity and applicability.
The long-term goal of the project is to use the multidisciplinary environment of the University College London to establish a new research group developing and analysing new PDE models, along with novel numerical techniques for emerging challenges in Mathematical Biology and Mathematical Physics. The core of this environment will be a team composed of the PI, two PDRAs and the UCL funded PhD student. Results will be widely disseminated in diverse environments, including the mathematical fluid mechanics and kinetic theory communities, but also within other disciplines such as computational science, or civil engineering.
Instead, in this project we propose to focus on the macroscopic models described by systems of Partial Differential Equations (PDEs). These equations allow us to capture interactions between multiple agents/species/phases in an elegant reformulation involving a small number of nonlocal hydrodynamic conservation laws. It often leads to completely unexplored classes of systems, but also opens the possibility of using a wide range of tools and techniques from the modern theory of PDEs to provide essential insight into the dynamics of large complex systems of interacting agents.
Solving the fundamental problems from the objectives of this proposal will provide a profound mathematical understanding of PDEs emerging in the modelling of collective behaviour. It will allow us to characterise the qualitative properties of the macroscopic models, and to asses whether they are fit to describe the achieving of consensus or emergence of complex patterns observable in nature. We are especially excited to gain this knowledge for the formally derived macroscopic models, as this could provide the arguments for their validity and applicability.
The long-term goal of the project is to use the multidisciplinary environment of the University College London to establish a new research group developing and analysing new PDE models, along with novel numerical techniques for emerging challenges in Mathematical Biology and Mathematical Physics. The core of this environment will be a team composed of the PI, two PDRAs and the UCL funded PhD student. Results will be widely disseminated in diverse environments, including the mathematical fluid mechanics and kinetic theory communities, but also within other disciplines such as computational science, or civil engineering.
Planned Impact
This project has a potential to influence especially medicine, biology and social science, where collective behaviour and self organisation of interacting agents is ubiquitous. The important examples include growth of biological tissues, patterns in motions of animals, or road traffic. But surprisingly, very similar mechanisms are observable also in variety of economic, engineering, and social settings - the distribution of goods, spacecraft formation, sensor networks, digital media arts, and the emergence of languages in primitive societies. Mathematical modelling and analysis are indispensable in discovering conditions leading, for instance, to creation of congestions in a system. It is of key importance to understand these factors, as it might help to prevent, for example, diseases such as cancer, formation of traffic jams on road, or variety of financial and social crises.
Investigation of mathematical models of interacting agents triggered out substantial progress in many different fields of mathematics, especially in: measure theory, optimal transportation, probability theory, functional, harmonic, stochastic and numerical analysis, dynamical systems, and ordinary and partial differential equations. Enormous interest in this area can be confirmed by awarding highest honours to mathematicians who contributed most significantly to this progress, but also by exponential growth of special sessions, conferences, or even whole thematic programs devoted to emerging problems.
The focus of this project is on PDE models that can be, at least formally, derived from the microscopic agent-based systems. Such formulation opens the possibility of using a wide range of tools and techniques from the modern theory of PDEs in order to supply us with essential insight into dynamics of large complex systems of interacting agents. However, despite all the excellent progress that has been made already, some of the fundamental problems like existence and stability of solutions remain open even for the simplest systems. The techniques developed in this research will deepen our theoretical understanding of these PDEs models and to establish whether they are fit to describe arriving to consensus or emergence of complex patterns observable in nature. It is an important argument in favour of macroscopic PDEs models which are much more convenient for the purposes of further numerical analysis. Having a reliable model along with the numerical scheme is of key importance to forecast, for example, the most efficient drug, or the safest walkways in densely populated areas. This project aims at making significant advances not only for these two selected applications, but for a whole class of underlying PDE systems.
Investigation of mathematical models of interacting agents triggered out substantial progress in many different fields of mathematics, especially in: measure theory, optimal transportation, probability theory, functional, harmonic, stochastic and numerical analysis, dynamical systems, and ordinary and partial differential equations. Enormous interest in this area can be confirmed by awarding highest honours to mathematicians who contributed most significantly to this progress, but also by exponential growth of special sessions, conferences, or even whole thematic programs devoted to emerging problems.
The focus of this project is on PDE models that can be, at least formally, derived from the microscopic agent-based systems. Such formulation opens the possibility of using a wide range of tools and techniques from the modern theory of PDEs in order to supply us with essential insight into dynamics of large complex systems of interacting agents. However, despite all the excellent progress that has been made already, some of the fundamental problems like existence and stability of solutions remain open even for the simplest systems. The techniques developed in this research will deepen our theoretical understanding of these PDEs models and to establish whether they are fit to describe arriving to consensus or emergence of complex patterns observable in nature. It is an important argument in favour of macroscopic PDEs models which are much more convenient for the purposes of further numerical analysis. Having a reliable model along with the numerical scheme is of key importance to forecast, for example, the most efficient drug, or the safest walkways in densely populated areas. This project aims at making significant advances not only for these two selected applications, but for a whole class of underlying PDE systems.
Organisations
- Imperial College London (Lead Research Organisation)
- Aix-Marseille University (Collaboration)
- Gustave Eiffel University (Collaboration)
- Paris Dauphine University (Collaboration)
- Eindhoven University of Technology (Collaboration)
- Academy of Sciences of the Czech Republic (Collaboration)
- National Center for Scientific Research (Centre National de la Recherche Scientifique CNRS) (Collaboration)
- University of L'Aquila (Collaboration)
- Polish Academy of Sciences (Collaboration)
- University of Oxford (Collaboration)
- University of Texas (Collaboration)
- Anhui University (Collaboration)
- Yonsei University (Collaboration)
- Beijing Normal University (Collaboration)
- University of Warsaw (Collaboration)
- University of Strasbourg (Collaboration)
- Basque Center for Applied Mathematics (Collaboration)
Publications
A. Abbatiello, D. Basaric And N. Chaudhuri
(2023)
On a blow-up criterion for the Navier-Stokes-Fourier system under general equations of state
in arXiv:2310.12230
Breit D
(2022)
Compressible Navier--Stokes System with Transport Noise
in SIAM Journal on Mathematical Analysis
Breit D
(2021)
Compressible Navier--Stokes system with transport noise
Brzezniak Z
(2025)
Sequential stability of weak martingale solutions to stochastic compressible Navier-Stokes equations with viscosity vanishing on vacuum
in Journal of Differential Equations
Chaudhuri N
(2022)
A new construction of weak solutions to compressible Navier-Stokes equations
Chaudhuri N
(2022)
Hard congestion limit of the dissipative Aw-Rascle system
Chaudhuri N
(2024)
On thermally driven fluid flows arising in astrophysics
in Physica D: Nonlinear Phenomena
Chaudhuri N
(2024)
Duality solutions to the hard-congestion model for the dissipative Aw-Rascle system
in Communications in Partial Differential Equations
Related Projects
| Project Reference | Relationship | Related To | Start | End | Award Value |
|---|---|---|---|---|---|
| EP/V000586/1 | 13/09/2021 | 29/09/2023 | £1,003,793 | ||
| EP/V000586/2 | Transfer | EP/V000586/1 | 30/09/2023 | 12/09/2026 | £538,007 |
| Description | The two first objectives of the proposed research project are now completed. Our first objective was to prove the existence of solutions to hydrodynamic models of interacting agents under specific assumptions on the form of the pressure, and we have completed our work in one-space dimension. We have understood how the various local and non-local terms in the system are related with each other and how switching from one velocity formulation to two velocity formulation helps to draw analogies with microscopic counterparts of the systems. These findings were published in two recent papers J.A. Carrillo, G.-Q. Chen, D. Yuan, E. Zatorska: Global solutions of the one-dimensional compressible Euler equations with nonlocal interactions via the inviscid limit. To appear in Arch. Ration. Mech. Anal. and in N. Chaudhuri, Y.-P. Choi, O. Tse, E. Zatorska: Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming. To appear in J. London Math. Soc.. For the multidimensional setting we have discovered a fascinating connection between the dissipative Aw-Rascle system and the extensively studied Euler-alignment model, and we explored this connection in our overview paper N. Chaudhuri, J. Peszek, M. Szlenk, E. Zatorska: Non-local dissipative Aw-Rascle model and its relation with matrix-valued communication in Euler alignment, available on Arrive. Concerning the second objective -to investigate the stability of solutions and approximations- we are especially advanced in working o the key part of this proposal: rigorous justification of constrained two-phase compressible/incompressible Euler system. This system was derived within two framework of solutions (distributional and measure-valued) from the dissipative Aw-Rascle system in one spatial dimension. In multi-dimensional setting, we showed that the primitive equations have solutions represented by the parametrised Young measures, but it the weak setting the system is ill-posed. We have also studied the inviscid limit for general Euler system in one-dimensional setting. Finally, our works on agent-fluid coupling systems are now quite advanced. We are in the process of developing the relative entropy technique that will allow us to study various singular limits of the fluid-fluid systems seen as hydrodynamic modelled of the agent-fluid models. Of particular interest is a pressure less limit (which is analogues to Low Mach number limit) and therefore needs to be performed in conjunction with the vanishing viscosity limit. We expect to work on the direct methods to study such limit passages, and in particular, to derive the constrained fluid systems, towards the end of the project. |
| Exploitation Route | The outcomes of this project together with the associate publications bring solutions to important and very challenging mathematical problems. They are of high importance for the mathematical community working on the hydrodynamic conservation laws and they contribute with valuable scientific knowledge to the theory of mathematical analysis in general. The PhD students educated during this projects, the PDRAs and the collaborators are already working on the extensions of the obtained results making the scientific impact even bigger. |
| Sectors | Education |
| URL | https://warwick.ac.uk/fac/sci/maths/people/staff/zatorska/ |
| Description | EPSRC Early Career Forum |
| Geographic Reach | National |
| Policy Influence Type | Participation in a guidance/advisory committee |
| Description | Conference support |
| Amount | € 10,000 (EUR) |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 05/2025 |
| End | 06/2025 |
| Description | EPSRC Doctoral Award of Esther Bou-Dagher |
| Amount | ÂŁ41,388 (GBP) |
| Organisation | Imperial College London |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 09/2022 |
| End | 09/2023 |
| Description | Nelder Fellowship fro prof Eduard Feireisl |
| Amount | ÂŁ5,000 (GBP) |
| Organisation | Imperial College London |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 03/2022 |
| End | 04/2022 |
| Description | Research in Pairs (Scheme 4) Support of collaborative research with Maja Szlenk at London |
| Amount | ÂŁ1,200 (GBP) |
| Funding ID | 42208 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 02/2023 |
| End | 03/2023 |
| Description | Collaboration between the PhD students Muhammed Ali Mehmood and Emile Deleage |
| Organisation | Aix-Marseille University |
| Country | France |
| Sector | Academic/University |
| PI Contribution | Collaboration on proving the stability of partially congested travelling wave solutions to the extended Aw-Rascle system. Muhammed has brought to the project an expertise on analysing the asymptotic limits of the Aw-Rasce model in 1D. |
| Collaborator Contribution | Emil is a PhD student in the group of Charlotte Perrin. He has contributed the collaboration with knowledge of the similar techniques developed before by his supervisor for Euler models with congestion. " |
| Impact | Seminars given at Imperial College London and in Marseille by both collaborators, preprint: Stability of partially congested travelling wave solutions for the extended Aw-Rascle system available on arXiv:2404.17406. |
| Start Year | 2023 |
| Description | Collaboration off Mean-field models in game theory |
| Organisation | Paris Dauphine University |
| Country | France |
| Sector | Academic/University |
| PI Contribution | It is a research project between Ewelina Zatorska, Mikhail Perepelitsa (Texas) and Esther Bou Dagher (Paris, before postdoc of E. Zatorska at Imperial College London). We are working on the construction of solutions and on estimates that would allow us to prove that the long-time asymptotic of solution is a delta formed at one of the infinities, which corresponds to formation of one strategy/opinion. |
| Collaborator Contribution | Mikhail Perepelista proposed this model along with some simplifications that we investigate at the moment. |
| Impact | Very advanced work on the first joint paper. |
| Start Year | 2023 |
| Description | Collaboration off Mean-field models in game theory |
| Organisation | University of Texas |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | It is a research project between Ewelina Zatorska, Mikhail Perepelitsa (Texas) and Esther Bou Dagher (Paris, before postdoc of E. Zatorska at Imperial College London). We are working on the construction of solutions and on estimates that would allow us to prove that the long-time asymptotic of solution is a delta formed at one of the infinities, which corresponds to formation of one strategy/opinion. |
| Collaborator Contribution | Mikhail Perepelista proposed this model along with some simplifications that we investigate at the moment. |
| Impact | Very advanced work on the first joint paper. |
| Start Year | 2023 |
| Description | Collaboration on the nonlocal Euler Equations as inviscid limit of the nonlocal Navier-Stokes Equations |
| Organisation | University of Oxford |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | This is a project in collaboration with J.A Carrillo, G.Q. Chen and D.Y from Oxford. I have been working on the incorporation of the nonlocal terms in the construction of solutions to the approximate Navir-Stokes equations. |
| Collaborator Contribution | My collaborators were adapting the Chen-Perepelitsa technique to prove the compactness of solutions in our case, they also generelised our initial result for the general equation of state and more general interaction term (including alignment). |
| Impact | J.A. Carrillo, G.-Q. Chen, D. Yuan, E. Zatorska: Global solutions of the one-dimensional compressible Euler equations with nonlocal interactions via the inviscid limit. To appear in Arch. Ration. Mech. Anal. arXiv:2403.08576. |
| Start Year | 2023 |
| Description | Collaboration on two-velocity formulation of nonlocal Navier-Stokes alignment model |
| Organisation | Eindhoven University of Technology |
| Country | Netherlands |
| Sector | Academic/University |
| PI Contribution | I have proposed the problem, and together with Nilasis Chaudhuri we've been working on a-priori estimates and on the construction of the approximate solutions adapting the proof of P. Constantin et al. to the nonlocal case. |
| Collaborator Contribution | Oliver Tse (Eindhoven) was responsible for the proof of convergence of the approximate solutions using measure-theoretical approach. Young-Pil Choi (Seul) was responsible for lon-time asymptotics of solutions. Both of them visited me at Imperial College London and also at the University of Warwick. |
| Impact | N. Chaudhuri, Y.-P. Choi, O. Tse, E. Zatorska: Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming. To appear in J. London Math. Soc. arXiv:2402.07130. |
| Start Year | 2022 |
| Description | Collaboration on two-velocity formulation of nonlocal Navier-Stokes alignment model |
| Organisation | Yonsei University |
| Country | Korea, Republic of |
| Sector | Academic/University |
| PI Contribution | I have proposed the problem, and together with Nilasis Chaudhuri we've been working on a-priori estimates and on the construction of the approximate solutions adapting the proof of P. Constantin et al. to the nonlocal case. |
| Collaborator Contribution | Oliver Tse (Eindhoven) was responsible for the proof of convergence of the approximate solutions using measure-theoretical approach. Young-Pil Choi (Seul) was responsible for lon-time asymptotics of solutions. Both of them visited me at Imperial College London and also at the University of Warwick. |
| Impact | N. Chaudhuri, Y.-P. Choi, O. Tse, E. Zatorska: Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming. To appear in J. London Math. Soc. arXiv:2402.07130. |
| Start Year | 2022 |
| Description | Collaboration with E. Feireisl on heat-conducting flows on external domains |
| Organisation | Academy of Sciences of the Czech Republic |
| Country | Czech Republic |
| Sector | Academic/University |
| PI Contribution | That is the project involving Nilasis Chaudhuri, Ewelina Zatorska, Boguslaw Zegarlinski from Warsaw and Eduard Feireisl from Prague. We came up with the proof of existence of approximate solutions along with a preliminary version of the relative entropy estimate. |
| Collaborator Contribution | Boguslaw Zegarlinski proposed the model to study. Eduard Feireis is an expert in Mathematical Fluid Mechanics, his expertise was essential when trying to find the right assumptions for the equation of state in order to close the estimates. In the end thanks to his suggestions we assumed Third Law of Thermodynamics being valid in the degenerate region (of low densities) and we replaced the standard weak formulation based on the entropy inequality by the ballistic energy inequality. |
| Impact | N. Chaudhuri, E. Feireisl, E. Zatorska, B. Zegarlinski: On thermally driven fluid flows arising in astrophysics. Physica D 470, 134401 (2024). |
| Start Year | 2022 |
| Description | Collaboration with T. Piasecki |
| Organisation | University of Warsaw |
| Country | Poland |
| Sector | Academic/University |
| PI Contribution | The purpose of the project with T. Piasecki (University of Warsaw, Warsaw, Poland) is to investigate the maximal regularity of strong solutions to the generalised Aw-Rascle system. The contribution of Ewelina Zatorska and of Nilasis Chaudhuri was to propose the model, and to propose the formulation in terms of hyperbolic-parabolic system with regularisation in the continuity equation. We have completed the result for certain local systems and are now eager to study the non-local Aw-Rascle system as well. |
| Collaborator Contribution | Tomasz Piasecki is an expert in maximal regularity of solutions to equations of compressible (and possibly complex) flow. His expertise allowed us to use elements of technique of R-sectorials operators developed in common works withY . Shibata. He performed most of the estimates for the linearised system. |
| Impact | First publication submitted to the journal and available on arXiv: N. Chaudhuri, T. Piasecki, E. Zatorska: Regular solutions to the dissipative Aw-Rascle system. arXiv:2411.03024. Habilitation of Tomasz Piasecki at the University of Warsaw. |
| Start Year | 2022 |
| Description | Collaboration with Y. Li and F. Fanelli |
| Organisation | Anhui University |
| Country | China |
| Sector | Academic/University |
| PI Contribution | We initiated a project on the singular limit problems of the degenerate compressible Navier-Stokes system with Y. Li (Anhui University, Anhui, China) and F. Fanelli (BCAM, Bilbao, Spain). This project began during their visits to Imperial College London in April and August 2023, respectively, hosted by Ewelina Zatorska and Nilasis Chaudhuri. The project is still at a preliminary level, where we are able to explore the role of a new relative entropy (termed as kappa(?)-relative entropy) and its application to the system. |
| Collaborator Contribution | Our collaborators are having necessary expertise in the derivation of relative entropy type estimates and in direct method of proving the asymptotic limits (singular limits) for the Navier-Stokes system with degenerate viscosity.. |
| Impact | We are still working on our first article, which is now near completion. |
| Start Year | 2023 |
| Description | Collaboration with Y. Li and F. Fanelli |
| Organisation | Basque Center for Applied Mathematics |
| Country | Spain |
| Sector | Academic/University |
| PI Contribution | We initiated a project on the singular limit problems of the degenerate compressible Navier-Stokes system with Y. Li (Anhui University, Anhui, China) and F. Fanelli (BCAM, Bilbao, Spain). This project began during their visits to Imperial College London in April and August 2023, respectively, hosted by Ewelina Zatorska and Nilasis Chaudhuri. The project is still at a preliminary level, where we are able to explore the role of a new relative entropy (termed as kappa(?)-relative entropy) and its application to the system. |
| Collaborator Contribution | Our collaborators are having necessary expertise in the derivation of relative entropy type estimates and in direct method of proving the asymptotic limits (singular limits) for the Navier-Stokes system with degenerate viscosity.. |
| Impact | We are still working on our first article, which is now near completion. |
| Start Year | 2023 |
| Description | Derivation of a soft congestion model with a variable constraint |
| Organisation | National Center for Scientific Research (Centre National de la Recherche Scientifique CNRS) |
| Country | France |
| Sector | Academic/University |
| PI Contribution | Together with the PhD student the PI proposed the project, and we have also made the first attempt at derivation of the system, which turned out to be wrong. Thanks to our collaboration we realised that the constraint is not present in the viscosity term, but in the additional local forcing term. After this step we have been responsible for main parts of the derivation in the stationary, semi-stationary and fully evolutionary part. |
| Collaborator Contribution | Aline Lefebvre-Lepot is an expert in modelling of flows with lubrication effect, she is also an author of the previous result which derived the model without constraint. She made critical observations for our modelling assumptions that we use in the final version. Charlotte Perrin connect interest in modelling aspects of constrained flows with the analytical rigour, she was responsible for key ideas related to estimates that allow to extend the solutions from local to global in time. |
| Impact | First preprint near completion |
| Start Year | 2024 |
| Description | Extension of the convex integration technique to degenerate hyperbolic equations |
| Organisation | Academy of Sciences of the Czech Republic |
| Country | Czech Republic |
| Sector | Academic/University |
| PI Contribution | The PDRA and PI came with the idea of how to solve the problem. We figured out that the convex integration technique could be applied here although the system includessome degenerate dissipative term. |
| Collaborator Contribution | Prof Eduard Feireisl from Czech Academy of Sciences specialises in analysis of compressible fluid equations of the Navier-Stokes and Euler type. He contributed to the collaboration by explaining the rest of the team the foundations of the so-called "baby convex integration". He is also involved into another project on hydrodynamic model of solar system, and on the transport noise in the hydrodynamic equations. |
| Impact | N. Chaudhuri, E. Feireisl, E. Zatorska: Nonuniqueness of weak solutions to the dissipative Aw-Rascle model. Appl Math Optim 90, 19 (2024). |
| Start Year | 2022 |
| Description | Matrix-valued Euler-alignment system |
| Organisation | University of Warsaw |
| Country | Poland |
| Sector | Academic/University |
| PI Contribution | This collaboration includes myself, and three colleagues from the University of Warsaw: Nilasis Chaudhuri, Jan Peszek and Maja Szlenk. At the time of commencing this collaboration, N. Chaudhuri was still a member of my team at Imperial College. We have identified the connection between multi-dimensional dissipative Aw-Rascle model and the Euler-alignment model with singular matrix-valued communication weight. We came with some ideas of how to prove the existence of solutions in the situation when the interaction kernel in the Aw-Rascle system is nonlocal and given as a convolution with Newtonian potential. |
| Collaborator Contribution | J.Peszek is an expert in the kinetic theory, and in particular with analysis of Cucker-Smale system of particle interactions which is a microscopic analogue of the Euler-alignment model. His role is to adapt the language developed with D. Poyato to be able to analyse, using measure theoretical approach, the dissipative Aw-Rascle model. In the first instance, he will also navigate us how to generalise the existing results for Euler-alignment system to the matrix-valued communication kernel. Maja Szlenk has an expertise in analysis of fluid equations with degenerate viscous dissipation and nonlocal terms. She will be working on generalisation of the celebrated Bresch-Desjardins and Millet-Vasseur inequalities to the nonlocal case. |
| Impact | First prepping: N. Chaudhuri, J. Peszek, M. Szlenk, E. Zatorska: Non-local dissipative Aw-Rascle model and its relation with matrix-valued communication in Euler alignment. arxiv:2409.07593. |
| Start Year | 2023 |
| Description | Measure-valued solutions to the local and non-local Aw-Rascle equations in multi-d |
| Organisation | Polish Academy of Sciences |
| Department | Institute of Mathematics of the Polish Academy of Science |
| Country | Poland |
| Sector | Academic/University |
| PI Contribution | We considered the multi-dimensional generalization of the Aw-Rascle system for vehicular traffic. For an arbitrary large class of initial data and the periodic domain, we proved the existence of global-in-time measure-valued solutions. Moreover, using the relative energy technique, we showed that the measure-valued solutions coincide with the classical solutions as long as the latter exist. These parts were achieved thanks to the PDRA Nilasis Chaudhuri hired in this award, whose expertise on measure-valued solutions for fluid equations was essential to justify the estimates rigorously. The heuristic a-priori estimates and the idea for the notion of solutions were proposed by the PI. Considering a non-local kernel that describes the Newtonian repulsion, we already introduced a concept of generalised(measure-valued) solution. We now intend to study the generalised-strong uniqueness property. |
| Collaborator Contribution | Prof. Piotr Gwiazda from Polish Academia of Sciences is a recognised expert on measure-valued solutions to hyperbolic conservation laws. He was overviewing the parts of the proof dedicated to compatibility of various defect measures in the definition of measure-valued solutions. He also came up with the argument justifying strong convergence of the density. Recently, our group was extended by Dr had. Aneta Wroblewska-Kaminska, who was working with the PI beforehand on non-local fluid models. Dr. Wroblewska-Kaminska works with us on extension of the result by Chaudhuri, Gwiazda and Zatorska to the case when the offset function is a non-local function of the density. |
| Impact | N. Chaudhuri, P. Gwiazda, E. Zatorska: Analysis of the generalised Aw-Rascle model. Comm. PDEs, 48(3) 440-477, (2023). |
| Start Year | 2021 |
| Description | Measure-valued solutions to the local and non-local Aw-Rascle equations in multi-d |
| Organisation | Polish Academy of Sciences |
| Country | Poland |
| Sector | Public |
| PI Contribution | We considered the multi-dimensional generalization of the Aw-Rascle system for vehicular traffic. For an arbitrary large class of initial data and the periodic domain, we proved the existence of global-in-time measure-valued solutions. Moreover, using the relative energy technique, we showed that the measure-valued solutions coincide with the classical solutions as long as the latter exist. These parts were achieved thanks to the PDRA Nilasis Chaudhuri hired in this award, whose expertise on measure-valued solutions for fluid equations was essential to justify the estimates rigorously. The heuristic a-priori estimates and the idea for the notion of solutions were proposed by the PI. Considering a non-local kernel that describes the Newtonian repulsion, we already introduced a concept of generalised(measure-valued) solution. We now intend to study the generalised-strong uniqueness property. |
| Collaborator Contribution | Prof. Piotr Gwiazda from Polish Academia of Sciences is a recognised expert on measure-valued solutions to hyperbolic conservation laws. He was overviewing the parts of the proof dedicated to compatibility of various defect measures in the definition of measure-valued solutions. He also came up with the argument justifying strong convergence of the density. Recently, our group was extended by Dr had. Aneta Wroblewska-Kaminska, who was working with the PI beforehand on non-local fluid models. Dr. Wroblewska-Kaminska works with us on extension of the result by Chaudhuri, Gwiazda and Zatorska to the case when the offset function is a non-local function of the density. |
| Impact | N. Chaudhuri, P. Gwiazda, E. Zatorska: Analysis of the generalised Aw-Rascle model. Comm. PDEs, 48(3) 440-477, (2023). |
| Start Year | 2021 |
| Description | New proof of existence of solutions to Compressible Navier-Stokes equations |
| Organisation | University of Warsaw |
| Department | Faculty of Mathematics, Informatics and Mechanics (MIM) |
| Country | Poland |
| Sector | Academic/University |
| PI Contribution | The PDRA and PI were working on few technical lemmas in later parts of the construction argument, as well as on the compensated compactness technique used to justify the main result |
| Collaborator Contribution | Prof Piotr Mucha from the University of Warsaw came up with the idea of proof of strong solutions to the approximate Navier-Stokes solutions compatible with the compactness criterion of Bresch and Jabin. |
| Impact | N. Chaudhuri, P. B. Mucha and E. Zatorska, A new construction of weak solutions to compressible Navier-Stokes equations, Math. Ann. (2024). |
| Start Year | 2021 |
| Description | Rigorous derivation of the lubrication model through hard congestion limit of the dissipative Aw-Rascle model. |
| Organisation | National Center for Scientific Research (Centre National de la Recherche Scientifique CNRS) |
| Department | Centre National de la Recherche Scientifique Marseille |
| Country | France |
| Sector | Academic/University |
| PI Contribution | The PDRA and PI proposed the problem and the idea of approximating the limiting two-phase system by degenerate dissipative Aw-Rascle model with singular offset potential. We also prepared most of the a-priori estimates and construction of local-in-time solution. The PhD student prepared most of the theory for the paper on duality formulation and limit passage. |
| Collaborator Contribution | Dr Charlotte Perrin from CNRS came up with the idea of application of the Oleinik condition to prove the existence of weak solutions. She also proposed the framework of duality solutions which we will consider in the forthcoming publication. Dr Laurent Cavort was responsible for implementation of Finite Volume numerical scheme for the approximate system and for running the numerical tests comparing the limiting system with the constrained compressible Euler Equations. |
| Impact | N. Chaudhuri, L. Navoret, C. Perrin, E. Zatorska: Hard congestion limit of the dissipative Aw-Rascle system. Nonlinearity, 37 045018 (2024) N. Chaudhuri, M. Ali Mehmood, C. Perrin and E. Zatorska, Duality solutions to the hard-congestion model for the dissipative Aw-Rascle system. Comm. PDEs 49(7-8), 671-697 (2024) M.A. Mehmood. "Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function". J. Math. Anal. Appl. 533.1 (2024), 128028 |
| Start Year | 2022 |
| Description | Rigorous derivation of the lubrication model through hard congestion limit of the dissipative Aw-Rascle model. |
| Organisation | University of Strasbourg |
| Country | France |
| Sector | Academic/University |
| PI Contribution | The PDRA and PI proposed the problem and the idea of approximating the limiting two-phase system by degenerate dissipative Aw-Rascle model with singular offset potential. We also prepared most of the a-priori estimates and construction of local-in-time solution. The PhD student prepared most of the theory for the paper on duality formulation and limit passage. |
| Collaborator Contribution | Dr Charlotte Perrin from CNRS came up with the idea of application of the Oleinik condition to prove the existence of weak solutions. She also proposed the framework of duality solutions which we will consider in the forthcoming publication. Dr Laurent Cavort was responsible for implementation of Finite Volume numerical scheme for the approximate system and for running the numerical tests comparing the limiting system with the constrained compressible Euler Equations. |
| Impact | N. Chaudhuri, L. Navoret, C. Perrin, E. Zatorska: Hard congestion limit of the dissipative Aw-Rascle system. Nonlinearity, 37 045018 (2024) N. Chaudhuri, M. Ali Mehmood, C. Perrin and E. Zatorska, Duality solutions to the hard-congestion model for the dissipative Aw-Rascle system. Comm. PDEs 49(7-8), 671-697 (2024) M.A. Mehmood. "Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function". J. Math. Anal. Appl. 533.1 (2024), 128028 |
| Start Year | 2022 |
| Description | Singular limits of two-fluid model in one space dimension |
| Organisation | University of L'Aquila |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | PI and PDRAII (Dr Shrish Parmeshwar) have initiated the discussion of the problem, and prepared some preliminary version of the two-velocity formulation of the problem, along with the relative energy estimate based on this formulation. The PDRAII has also been working on the construction of weak solutions and alternative relative entropies. |
| Collaborator Contribution | Dr Felisia Angela Chiarello (L' Aquila) is an expert in multi scale conservation laws, and she has explained us why the standard entropy formulations for the two-fluid fluids with non-convex implicit pressure will nowt work. Prof. Donatella Donatelli (L'Aquila) has a keen interest in two-fluid compressible-compressible models with various closure assumptions. She helped us to understand which singular limits in these models need to be done simultaneously, and she is also an expert in the direct approach (without the use of relative entropy method) which will be key in further stages of the project. |
| Impact | First paper in preparation, and ideas for further collaboration. |
| Start Year | 2024 |
| Description | Strong solutions to the Aw-Rascle system around Burgers' profiles |
| Organisation | Beijing Normal University |
| Country | China |
| Sector | Academic/University |
| PI Contribution | I have invited collaborators and initiated a discussion. I have also found a PhD student from Beijing Normal University who will be responsible for checking our ideas during her internship with me. |
| Collaborator Contribution | Raphael Danchin (Paris) is an expert in harmonic analysis and he is also an author of some previous results on compressible Euler equations near the Burgers' profiles. Piotr Mucha (Warsaw) is close collaborator of Raphael Danchin, but he is also more interested in hydrodynamic models of collective motion, not only the fluid equations. Jiakun Yang is a PhD student of prof Liutang Sue, and she has a keen interest and preparation in maximal regularity theory for complex fluid systems. |
| Impact | We are at the beginning of the collaboration, the application for the funding from Chinese government to cover Jiakun's internship has been prepared and submitted, but we don't yet know the outcome. |
| Start Year | 2024 |
| Description | Strong solutions to the Aw-Rascle system around Burgers' profiles |
| Organisation | Gustave Eiffel University |
| Country | France |
| Sector | Academic/University |
| PI Contribution | I have invited collaborators and initiated a discussion. I have also found a PhD student from Beijing Normal University who will be responsible for checking our ideas during her internship with me. |
| Collaborator Contribution | Raphael Danchin (Paris) is an expert in harmonic analysis and he is also an author of some previous results on compressible Euler equations near the Burgers' profiles. Piotr Mucha (Warsaw) is close collaborator of Raphael Danchin, but he is also more interested in hydrodynamic models of collective motion, not only the fluid equations. Jiakun Yang is a PhD student of prof Liutang Sue, and she has a keen interest and preparation in maximal regularity theory for complex fluid systems. |
| Impact | We are at the beginning of the collaboration, the application for the funding from Chinese government to cover Jiakun's internship has been prepared and submitted, but we don't yet know the outcome. |
| Start Year | 2024 |
| Description | Strong solutions to the Aw-Rascle system around Burgers' profiles |
| Organisation | University of Warsaw |
| Country | Poland |
| Sector | Academic/University |
| PI Contribution | I have invited collaborators and initiated a discussion. I have also found a PhD student from Beijing Normal University who will be responsible for checking our ideas during her internship with me. |
| Collaborator Contribution | Raphael Danchin (Paris) is an expert in harmonic analysis and he is also an author of some previous results on compressible Euler equations near the Burgers' profiles. Piotr Mucha (Warsaw) is close collaborator of Raphael Danchin, but he is also more interested in hydrodynamic models of collective motion, not only the fluid equations. Jiakun Yang is a PhD student of prof Liutang Sue, and she has a keen interest and preparation in maximal regularity theory for complex fluid systems. |
| Impact | We are at the beginning of the collaboration, the application for the funding from Chinese government to cover Jiakun's internship has been prepared and submitted, but we don't yet know the outcome. |
| Start Year | 2024 |
| Description | 05/09/2023 Workshop Singularities and patterns in evolution equations, Bath, UK. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | This event as part of the activities of the EPSRC network in Generalised and low-regularity solutions for nonlinear PDEs. It was a chance to foster collaboration and exchange of ideas between the researchers in the UK, but also with the lectures from some invited guests from France and Germany. For my group it was also an opportunity to meet together, all my postdocs and PhD students were participating and giving talks or preparing posters with presentations. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://people.bath.ac.uk/km230/singpattern2023.html |
| Description | 07/06/2023 Workshop on Compressible Multiphase Flows, Strasbourg (IRMA) France. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | In this workshop, the modelling of multiphase flows were addressed. The goal was to share modelling methods, difficulties, (rigorous or more phenomenological) analysis, allowing the description of multiphase flows with exchanges (mass transfer, energy exchange) and apparition of shock waves. There were about 30 participants. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://indico.math.cnrs.fr/event/9225/ |
| Description | 08/11/2022 Kinetic and hydrodynamic descriptions in collective behavior, Granada, Spain. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This workshop was dedicated to bring together experienced researchers and early stage scientists on the mathematical theory of collective dynamics (gravitation, plasmas, flocking, swarming, synchronization, etc). Topics focused on recent developments connecting the various scales of description of these phenomena (including micro, meso and macroscopic), and cutting-edge techniques stemming from PDEs, probability, kinetic theory, fluid mechanics and graph theory. The conference was organized on a series of extended talks by experts and some short talks facilitation engagement of young researchers to communicate their results. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://sites.google.com/view/kinetic-hydrodynamic/ |
| Description | 10/06/2022 Imperial-ENSTA-Fresnel workshop, Paris, France |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The objective of this workshop was to discuss recent advances on modelling of wave and diffusion phenomena in complex media through the lenses of asymptotic and numerical methods for partial differential equations (PDEs) and stochastic differential equations (SDEs). The applications of these research areas are versatile and include notably a better understanding of natural phenomena and living organisms, as well as control of wave and diffusion in man-made composite structures known as metamaterials. This was reflected in the choice of speakers who range from applied mathematicians to theoretical physicists and engineers from the Imperial College London, POEMS team (CNRS-ENSTA-INRIA; Palaiseau, France) and Institut Fresnel (CNRS-Aix-Marseille University-Centrale Marseille; France). |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://www.imperial.ac.uk/events/149316/imperial-cnrs-diffusion/ |
| Description | 11/01/2024 Invited speaker at Winter Workshop on Fluid-Structure Interaction Problems, Univer- site Libre de Bruxelles. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Winter Workshop on Fluid-Structure Interaction Problems was held in Brussels (ULB) and was organised by D. Bonheure, C. Grandmont and M. Hillairet. The workshop was be mostly centered around fluid-structure interaction problems but will also address more generally mathematical methods devoted to fluid dynamics issues. It was an occasion for me and for my PhD student Muhammed Ali Mehmood to discuss with our collaborator Charlotte Perrin from Marseille and to start a collaboration with Aline Lefebvre-Lepot from Paris Orsey. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://pde.ulb.be/WFSI/index.html |
| Description | 11/07/2023 Contemporary advances in nonlinear PDE and the Calculus of Variations, University of Reading. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | This network event, part of the EPSRC network in Generalised and low-regularity solutions for nonlinear PDEs, brought together experts working in various areas of the theory of nonlinear PDE and the Calculus of Variations, in order to discuss new progress in the area, modern methods developed in recent years, open problems and future challenges. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://indico.math.cnrs.fr/event/9225/ |
| Description | 14/07/2022 Keynote speaker at Equadiff 15, Brno, Czech Republic. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The tradition of the Czechoslovak Equadiff dates back to 1962 when Equadiff 1 took place in Prague. Subsequent Czechoslovak Equadiff conferences were held in Prague (1977, 1989, 2001, 2013), Bratislava (1966, 1981, 1993, 2005, 2017), and Brno (1972, 1985, 1997, 2009). All proceedings from these conferences are available via Czech Digital Mathematics Library. The Western Equadiff conferences were held in Marseille (1970), Brussels/Louvain-la-Neuve (1973), Firenze (1978), Wurzburg (1982), Xanthi (1987), Barcelona (1991), Lisboa (1995), Berlin (1999), Hasselt (2003), Vienna (2007), Loughborough (2011), Lyon (2015), and Leiden (2019). Equadiff is therefore one of the oldest active series of mathematical conferences in the world. The coming Equadiff in Brno in summer 2022 will be the 15th conference within the Czechoslovak Equadiff series. The conference was rescheduled to the year 2022 from the original date in July 2021 due to an unstable pandemic situation in the world. The conference Equadiff 15 is organized by joint efforts of the following institutions: Masaryk University (Faculty of Science), Brno University of Technology (Faculty of Mechanical Engineering), Czech Academy of Sciences (Institute of Mathematics), The Union of Czech Mathematicians and Physicists (Brno branch). The conference is held under the auspices of the dean of the Faculty of Science of Masaryk University, prof. Tomáš Kašparovský. The PI Was invited as a Keynote speaker and to lead one of the thematic sessions on multi-phase flows. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://conference.math.muni.cz/equadiff15/ |
| Description | 16/09/2023 Energetic Methods for Multi-Component Reactive Mixtures Modelling, Stability, and Asymptotic Analysis, WIAS Berlin. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | EMRM 2023 was devoted to the mathematical analysis of PDE models for multi-component systems and reactive mixtures. The main topics were: diffusion models, hydrodynamic models, hyperbolic-parabolic systems, asymptotic analysis. The workshop placed particular emphasis on methods based on an energy or entropy structure, which have proved crucial for questions concerning existence, stability, and the study of asymptotic limits. EMRM 2023 specifically addressed problems related to: chemical reaction-diffusion processes, biological transport and cross-diffusion phenomena, multi-phase flows in hydrodynamics. A key challenge was the intrinsic interaction between different constituents with further effects such as heat conduction, electrostatic forces, and compressibility. EMRM 2023 provided a platform for experts and early-career researchers to discuss current research results and future directions. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.wias-berlin.de/workshops/EMRM2023/ |
| Description | 17/10/2023 Stability Analysis for Nonlinear PDEs across Multiscale Applications, Penn State, US. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This workshop was part of a series organized through a collaborative research project, a joint NSF-ESRC grant. The goal was to help develop innovative mathematical methods and techniques to solve some outstanding stability problems of nonlinear partial differential equations across the scales, including asymptotic, quantifying, and structural stability problems in hyperbolic conservation laws, kinetic equations, and related multiscale applications in fluid-particle (agent based) models. A main focus was on the following four interrelated objectives: -Stability analysis of shock wave patterns of reflections/diffraction with focus on the shock reflection-diffraction problem in gas dynamics; -Stability analysis of vortex sheets, contact discontinuities, and other characteristic discontinuities; -Stability analysis of particle to continuum limits including the quantifying asymptotic/mean-field/large-time limits for pairwise interactions and particle limits for general interactions among multi-agent or many-particle systems; -Stability analysis of asymptotic limits with emphasis on the vanishing viscosity limit of solutions from multi-dimensional compressible viscous to inviscid flows with large initial data. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://sites.psu.edu/jabinstability/home/ |
| Description | 18/03/2022 SIAM PDEs meeting |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The primary goal of this conference was to bring together scientists and mathematicians working in partial differential equations and related fields. Contemporary challenges raised by recent advances in engineering, industry, and bio-technology will be confronted with state-of-the-art mathematical and computational tools in PDEs. Advanced graduate students and young researchers were also encouraged to participate. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://www.siam.org/conferences/cm/conference/pd22 |
| Description | 18/11/2021 Asymptotic Behaviors of systems of PDEs arising in physics and biology - 4th edition, Lille, France. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The main goals of this workshop were the theoretical study of asymptotic behaviors (in large time or with respect to some parameters) of problems arising in physics and biology and the development of asymptotic preserving numerical methods. The fourth edition of this workshop featured ten plenary speakers (including the PI), several contributed talks and a poster session. |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://indico.math.cnrs.fr/event/6588/ |
| Description | 19/08/2022 Workshop on Stability Analysis for Nonlinear PDEs, Oxford, UK. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The workshop brought together leading experts in the stability analysis of nonlinear partial differential equations across multi-scale applications. Some of the topics addressed include: Stability analysis of shock wave patterns of reflections/diffraction. Stability analysis of vortex sheets, contact discontinuities, and other characteristic discontinuities for multidimensional hyperbolic systems of conservation laws. Stability analysis of particle to continuum limits including the quantifying asymptotic/mean-field/large-time limits for pairwise interactions and particle limits for general interactions among multi-agent systems Stability analysis of asymptotic limits with emphasis on the vanishing viscosity limit of solutions from multidimensional compressible viscous to inviscid flows with large initial data. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://www.maths.ox.ac.uk/node/60407 |
| Description | 20/09/2021 XII Forum of Partial Differential Equations, Bedlewo, Poland. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Forum of Partial Differential Equations is a series of conferences with 22 years tradition. The 12th edition was held in Bedlewo from September 19 till September 25, 2021. It was focused mostly on selected domains of Partial Differential Equations such as calculus of variations, computational mathematics, mathematical fluid mechanics and dynamical systems. Due to uncertain epidemic situation the conference was held in a hybrid form. The aim of the conference was to bring together polish researchers working in PDE and renowned foreign experts invited as plenary speakers. In addition to creating opportunity to exchange ideas and experience between Polish PDE community and other schools, the organisers aimed at enabling youngest Polish scientists working in PDE area to establish foreign contacts and become familiar with latest trends in PDE analysis. |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://www.impan.pl/en/activities/banach-center/conferences/21-xiiforumpde |
| Description | 23/08/2023 Minisymposium: Limit behavior and asymptotic properties in fluid mechanics, ICIAM Tokyo (online). |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The International Council for Industrial and Applied Mathematics (ICIAM) is a worldwide organisation for professional applied mathematics societies, and for other societies with a significant interest in industrial or applied mathematics. The Council works to advance the applications of mathematics in all parts of the world. The ICIAM Congresses, held every 4 years, are run under the auspices of the Council. In 2023 this congress was held in Tokyo at the Waseda University, and was a great opportunity to disseminate the results to a very broad audience, and to interact with experts in almost all the fields of applied mathematics. Unfortunately, due to family reasons the long journey to Tokyo was not possible for me, and in the end I participated in the conference on-line, including giving a talk on one of the mini symposia. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://iciam2023.org |
| Description | 24/01/2024 Invited speaker at Modeling, analysis, and control of multi-agent systems across scales, Pisa: Centro De Giorgi, Italy. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Complex systems of many interacting particles are ubiquitous in nature and account for several interesting phenomena, ranging from gas atoms to social dynamics, passing through chemical reaction networks. A powerful tool for the understanding of such systems and for the taming of their remarkable intrinsic complexity is the introduction of macroscopic descriptions: particles are not seen individually anymore, but are described through their density which, in the limit as the number of particles goes to infinity, solves a single kinetic integro-differential equation. The Boltzmann equation and mean-field limits of particle systems are prominent examples of this point of view, which, originating from statistical physics, has indeed proved to be a strikingly effective instrument with a large scope of possible applications in other branches of science and societal problems. The mathematical challenge relates to this approach are actually manyfold, as they not only involve the rigorous validation of the limit process, but also the effective simulation of large systems through the introduced approximations, as well as the designing of suitable control strategies to steer the system towards some desired state. The desired impact of these theories is not only confined to the theoretical understanding or the computations issues of these complex systems, but aims at suggesting viable and targeted interventions for improving real-life scenarios. The investigation of these systems lies at the interface among modelling, analysis, probability, and numerics. It may also require some advanced theoretical techniques which were developed independently of these applications in contexts such as measure theory, functional analysis, or differential geometry. This workshop brought together leading experts and young researchers in this fields, taking advantage in a synergic fashion of the different backgrounds and domains of expertise. Attention was given both to up-to-date applications to physical, biological, and social systems, and to cutting edge theoretical advances in the rigorous mathematical formulation of the theory, including -- Boltzmann and mean-field descriptions; -- well-posedness of control systems and optimality conditions; -- particle-based optimization, and learning; -- simulations and real-life applications; -- abstract functional settings and tools; -- convergence of numerical schemes. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://cvgmt.sns.it/event/800/ |
| Description | 27/03/2023 CNRS-Imperial Workshop on Waves and Imaging, London |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was a two-day-long workshop with 25 participants mostly from Imperial College London and the CNRs researchers from France. The topics of presentations were related to analysis of fluids equations, and in particular to waves and imaging. The workshop brought together very interesting group of researchers including some analysts, but most of all some applied researchers and possible industrial partners. It was a very good opportunity for discussion and exchange of ideas. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.imperial.ac.uk/events/159776/cnrs-imperial-workshop-on-waves-and-imaging-2023/ |
| Description | 27/06/2022 Summer school on fluids and turbulence, Lyon, France |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This event brought together recognized experts, young researchers and Ph.D. students working in Partial Differential Equations arising in fluid dynamics. The topics included issues like heterogeneity effects, singular behaviours and multi-scale processes and their interplay in real world phenomena related to fluid motion and turbulence. The format will consisted of three introductive lecture series, intended for a wide audience (Master and Ph.D. students), and more specialized talks, presenting some of the most recent developements in the field. The conference took place at the Institut Camille Jordan, in the campus of the University of Lyon 1 Claude Bernard, in Villeurbanne, close to Lyon. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://mathsfluid2022.sciencesconf.org |
| Description | 28/04/2022 Frontiers in kinetic equations for plasmas and collective behaviour, Cambridge, UK. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Plasma physics is one of the most classical applications of kinetic theory. Traditionally, the Vlasov-Maxwell and the Vlasov-Poisson-Landau equations are the central models, but real-world plasmas can often involve other physical effects that are difficult to model. Moreover, even for the traditional models, the development and theory of efficient numerical methods for these equations in realistic physical problems is still challenging. Collective behavior models have been ubiquitous in applications of kinetic approaches in mathematical biology: computational neuroscience, tissue remodelling, growth-fragmentation models, swarming and others. Finally, similar ideas in consensus or alignment models in swarming have recently been used to develop alternative methods to stochastic gradient descent in global optimization, sampling and data science. These three applications have in common basic tools of kinetic theory such as mean-field limits, gradient flows, hydrodynamic derivations, and numerical approaches. The cross pollination of ideas between classical and modern applications of kinetic theory was specifically encouraged during this workshop. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://www.newton.ac.uk/event/fktw03/ |
| Description | 28/07/2023 New Trends in Fluids and Collective Dynamics, Banff, Canada (online). |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The "New Trends in Fluids and Collective Dynamics" workshop in Banff from July 23 to July 28, 2023, was hosted by the Banff International Research Station. The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT). Mathematics of collective behavior and the subject of fluid dynamics undergo a rapid merger where most fascinating breakthroughs have been discovered in recent years. These breakthroughs are based on our novel understanding of collective phenomena from the viewpoint of the laws of fluid motion. Pattern formations in animal swarms or clustering in social networks or even technological applications to decentralized control of unmanned vehicles are just a few examples that can be studied with the new techniques. This workshop brought together a diversified group of researches from both fields to share the knowledge and promote collaboration in this emerging and fascinating area. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.birs.ca/events/2023/5-day-workshops/23w5002 |
| Description | Colloquium of the Mathematics Institute of Polish Academy of Sciences, Warsaw, Poland, 20 March 2024. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The title of the talk: Dissipative Aw-Rascle system: various notions of solutions During my talk I introduced the Aw-Rascle model of one line vehicular traffic and then it's dissipative version in multi-dimensions. I explained connections with other models of mathematical fluid mechanics and kinetic theory and introduce definitions of suitable weak solutions. I then discussed an interesting problem of singular limit leading to hard-congestion model, and presented the proof of this result in one-dimensional setting. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://sites.google.com/view/kolokwium/past-meetings-20232024 |
| Description | Conference and Seminar presentations of PDRA Shrish Parmeshwar |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Talk at Young Researchers in PDEs workshop, 28th September 2023, https://sites.google.com/view/2023-period-pdes-icmat-uam/events/young-researchers-in-pdes Talk at the Analysis and Probability seminar talk at Chalmers is 20th February 2024, https://www.chalmers.se/en/current/calendar/mv-analysis-and-probability-seminar-shrish-parmeshwar/ Contributed talk at Equadiff 2024 in Karlstad (Sweden), 11 June 2024 https://www.kau.se/files/2024-05/Book_of_Abstracts.pdf Invited a talk at the Analysis of PDEs in Mathematical Physics workshop at the University of Bath, 4-5 June 2024 |
| Year(s) Of Engagement Activity | 2023,2024,2025 |
| URL | https://shrishparmeshwar.wordpress.com |
| Description | Conference and Seminar presentations of the PhD Student Muhammed Ali Mehmood |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Sep 2023. Hard-congestion limit of the dissipative Aw-Rascle system. Invited talk at the "Singularities and patterns in evolution equations" conference held at the University of Bath. Oct 2023. Analysis of hard-congestion models. Invited talk at the Applied Analysis Seminar, Aix-Marseille University, FR. Oct 2023. Analysis of hard-congestion models. Invited talk at the Junior Analysis Seminar, Imperial College London Feb 2024. On the hard-congestion model. Invited talk at the Junior Analysis Seminar, University of Warwick. Jun2024. "Duality solutions for the hard-congestion model", talk given at Analysis of PDEs in Mathematical Physics, University of Bath, UK. Sept 2024. Attendance of Trends and perspectives in Analysis of PDEs, Sapienza University of Rome, Italy - 16-20 September 2024. Dec 2024. "Duality solutions and the hard-congestion model", talk given at 14th AIMS conference, NYU Abu Dhabi, UAE - 16-20 December 2024 Jan 2025. Attended: Winter School: Boundary and Singularity in Fluid Mechanics, Stony Brook NY, USA - January 5-12 2025. |
| Year(s) Of Engagement Activity | 2023,2024,2025 |
| URL | https://profiles.imperial.ac.uk/muhammed.mehmood21 |
| Description | Conference talks of PDRA Nilasis Chaudhuri |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | ? Research visit: April 2023, Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic. Talk title: A new construction of weak solutions to compressible Navier-Stokes system. ? Research visit: March 2023, University of Warsaw and IMPAN, Warsaw, Poland. Talk title: Navier-Stokes-Fourier system with Dirichlet boundary condition for the temperature. ? Mathematical Finance and Stochastic Analysis seminar, University of York, June 26, 2023, York, United Kingdom. Talk title: Navier-Stokes-Fourier system with Dirichlet boundary condition for the temperature. ? Necas Seminar on Continuum Mechanics, December 19, 2022, Prague, Czech Republic. ? MathFlows 2022, December 05-09, 2022, CIRM, Luminy, France. ? 'Against the Flow', October 20, 2022, Bedlewo, Poland. ? Mini-symposium: Equadiff 15, July 14, 2022, Brno, Czech Republic. |
| Year(s) Of Engagement Activity | 2022,2023 |
| Description | EMS Topical Activity Group kick-off meeting "Mixtures: Modeling, analysis and computing" Prague, Czech Republic, 5-7 February 2025. |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | In 2023 the European Mathematical Society (EMS) called for formation of topical activity groups (EMS-TAG) and in 2024 we established one---the topical activity group Mixtures. The objective of the topical activity group Mixtures is to provide a platform that will allow its members to, well, mix ideas about mixtures, including -physical background and mathematical modelling, -mathematical analysis of the corresponding governing partial differential equations, -scientific computing and numerical analysis of the corresponding governing partial differential equations. The conference Mixtures: Modeling, analysis and computing gave the attendees the opportunity to present their work and meet their peers in person. Each participant was expected to give a short talk on his/her work--we see the conference as the networking event in the mixtures community. Besides that we also spent some time on internal affairs (TAG chair election). |
| Year(s) Of Engagement Activity | 2025 |
| URL | https://euromathsoc.org/news/conference-%22mixtures:-modeling-analysis-and-computing%22-(prague-febr... |
| Description | Invited speaker at 19th International Conference onHyperbolicProblems, Shanghai Jiao Tong University, China, 1-5 July 2024 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The 19th HYP conference was a part of of a biannual cycle of conferences, most important in the field of analysis and applications of hyperbolic conservation laws. Being one of the invited speakers was a big distinction. The objective of this conference was to bring together researchers and students with interests in theoretical, applied, and computational aspects of hyperbolic PDEs and of other related evolutionary PDEs. The conference kept the traditional balance of the HYP series, blending theory, numerics and applications. Particular attention was devoted to the following topics: -Theory and application of conservation laws, -Navier Stokes, Euler equations, and other PDEs in fluid dynamics, -Nonlinear wave equations,- Kinetic equations,- Multi-physics models, -PDEs of mixed type which are all important for my project. My talk on Dissipative Aw-Rascle model resulted in many fruitful discussion and was a perfect occasion to start new collaborations. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://www.hyp2024.sjtu.edu.cn/article/welcome |
| Description | Invited speaker at 50th Journées Équations aux Dérivées Partielles, Aussois (Alps), 3-7 June 2024 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This traditional meeting plays a central role in structuring the French PDE community: one of its main goals is to facilitate meetings and discussions between researchers in the community by encouraging the mixing of generations and themes. A particular effort is made to engage with the the youngest colleagues. Every year the French community invites 1-2 guests from abroad for a special lecture, so it is quite a big distinction to be amongst the speakers. I have also prepared the overview paper that will be published in the conference proceedings. Title of my presentation was: Analysis of the dissipative Aw-Rascle model. I introduced and discuseds a generalization of the one-dimensional Aw-Rascle model of vehicular traffic, which has recently been proposed as a model for crowd dynamics. Mathematically, this system lies between the compressible Euler and compressible Navier-Stokes equations, featuring density-modulated dissipation. In one spatial dimension, the same system models the flow of rigid spheres of radius 1 surrounded by a viscous lubricant. At the level of classical solutions, the system is equivalent to the pressureless Navier-Stokes equations with the singular viscosity coefficient $\frac{\epsilon}{1-\rho}$. The first part of my talk addressed the questions of existence, uniqueness, and the singular limit of weak and duality solutions as $\epsilon\to 0$. I then explained the differences and new challenges that arise in the analysis of this system in the multi-dimensional case. Here, we are able to prove the existence and weak-strong uniqueness of measure-valued solutions, as well as the ill-posedness of the model in the class of weak solutions. I have received a few interesting questions and I initiated a collaboration during this conference. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://jedp2024.sciencesconf.org |
| Description | Invited speaker at Fluids@PoliMilano, Milano 8-10 January 2025 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The conference was a part of the initiatives within the MUR "Excellence Department Grant 2023-2027" assigned to the Department of Mathematics of Politecnico di Milano. It gathered experts on mathematical analysis, modelling, and numerical analysis of fluid systems. |
| Year(s) Of Engagement Activity | 2025 |
| URL | https://www.mate.polimi.it/events/Fluids-PoliMi/ |
| Description | Invited speaker at Stability and Multi-scale Analysis for PDEs, University of Texas at Austin, 13-15 May 2024 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This workshop was part of a series organized through a collaborative research project, a joint NSF-ESRC grant. The goal was to help develop innovative mathematical methods and techniques to solve some outstanding stability problems of nonlinear partial differential equations across the scales, including asymptotic, quantifying, and structural stability problems in hyperbolic conservation laws, kinetic equations, and related multiscale applications in fluid-particle (agent based) models. A main focus is on the following four interrelated objectives: Stability analysis of shock wave patterns of reflections/diffraction with focus on the shock reflection-diffraction problem in gas dynamics; Stability analysis of vortex sheets, contact discontinuities, and other characteristic discontinuities; Stability analysis of particle to continuum limits including the quantifying asymptotic/mean-field/large-time limits for pairwise interactions and particle limits for general interactions among multi-agent or many-particle systems; Stability analysis of asymptotic limits with emphasis on the vanishing viscosity limit of solutions from multi-dimensional compressible viscous to inviscid flows with large initial data. The title of the talk was: Recent developments for the one-dimensional compressible Euler system with local and non- local interactions and dissipation terms. I summarised our recent findings on the existence of regular and weak solutions for the compressible Euler equations with nonlocal interaction terms including attraction-repulsion and alignment. In particular, I presented the relative entropy structure based on the two- velocity reformulation of the system, the viscous approximation, and the long-time behaviour of solutions. I have received many interesting questions and initiated new collaborations. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://sites.utexas.edu/pdeworkshop2024/ |
| Description | Invited speaker at Trends and perspectives in the analysis of PDEs, Rome, 16-20 September 2024. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The main goal of this conference was to present recent advances in the theoretical study of PDEs, with a special emphasis on models from fluid and quantum mechanics and related systems of equations. On the other hand, ithe aim was to bring together leading experts and young researchers in the analysis of PDEs related to those different topics, with the purpose of creating and enforcing interactions between the various communities. It provided a broad perspective of problems and ideas arising in the mathematical study of models from fluid dynamics and quantum mechanics, as well as of modern tools and techniques which are employed in their study. Topics of the conference included well-posedness and ill-posedness results, regularity issues, singular perturbation problems, boundary methods, asymptotic analysis tools, dispersive techniques. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://sites.google.com/view/trendspdes/home?pli=1 |
| Description | Minisymposia Speaker (MS3, MS4) at Equadiff 2024, Karlstad, Sweeden, 10-14 June 2024. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | I gave talks at two mini symposia: [MS3] Application of the nonlocal Bresh-Desjardins estimate for a hydrodynamic model of interacting agents based on joint work with Nilasis Chaudhuri, Young-Pil Choi, and Oliver Tse. and [MS4] Global Solutions of the one-dimensional Compressible Euler Equations with Non-local Interactions based on the joint work with Jose A. Carrillo, Gui-Qiang G. Chen, and Difan Yuan The Equadiff conferences are a series of international meetings devoted to the field of differential equations in the broadest sense. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://www.kau.se/en/equadiff |
| Description | Minisymposium speaker at AMS-UMI Joint Meeting, Palermo 23-26 July 2024 |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was a 2nd International Joint Meeting co-organized by the Unione Matematica Italiana (UMI) and the American Mathematical Society (AMS). It is a big conference with multiple parallel sessions. I participated in Special Session B23: Dynamics of compressible Euler equations and complex flows. It was focused on analysis of PDEs arising in physics, which are nonlinear hyperbolic or systems that combine hyperbolic with parabolic features. Such equations are ubiquitous in multiple domains of applied sciences, ranging from high-speed flows in fluids, to flows of complex systems, to plasma physics, and astrophysics. The aim was to bring together people that work on compressible Euler equations (and related subjects) with specialists who work on flows of complex and multiscale systems and to stimulate an exchange between these subjects. We disussed theory, numerics, as well as modeling and applications, and their interplay. The session explored new directions, and to stimulated new collaborations. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://umi.dm.unibo.it/jm-umi-ams/ |
| Description | Organisation of an international conference: "MathFlows" CIRM Luminy, France. |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The meeting was dedicated to mathematical aspects of fluid mechanics. The main topics concerned uniqueness, regularity, structure of solutions, and the possibility of the construction of weak-type solutions to the Navier-Stokes and Euler systems. This was supplemented with new trends related to models from collective behavior that led to interesting classes of PDEs, and the study of fluids in domains with rough boundaries. Both topics have been at the origin of a number of mathematical works in the last years. Our idea of the meeting was to invite a large group of excellent young researchers, as well as more experienced colleagues that are experts in the field, and to propose a scientific program with a reasonable number of talks (proposed both to young and experienced researchers), in order to leave some room for discussion. Our hope was to initiate an exchange of ideas between different schools which (hopefully) will lead to fruitful scientific collaborations. The meeting was the 7-th edition of a series of so-called MathFlows conferences that have been organised since 2012, either in Porquerolles (France) (2015 - 2018) or in the Banach Center of Bedlewo (Poland) (2014 - 2017 - 2020). By organizing the meeting in the famous CIRM centre for the first time, we renewed the list of participants and attracted experts from more countries and mathematical schools. |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://conferences.cirm-math.fr/2638.html |
| Description | Plenary speaker at 9th Polish Forum of Mathematics, Katowice, 9-13 September 2024. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | The Forum of Polish Mathematicians is coorganised by Polish Maths Society in cooperation with universities operating in academic centers selected as venues for the conferences, in this case Katowice. This is primarily cyclical meeting of mathematicians working both in Poland and abroad, dealing with various scientific topics; an event that encourages the exchange of ideas and discussions on the current tasks of contemporary mathematics; an opportunity for debates on problems troubling the mathematical community in Poland. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://9forum-ptm.us.edu.pl/informacje-ogolne/ |
| Description | Seminar talks of the PI Ewelina Zatorska |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The PI (Ewelina Zatorska) was invited to give talks at the following national and international seminars and colloquia: 23/01/23 PDEs Seminar, University of Oxford. 24/11/2022 Warwick Analysis Seminar. 03/11/2022 London Analysis Seminar (unable to take up). 25/10/2022 Joint UCLA-Caltech-USC Analysis and PDE seminar (online). 18/10/2022 Geometry, Analysis and Gravitation Seminar at Queen Mary University of London. 07/06/2022 Mathematics for Computer Science and Applications Online Seminar Of Cracow University of Technology and University of Warmia and Mazury in Olsztyn (online). 13/05/2022 Mathematics Colloquium, University of Warwick. 04/05/2022 PDE Seminar, South China University of Technology, Guangzhou, China (online). 09/02/2022 Virtual seminar: PolWoMaths Seminar (online). |
| Year(s) Of Engagement Activity | 2022 |