National research project founded by the UEFISCDI (Ministerul Cercetarii si Inovarii,

               Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii)

                                                  PN-IV-P2-2.1-T-TE-2023-1704

                              Project leader: Oana-Valeria STAMATE (LUPASCU)

                               Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics

                                                                                         of the Romanian Academy

Period : August 2025 - July 2027

Raportul Stiintific

https://drive.google.com/file/d/1YMULJ1lXrE1fILAPIs7teQTwDxfkFqBo/view?usp=sharing

Abstract

     The project will develop stochastic and numerical methods for the study of boundary effects in nonlinear partial differential equations with physical relevance. We will use infinite dimensional stochastic analysis tools, in particular, measure-valued branching Markov processes. 

     We will provide probabilistic representations for solutions of Navier-Stokes equations in bounded planar domains  through an appropriate nonlocal branching process and prove the convergence of the corresponding numerical algorithm. In the case of less regular domains, such as Lipschitz or polygonal domains we will provide an intuitive understanding of how the loss of regularity is related to the geometry of the domain through the branching process. 

     Moreover,  we will provide a probabilistic representation for the solutions of nonlinear Neumann boundary problems and corresponding numerical algorithms. Using stochastic differential equations we will derive probabilistic interpretations fragmentation and coagulation processes in duality. These will be applied to the study of avalanche models.

Research team

Project Leader:

Dr. Oana-Valeria STAMATE (LUPASCU) 

Researcher Members: 

Drd. Ana-Maria Boengiu 

Dr. Iulian Cimpean

Dr. Catalin Vrabie

Dr. Arghir Dani Zarnescu 

Activities and results

Papers:

1. M. Deaconu, O. Lupascu-Stamate, Asymptotic behavior of a one-dimensional avalanche model through a particular stochastic process, acceptat pentru publicare in REVUE ROUMAINE DE MATHÉMATIQUES PURES ET APPLIQUÉES , 2025.

2. Xingyu Li and Arghir Zarnescu, ``Patterns in a Smoluchowski Equation", acceptat la Communications in Contemporary Mathematics, 2025

Preprints:

1.  Iulian Cimpean, Ionut Popescu,  Arghir Zarnescu,  "Quantitative boundary Holder estimates for the inhomogeneous, Poisson problem through a probabilistic approach", arXiv:2510.06906

2.  Simone Rusconi,  Christina Schenkb,  Razvan Ceuca, Arghir Zarnescu,  Elena Akhmatskaya, ``Complexity reduction of physical models: an equation-free approach by means of scaling", arXiv:2506.05029

3.  G. Canevari, K. Cerednichenko,  Arghir Zarnescu, ``A not-so-strange term coming from somewhere", arXiv:2511.20071

4.  O. Lupascu-Stamate, V. Staniculescu, Numerical methods for linear parabolic  SPDEs   with boundary conditions by probabilistic approaches, preprint 2025.

5. Lucian Beznea,O. Lupascu-Stamate, R. Stoica, Levy driven branching Cox processes, preprint 2025.

Talks in international conferences:

1.  Arghir Zarnescu, Electric field effects in nematic liquid crystals, COMPLAS 2025 Conference, Barcelona, Spania, Septembrie 2025

2.  Arghir Zarnescu, A Variational Approach to Fluids, WADE25 Conference, Varsovia, Polonia, Septembrie 2025

3. Oana Lupascu-Stamate, Mathematical modeling of the dense avalanche onset , The 32st Conference on Applied and Industrial Mathematics, 18-21 septembrie 2025, Bucuresti

4. Oana Lupascu-Stamate, Mathematical modeling of the dense avalanche onset ,The 41st “Caius IACOB” Conference on Fluid Mechanics and its Technical Applications, 30 - 31 Octombrie, 2025, Bucuresti

5. Oana Lupascu-Stamate, Mathematical modeling of the dense avalanche onset on a surface with topography, Seminar ISMMA, 14 October, 2025

6.  Iulian Cimpean,  From Fokker-Planck equations to conservative right (strong Markov) diffusion processes}, The 12th International Conference on Stochastic Analysis and its Applications, Bucharest, Romania, 8-12 septembrie, 2025. 

Talks in outreach conferences:

1. Oana Lupascu-Stamate, Curgerea unei avalanse de zapada ,Matematica si Scoala, III, Scoala Gimnaziala Mihai Eminescu, 20 noiembrie 2025, Pitesti (outreach conference)

2. Iulian Cimpean, Polenul, Einstein si clopotul lui Gauss ,Matematica si Scoala, III, Scoala Gimnaziala Mihai Eminescu, 20 noiembrie 2025, Pitesti (outreach conference)

Organizarea de colocvii sau workshopuri:

1.  2nd Bilbao Meeting on Mathematical Fluid Dynamics, co-organizata de Arghir Zarnescu, la Basque Center for Applied Mathematics,

in Bilbao, Spain, 3-5 Novembre 2025 

Participari la stagii de cercetare, cursuri, seminarii sau scoli:

1. Iulian Cimpean, vizita de cercetare la Universitatea Bielefeld, Germania, perioada 30.11-13.12.2025  

2.  Iulian Cimpean, participare la seminarul  stiintific de "Teoria Potentialului" organizat regulat de Institutul de Matematica Simion Stoilow al Academiei Romane si Facultatea de Matematica si Informatica, Universitatea Bucuresti

3.  Iulian Cimpean, participare la seminarul  stiintific de "Probabilitati si teme conexe" organizat regulat de Facultatea de Matematica si Informatica, Universitatea Bucuresti si Institutul de Matematica Simion Stoilow al Academiei Romane