Activity of the EDPs<sup>2</sup> team

Acronym:

EDPs² : "Partial derivative equations : deterministic and probabilistic studies"

Team description:

The EDPs² team is based on the federative topic of non linear partial derivative equations. This team is made up of several professors and researchers, all of them completing one another, specialists in analysis and/or scientific calculation of partial derivative equations, theory of stochastic processes and/or numerical probabilities.

This recent grouping (mixing deterministic and sotchastic aspects : theory and scientific calculation) is both original and ambitious, and matches the real need of a better understanding of complex systems all around us (environment, biology-medicine, industry, ...). It allows for example to tackle with different points of view : free boundary problems (dynamic and stable states, image processing, support evolution), scale and multiple physics problems (complex fluids, image processing), problems about homogenisation effects and defect measures (roughness, breaks), optimization and identification problems (optimal transport, data assimilation, parameter identification).

Publications of EDPs² team

One can find in this file Publications the recent list of publications (2014-2019).

Recent results:

Non-linear analysis of PDE

Theoretical results

D. Bresch et P.-E. Jabin. Global existence of weak solutions for compressible Navier-Stokes equations: Thermodynamical unstable pressure and anisotropic viscous stress tensor. Annals of Maths, (188) :577–684, 2018.

C. Bourdarias, M. Gisclon, S. Junca. Fractional bv spaces and first applications to scalar conservation laws. Journal of Hyperbolic Differential Equations, 11(4):655–677, 2014.

D. Bresch, M. Hillairet. A compressible multi-fluid system with a new physical relaxation term. Annales Scientifiques ENS, (52), 255-295 (2019).

D. Bresch, M. Gisclon, I. Lacroix-Violet. On Navier-Stokes-Korteweg and Euler-Korteweg Systems: Application to quantum fluids models. Arch. Rational Mech. Anal, (2019), 233, Issue 3, 975--1025.

Modelling and numerical simulation

Dispersive shallow water wave modelling

Khakimzyanov, G. S., Dutykh, D., Fedotova, Z. I. & Mitsotakis, D. E. Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space. Commun. Comput. Phys. 23, 1–29 (2018).
Khakimzyanov, G. S., Dutykh, D., Gusev, O. & Shokina, N. Y. Dispersive shallow water wave modelling. Part II: Numerical modelling on a globally flat space. Commun. Comput. Phys. 23, 30–92 (2018).
Khakimzyanov, G. S., Dutykh, D. & Fedotova, Z. I. Dispersive shallow water wave modelling. Part III: Model derivation on a globally spherical geometry. Commun. Comput. Phys. 23, 315–360 (2018).
Khakimzyanov, G. S., Dutykh, D. & Gusev, O. Dispersive shallow water wave modelling. Part IV: Numerical simulation on a globally spherical geometry. Commun. Comput. Phys. 23, 361–407 (2018).

Maximal value of the surface elevation during the event.

Mixed flows in closed pipes with air pocket

C. Demay, C. Bourdarias, B. de Laage de Meux, S. Gerbi, J.-M. Hérard A splitting method adapted to the simulation of mixed flows in pipes with a compressible two-layer model, ESAIM: M2AN, Mathematical Modelling and Numerical Analysis, Volume 53, No 2, 2019, pp. 405-442.
C. Demay, C. Bourdarias, B. de Laage de Meux, S. Gerbi, J.-M. Hérard Numerical simulation of a compressible two-layer model: A first attempt with an implicit-explicit splitting scheme, Journal of Computational and Applied Mathematics, Volume 346, 2019, pp. 357–377.
C. Demay, J.-M. Hérard, "A compressible two-layer model for transient gas–liquid flows in pipes", Continuum Mechanics and Thermodynamics, Volume 29, Issue 2, 2017, pp. 385–410.

Test case from de F. Aureli, A. Dazzi, A. Maranzoni, and P. Mignosa, "Validation of single- and two-equation models for transient mixed flows:
a laboratory test case" Journal of Hydraulic Research, 53(4):440–451, 2015.

PDE Control

Stability and controlability of locally coupled systems

C. Kassem, A. Mortada, L. Toufayli and A. Wehbe, "Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions", Comptes Rendus Mathematiques de l'Académie des Sciences, Volume 357, Issue 6, 2019, pp. 494-512.
S. Gerbi, C. Kassem, A. Mortada and A. Wehbe, "Locally internal exact controllability and stabilization of coupled wave equations", Soumis.

Exponential stability with equal wave speed and polynomial otherwise

Control of the wave equation with dynamic boundary conditions and a delay term

K. Ammari and S. Gerbi, "Interior feedback stabilization of wave equations with dynamic boundary delay", ZAA-Zeitschrift für Analysis und ihre Anwendungen, Volume 36, Issue 3, 2017, pp. 297–327.

Exponential stability for a Kelvin-Voigt dissipation with a delay time equal to 2 s.

Stochastic analysis and probabilities

[1] Random walk approximation of BSDEs with Hölder continuous terminal condition, Christel Geiss, Céline Labart and Antti Luoto. To appear in Bernoulli, 2019.
[2] L2-Approximation rate of forward-backward SDEs using random walk, Christel Geiss, Céline Labart and Antti Luoto. 2018, minor revision in Applied Probability trust.
[3] Donsker-Type Theorem for BSDEs: Rate of Convergence, Philippe Briand, Christel Geiss, Stefan Geiss and Céline Labart. 2019, submitted.

[4] Simulation of BSDEs by Wiener Chaos Expansion, Philippe Briand and Céline Labart. Annals of Applied Probability, Vol 24, Issue 3, 1129-1171, May 2014.

[5] Simulation of BSDEs with jumps by Wiener Chaos Expansion, Christel Geiss and Céline Labart. Stochastic Processes and their Applications, Vol 126, pp. 2123-2162, 2016.

[6] Simulation of McKean-Vlasov BSDEs by Wiener Chaos Expansion, Celine Acary-Robert, Philippe Briand and Abir Ghannoum and Céline Labart. 2019, submitted.

[7] Particle systems and Numerical Schemes for Mean Reflected Stochastic Differential Equations, Philippe Briand, Paul-Eric Chaudru de Raynal, Arnaud Guillin and Céline Labart. To appear in Annals of Applied Probability, 2019.

[8] Mean Reflected Stochastic Differential Equations with Jumps, Philippe Briand, Abir Ghannoum and Céline Labart. 2018, minor revision in Applied Probability Trust.

[9] Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles, Roxana Dumitrescu and Céline Labart. Journal of Mathematical Analysis and Applications, 2016, Vol 442, Issue 1, pp. 206-243.

[10] Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles, Roxana Dumitrescu and Céline Labart. Journal of computational and Applied Mathematics, Vol 296, pp. 827-839, 2016.

Calculus of variations et spectral analysis

D. Bucur and A. Henrot, Maximization of the second non-trivial Neumann eigenvalue, Acta Math. 222 (2019), no. 2, 337–361

D. Bucur and I. Fragalà, Proof of the honeycomb asymptotics for optimal Cheeger clusters, Advances in Mathematics 350 (2019), 97–129

D. Bucur, B. Bogosel and A. Giacomini, Optimal shapes maximizing the Steklov eigenvalues, SIAM J. Math. Analysis 49 (2017), no. 2, 1645-1680

D. Bucur and A. Giacomini, Faber-Krahn inequalities for the Robin-Laplacian: a free discontinuity approach, Arch. Ration. Mech. Anal. 218 (2015), no. 2, 757–824

Optimal packing

Interactions with others fields

P. Sollich, J. Olivier, D. Bresch. Aging and linear response in the Hébraud-Lequeux model for amorphous rheology. Phys. Rev. A 50(16) 2016.

D. Bresch, N. Cellier, F. Couderc, M. Gisclon, P. Noble, G. Richard, C. Ruyer-Quil, J.P. Vila. Augmented skew-symmetric system for shallow-water system with surface tension allowing large gradient of density. Arxiv:1911.12217

S. Jenouvrier, J. Garnier, F. Patout et L. Desvillettes. Influence of dispersal processes on the global dynamics of emperor penguin, a species threatened by climate change. Biol. Conserv., 212:63–73, 2017.

Click on the image to watch the video: What future do emperor penguins face ?

J. Garnier, M. Lewis. Expansion under climate change : The genetic consequences. Bull. Math. Biol., 78(11):2165–2185, 2016.