Projet « Jeunes Chercheurs »
N° ANR-11-JS01-006-01
Funded by l’ANR from 2011 to 2014

Presentation



The project aims, firstly, at solving several concrete questions in the modern theory of
conservation laws and related convection-diffusion problems. These questions are concerned with actual techniques
of nonlinear analysis (some of the questions seemed not accessible only a few years ago) ; they originate from
recently identified of from long-standing important applications. The Tasks are all relevant of the
domains of expertise of one of the members of the project and at the same time,
there is a strong connection to at least one of the other members
and to our national and international collaborations.

The second goal is to make advance our understanding of the theory and the crucial tools
(including the most modern ones) for analysis of conservation laws and related nonlinear PDEs.
Indeed, the questions we have selected are representative of several fundamental issues relevant to the problems in hand, such as :
notions of solution and well-posedness ; analysis of non-local terms in conservation laws ; cooperation of
semigroups in evolution equations ; convection-diffusion problems of mixed type ;
boundary and interface problems for conservation laws ;
(ir)regularity and qualitative behaviour of solutions ; converging and efficient numerical approximations.

The scientific program is divided into eight main parts :

• Triangular systems of conservation laws
  
• Fractionnal conservation laws and convection-diffusion equations
  
• Viscoelasticity system with memory
  
• Conservation laws with non local fluxes
  
• Solution profiles for general conservation laws with dissipative source terms
  
• Simple 1D and 2D fluide-particule interaction models
  
• Relativistic heat equation
  
• Initiation to control of hyperbolic conservation laws
  
  A detailed description of these tasks is available here.

International journals

[1] Adimurthi, S. S. Ghoshal, G. D. Veerappa Gowda,
Finer regularity of an entropy solution for 1-d scalar conservation laws with non uniform convex flux,
Rendiconti Sem. Mat. Univ. di Padova, 132 (2014), 1—24.

[2] Adimurthi, S. S. Ghoshal, G. D. Veerappa Gowda,
$L^p$ stability for entropy solutions of scalar conservation laws with strict convex flux,
J. Differ. Equ. 256, (2014), 3395—3416.

[3] N. Alibaud, S.Cifani, E. R. Jakobsen,
Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations
SIAM J. Math. Anal. 44, (2012), 603—632.

[4] N. Alibaud, S. Cifani, E. R. Jakobsen,
Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations,
Arch. Rational Mech. Anal. 213, (2014), 705—762.

[5] B. Andreianov,
New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux,
ESAIM : Proc. and Surveys 50, (2015), 40—65.

[6] B. Andreianov, K. Brenner, C. Cancès,
Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium.
ZAMM Zeitshr. Angew. Math. Mech., 94, (2014), 655—667.

[7] B. Andreianov, C. Cancès,
The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions,
Appl. Math. Lett. 25, (2012), no.11, 1844—1848.

[8] B. Andreianov, C. Cancès,
Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks’ medium,
Comput. Geosci. 17, (2013), 551—572.

[9] B. Andreianov, C. Cancès,
A phase-by-phase upstream scheme that converges to the vanishing capillarity solution for countercurrent two-phase flow in two-rocks’ media ,
Comput. Geosci. 18, (2014), 211—226.

[10] B. Andreianov, C. Cancès,
On interface transmission conditions for conservation laws with discontinuous flux of general shape,
J. Hyperbolic Differ. Equ. 12 (2) (2015), 343—384.

[11] B. Andreianov, C. Donadello, S. S. Ghoshal, U. Razafison,
On the attainable set for a class of triangular systems of conservation laws,
J. Evol. Equ. 15 (3) (2015), 503—532.

[12] B. Andreianov, C. Donadello, U. Razafison, J.-Y. Rolland, M.D. Rosini,
Solutions of the Aw-Rascle-Zhang system with point constraints,
Netw. Heter. Media 11 (1) (2016), 29—47 (numéro spécial CoToCoLa)

[13] B. Andreianov, C. Donadello, U. Razafison, M. D. Rosini,
Riemann problems with non-local point constraints and capacity drop,
Math. Biosci. Eng., 12 (2) (2015), 259—268.

[14] B. Andreianov, C. Donadello, U. Razafison, M. D. Rosini,
Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks,
ESAIM M2AN Math. Modeling Numer. Anal., 50 (5) (2016), 1269—1287.

[15] B. Andreianov, C. Donadello, M. D. Rosini,
Crowd dynamics and conservation laws with non-local constraints and capacity drop,
M3AS Math. Models Meth. Appl. Sci. 24, (2014), 2685—2722.

[16] B. Andreianov, C. Donadello, M. D. Rosini,
A second order model for vehicular traffics with local point constraints on the flow,
M3AS Math. Models Meth. Appl. Sci. 26 (4) (2016), 751—802.

[17] B.Andreianov, M. K. Gazibo,
Entropy formulation of degenerate parabolic equation, with zero- flux boundary condition,
ZAMP Zeitshr. Angew. Math. Phys. 64, (2013), 1471—1491.

[18] B.Andreianov, M. K. Gazibo,
Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws,
Netw. Heter. Media 11 (2) (2016), 203—222 (numéro spécial CoToCoLa)

[19] B. Andreianov, F. Lagoutière, N. Seguin, T. Takahashi,
Well-posedness for a one-dimensional fluid-particle interaction model,
SIAM J. Math. Anal. 46, (2014), 1030—1052.

[20] B. Andreianov, D. Mitrovic,
Entropy conditions for scalar conservation laws with discontinuous flux revisited,
Annales Inst. H. Poincaré C – Anal. Non Linéaire 32 (6) (2015),1307-1335.

[21] B. Andreianov, K. Sbihi,
Well-posedness of general boundary-value problems for scalar conservation laws,
Transactions AMS 367 (6) (2015), 3763—3806.

[22] B. Andreianov, N. Seguin,
Analysis of a Burgers equation with singular resonat source term and convergence of well-balanced schemes,
Discrete Contin. Dyn. Syst. A 32 (2012), no.6, 1939—1964.

[23] M. Brassart,
Non-critical fractional conservation laws in domains with boundary
Netw. Heter. Media 11 (2) (2016), 251—262 (numéro spécial CoToCoLa)

[24] G. Crippa, C. Donadello, L. V. Spinolo,
Initial-boundary value problems for continuity equations with BV coefficients,
J. Math. Pures Appl. 102, (2014), 79—98.

[25] S.S. Ghoshal,
BV regularity near the interface for nonuniform convex discontinuous flux,
Netw. Heter. Media 11 (2) (2016), 331—348 (numéro spécial CoToCoLa)

Book chapter

[26] B. Andreianov,
Semigroup approach to conservation laws with discontinuous flux.
Springer Proc. In Math. and Stat. Series Vol. 49, (2014), 1—22.
PDF

[27] B. Andreianov,
One-dimensional conservation law with boundary conditions : general results and spatially inhomogeneous case.
HYP 2012 conf. proc, AIMS series in Appl. Math. Vol. 8, 259—267.
PDF

[28] B. Andreianov, M.K. Gazibo,
Convergence of finite volume method for degenerate parabolic problem with zero flux boundary condition.
FVCA7 Conference proc., Springer Proc. in Math. and Stat., Vol. 77 (2014), 303—311.

[29] G. Crippa, C. Donadello, L. V. Spinolo.
A note on the initial-boundary value problem for continuity equations with rough coefficients,
HYP 2012 conf. proc, AIMS series in Appl. Math. Vol. 8, 957—966.

[30] M.K. Gazibo,
Degenerate Convection-Diffusion Equation with a Robin boundary condition,
HYP 2012 conf. proc, AIMS series in Appl. Math. Vol. 8, 583—591.

[31] O. Delestre, U. Razafison, 
Numerical scheme for a viscous shallow water system including new friction laws of second order : validation, and application,
Adv. Hydroinformatics : SIMHYDRO 2014, Springer Water series (2016), 227—239.

Visitors

• Doctor Massimiliano ROSINI (ICM, University of Warsaw, Poland), March 1st-April 30th, 2014.

• Doctor Olivier DELESTRE (University of Nice-Sophia Antipolis, France), January 27th-31th, 2014.

• Professor Claire CHAINAIS-HILLAIRET (University of Lille 1), December 12th, 2013.

• Professor ADIMURTHI (TIFR Centre for Applicable Mathematics, Bangalore, India), May 25th-June 26th, 2013.

• Doctor Laura CARAVENA (Oxford Centre for Nonlinear PDE), May 15th-20th, 2013.

• Doctor Massimiliano ROSINI (ICM, University of Warsaw, Poland), February 20th-April 19th, 2013.

• Doctor Olivier DELESTRE (University of Nice-Sophia Antipolis, France), May 28th-June 1st, 2012.

• Professor Espen R. JAKOBSEN (Norwegian University of Science and Technology), May 21th-25th, 2012.

• Doctor Olivier DELESTRE (University of Nice-Sophia Antipolis, France), April 16th-20th, 2012.

• Doctor Clément CANCÈS (University Pierre et Marie Curie-ParisVI, France), April 10th-11th, 2012.

• Doctor Adama OUÉDRAOGO (University of Bobo-Dioulasso, Burkina-Faso), March 20th - April 20th, 2012.

• CNRS Research Director Cyril IMBERT (University of Creteil, France), March 27th-29th, 2012.

• Doctor Shyam GOSHAL (University of Bangalore, India), March 24th-31th, 2012.

• Professor Gianluca CRIPPA (Univeristy of Basel, Switzerland), March 5th-6th, 2012.

• Doctor Fabrio PRIULI (University of Padova, Italy), February 12th-16th, 2012.

• Professor Colombo RINALDO (University of Brescia, Italy), January 23th-27th, 2012.
  
  
  
  
  Schedule and events
  
  
  

• April, 27th-30th 2015, 5ème École de printemps EDP Non linéaire : Mathématiques et Interactions : modèles non locaux et applications.

• February, 9th-12th 2015, CoToCoLa week in the Trimestres du LMB.

  

• June, 19th-20th, Journée d’analyse non linéaire de Besançon.

• December, 6th, 2013, Claire CHAINAIS-HILLAIRET, "Inégalités de Beckner et méthode d’entropie pour l’équation des milieux poreux : du continu au discret "

• May, 29th, 2013, Journée d’analyse non linéaire de Besançon

• April, 11th, 2013, seminar, Massimiliano ROSINI, "Optimal traffic flow"

• September, 19th, 2012, seminar, Ulrich RAZAFISON, "Introduction to finite volume methods for hyperbolic scalar of conservation laws : part II"

• September, 12th, 2012, seminar, Ulrich RAZAFISON, "Introduction to finite volume methods for hyperbolic scalar of conservation laws : part I"

• June, 1st, 2012, Journée Numérique de Besançon

• May, 31th, 2012, seminar, Mohamed KARIMOU GAZIBO, "Entropy formulation of degenerate parabolic equation, with zero-flux boundary condition"

• May, 24th, 2012, seminar, Espen R. JAKOBSEN, "On numerical methods for fractional conservation laws and fractional degenerate parabolic equations"

• April, 17th, 2012, seminar, Olivier DELESTRE, "Méthode numérique pour le ruissellement et des écoulements sanguins"

• April, 10th, 2012, seminar, Clément CANCÈS, "Long-time asymptotic preserving finite volume-schemes for hyperbolic systems with strong relaxation"

• March, 27th 2012, seminar, Shyam GHOSHAL, "Finer analysis of characteristic curves and its application to exact and optimal controllability structure theorem of a scalar conservation law with strict convex flux"

• March, 13th 2012, seminar, Carlotta DONADELLO, "Introduction to hyperbolic system of conservation law, part II"

• March, 6th 2012, seminar, Gianluca CRIPPA, "ODES and singular integrals"

• February 14th, 2012, seminar, Fabio PRIULI, "Control problems for hyperbolic conservation laws"

• February, 7th, 2012, seminar, Carlotta DONADELLO, "Introduction to hyperbolic system of conservation law, part I"

• January, 24th, 2012, seminar, Colombo RINALDO, "Hyperbolic conservation laws and the macroscopic modeling of crowd"

• November, 24th-25th, 2011, Journées d’analyse non linéaire de Besançon