Asymptotic Behavior of systems of PDE arising in physics and biology: theoretical and numerical points of view (ABPDE II)
from
Wednesday, 15 June 2016 (09:00)
to
Friday, 17 June 2016 (16:00)
Monday, 13 June 2016
Tuesday, 14 June 2016
Wednesday, 15 June 2016
09:30
Welcome
Welcome
09:30  10:10
Room: Salle Kampé de Fériet  Bâtiment M2
10:10
Global Existence of Solutions to the 3D NavierStokes Equations with Degenerate Viscosities

Alexis Vasseur
(
Department of Mathematics
)
Global Existence of Solutions to the 3D NavierStokes Equations with Degenerate Viscosities
Alexis Vasseur
(
Department of Mathematics
)
10:10  10:55
Room: Salle de Réunion  Bâtiment M2
We prove the existence of global weak solutions for 3D compressible NavierStokes equations with degenerate viscosities. The method is based on the Bresch and Desjardins entropy. The solutions are obtained as limits of the quantic NavierStokes system. The main contribution is to derive the MelletVasseur type inequality for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible NavierStokes equations, for any gamma bigger than one, in three dimensional space, with large initial data, possibly vanishing on the vacuum. This is a joint work with Cheng Yu. The paper will appear in Inventiones.
10:55
Coffee break
Coffee break
10:55  11:30
Room: Salle Kampé de Fériet  Bâtiment M2
11:30
A (mainly numerical) study of a hyperbolic model for chemotaxis

Magali Ribot
(
Laboratoire de Mathématiques  Analyse, Probabilités, Modélisation  Orléans
)
A (mainly numerical) study of a hyperbolic model for chemotaxis
Magali Ribot
(
Laboratoire de Mathématiques  Analyse, Probabilités, Modélisation  Orléans
)
11:30  12:15
Room: Salle de Réunion  Bâtiment M2
The aim of this talk is to give some first results on the behaviour of the solutions of a 1D hyperbolic type chemotaxis system, based on incompressible Euler equation. More precisely, I will completely describe the stationary solutions with vacuum for this system and I will study numerically the stability of these steady states after the presentation of an adapted numerical scheme. A comparison with a limit parabolic system will also be performed.
12:15
Modelling and numerical approximation for the nonconservative bitemperature MHD model

Xavier Lhebrard
(
Centre Lasers Intenses et Applications
)
Modelling and numerical approximation for the nonconservative bitemperature MHD model
Xavier Lhebrard
(
Centre Lasers Intenses et Applications
)
12:15  12:50
Room: Salle de Réunion  Bâtiment M2
In order to achieve inertial confinement fusion, one has to improve the knowledge of the laserplasma interaction. There exists two main ways of describing this phenomenon, the microscopic (kinetic) approach and the macroscopic (hydrodynamic) approach. The kinetic approach is not competitive since it is too expensive in computational time. This is why we investigate an intermediate model in thermal nonequilibrium, which is between the kinetic model and the hydrodynamic model. In the first stage of the confinement the magnetic field is negligible, the relevant intermediate model is than the nonconservative bitemperature Euler model. Recently in [1], an entropic approximation of this system has been derived thanks to numerical schemes based on an underlying conservative kinetic model. However in the last stage of the confinement the target is penetrated by relativistic electrons, which induces a strongly variable magnetic field. This is why we want to deal with an intermediate model which takes into account the magnetic field. In this work we propose to study a bitemperature MHD model. This system consists in four conservation equations for mass, impulsion and magnetic field and two nonconservation equations, that is to say, one for each energy. Physically, the model describes the interaction of a mixture of one species of ions and one species of electrons in thermal nonequilibrium subjected to a transverse variable magnetic field. A first result is to have been able to established the hydrodynamic model from an underlying kinetic model. More precisely, using an out of equilibrium ChapmanEnskop procedure, the bitemperature MHD model is constructed from a BGK model coupled with Maxwell equations with full Lorentz force, which includes the magnetic field. Finally, we approximate the weak solutions of the bitemperature MHD model using a kinetic scheme, based on the underlying kinetic model. References [1] D. AregbaDriollet, S. Brull, J.Breil, B. Dubroca and E. Estibal, Modelling and numerical ap proximation for the nonconservative bitemperature Euler model, preprint, 2015.
12:50
Lunch
Lunch
12:50  14:35
14:35
A multiscale numerical approach for a class of timespace oscillatory problems

Mohammed Lemou
(
Institut de Recherche Mathématiques de Rennes
)
A multiscale numerical approach for a class of timespace oscillatory problems
Mohammed Lemou
(
Institut de Recherche Mathématiques de Rennes
)
14:35  15:20
Room: Salle de Réunion  Bâtiment M2
High oscillations may arise in many physical problems: Schrödinger equations, kinetic equations, or more generally high frequency waves. In this talk, we will present a general strategy that allows the construction of uniformly (with respect to the oscillation frequency) accurate numerical schemes in the following situations: i) time oscillations with applications to kinetic and Schrödinger equations. ii) timespace oscillations with applications to some high frequency waves and semiclassical quantum models. Some numerical tests will be presented to illustrate the efficiency of the strategy.
15:20
Global existence for small data of the viscous GreenNaghdi equations

Dena Kazerani
(
Laboratoire JacquesLouis Lions
)
Global existence for small data of the viscous GreenNaghdi equations
Dena Kazerani
(
Laboratoire JacquesLouis Lions
)
15:20  15:55
Room: Salle de Réunion  Bâtiment M2
We consider the Cauchy problem for the GreenNaghdi equations with viscosity, for small initial data. It is wellknown that adding a second order dissipative term to a hyperbolic system leads to the existence of global smooth solutions, once the hyperbolic system is symmetrizable and the socalled KawashimaShizuta condition is satisfied. We first show that the GreenNaghdi equations can be written in a symmetric form, using the associated Hamiltonian. This system being dispersive, in the sense that it involves third order derivatives, the symmetric form is based on symmetric differential operators. Then, we use this structure for an appropriate change of variable to prove that adding viscosity effects through a second order term leads to global existence of smooth solutions, for small data. We also deduce that constant solutions are asymptotically stable.
15:55
Coffee break
Coffee break
15:55  16:30
Room: Salle Kampé de Fériet  Bâtiment M2
16:30
Convergence to equilibrium for gradientlike systems with analytic features

Morgan Pierre
(
Laboratoire de Mathématiques et Applications
)
Convergence to equilibrium for gradientlike systems with analytic features
Morgan Pierre
(
Laboratoire de Mathématiques et Applications
)
16:30  17:05
Room: Salle de Réunion  Bâtiment M2
A celebrated result of S. Lojasiewicz states that every bounded solution of a gradient flow associated to an analytic function converges to a steady state as time goes to infinity. Convergence rates can also be obtained. These convergence results have been generalized to a large variety of finite or infinite dimensional gradientlike flows. The fundamental example in infinite dimension is the semilinear heat equation with an analytic nonlinearity. In this talk, we show how some of these results can be adapted to time discretizations of gradientlike flows, in view of applications to PDEs such as the AllenCahn equation, the sineGordon equation, the CahnHilliard equation, the SwiftHohenberg equation, or the phasefield crystal equation.
17:05
Numerical convergence rate for the diffusive limit of the psystem with damping

Hélène Mathis
(
Laboratoire de Mathématiques Jean Leray
)
Numerical convergence rate for the diffusive limit of the psystem with damping
Hélène Mathis
(
Laboratoire de Mathématiques Jean Leray
)
17:05  17:40
Room: Salle de Réunion  Bâtiment M2
We are interested in the study of the diffusive limit of the $p$system with damping and its approximation by an Asymptotic Preserving (AP) Finite Volume scheme. Provided the system is endowed with an entropyentropy flux pair, we give the convergence rate of classical solutions of the psystem with damping towards the smooth solutions of the porous media equation using a relative entropy method. Adopting a semidiscrete scheme, we establish that the convergence rate is preserved by the approximated solutions. Several numerical experiments illustrate the relevance of this result.
Thursday, 16 June 2016
08:45
Coffee
Coffee
08:45  09:30
Room: Salle Kampé de Fériet  Bâtiment M2
09:30
Implicitexplicit linear multistep methods for stiff kinetic equations

Giacomo Dimarco
(
Department of Mathematics and Computer Science
)
Implicitexplicit linear multistep methods for stiff kinetic equations
Giacomo Dimarco
(
Department of Mathematics and Computer Science
)
09:30  10:15
Room: Salle de Réunion  Bâtiment M2
We consider the development of high order asymptoticpreserving linear multistep methods for kinetic equations and related problems. The methods are first developed for BGKlike kinetic models and then extended to the case of the full Boltzmann equation. The behavior of the schemes in the NavierStokes regime is also studied and compatibility conditions derived. We show that, compared to IMEX RungeKutta methods, the IMEX multistep schemes have several advantages due to the absence of coupling conditions and to the greater computational efficiency. The latter is of paramount importance when dealing with the time discretization of multidimensional kinetic equations.
10:15
From particle methods to hybrid semiLagrangian schemes

Frédérique Charles
(
Laboratoire JacquesLouis Lions
)
From particle methods to hybrid semiLagrangian schemes
Frédérique Charles
(
Laboratoire JacquesLouis Lions
)
10:15  11:00
Room: Salle de Réunion  Bâtiment M2
Particle methods for transport equations consist in pushing forward particles along the characteristic lines of the flow, and to describe then the transported density as a sum of weighted and smoothed particles. Conceptually simple, standard particle methods have the main drawback to produce noisy solutions or to require frequent remapping. In this talk we present two classes of particle methods which aim at improving the accuracy of the numerical approximations with a minimal amount of smoothing. The idea of the Linearly Transformed Particle method is to transform the shape functions of particles in order to follow the local variation of the flow. This method has been adapted and analyzed for the Vlasov Poisson system and for a compressible aggregation equation. In both cases the error estimate is improved compared to classical particle methods, with the gain of a strong convergence of the numerical solution. However, for long remapping periods, shapes of particles could become to much stretched out. The second method solve this problem of locality by combining a backward semiLagrangian approach and local linearizations of the flow. The convergence properties are improved and validated by numerical experiments. This is a joint work with Martin CamposPinto (LJLL, UPMC).
11:00
Coffee break
Coffee break
11:00  11:30
Room: Salle Kampé de Fériet  Bâtiment M2
11:30
Time splitting methods and the semiclassical limit for nonlinear Schrödinger equations

Rémi Carles
(
Institut Montpelliérain Alexander Grothendieck
)
Time splitting methods and the semiclassical limit for nonlinear Schrödinger equations
Rémi Carles
(
Institut Montpelliérain Alexander Grothendieck
)
11:30  12:15
Room: Salle de Réunion  Bâtiment M2
We consider the time discretization based on LieTrotter splitting, for the nonlinear Schrödinger equation, in the semiclassical limit, with initial data under the form of WKB states. Both the exact and the numerical solutions keep a WKB structure, on a time interval independent of the Planck constant. We prove error estimates, which show that the quadratic observables can be computed with a time step independent of the Planck constant. We give a flavor of the functional framework, based on timedependent analytic spaces.
12:15
Dimensional reduction of a multiscale model based on long time asymptotics

Marie Postel
(
Laboratoire JacquesLouis Lions
)
Dimensional reduction of a multiscale model based on long time asymptotics
Marie Postel
(
Laboratoire JacquesLouis Lions
)
12:15  12:50
Room: Salle de Réunion  Bâtiment M2
Depending on their velocity field, some models lead to moment equations that enable one to compute monokinetic solutions economically. We detail the example of a multiscale structured cell population model, consisting of a system of 2D transport equations. The reduced model, a system of 1D transport equations, is obtained by computing the moments of the 2D model with respect to one variable. The 1D solution is defined from the solution of the 2D model starting from an initial condition that is a Dirac mass in the direction removed by reduction. Long time properties of the 1D model solution are obtained in connection with properties of the support of the 2D solution for general case initial conditions. Finite volume numerical approximations of the 1D reduced model can be used to compute the moments of the 2D solution with proper accuracy. The numerical robustness is studied in the scalar case, and a full scale vector case is presented.
12:50
Lunch
Lunch
12:50  14:40
14:40
Global existence and largetime behaviour for reactiondiffusion models

Klemens Fellner
(
Institut für Mathematik und Wissenschaftliches Rechnen
)
Global existence and largetime behaviour for reactiondiffusion models
Klemens Fellner
(
Institut für Mathematik und Wissenschaftliches Rechnen
)
14:40  15:25
Room: Salle de Réunion  Bâtiment M2
Systems of nonlinear reactiondiffusion equations are encountered frequently as models in chemistry, physics, populations dynamics and biology. However, due to the lack of comparison principles for general reactiondiffusion systems, already the existence of global weak/classical solutions poses many open problems, in particular in 3D. In the absence of comparison principles, so called duality methods have recently proven to be one of the most powerful tools in obtaining global solutions for nonlinear reactiondiffusion systems. The first part of this talk will present recent advances and results concerning the existence of global solutions via duality methods. The second part of the talk will then consider reactiondiffusion systems, which feature an entropy functional and discuss the convergence to equilibrium states with computable rates for large classes of such reactiondiffusion models.
15:25
Asymptotic analysis for a simplified model of model of dynamical perfect plasticity

Clément Mifsud
(
Laboratoire Jacques Louis Lions
)
Asymptotic analysis for a simplified model of model of dynamical perfect plasticity
Clément Mifsud
(
Laboratoire Jacques Louis Lions
)
15:25  16:00
Room: Salle de Réunion  Bâtiment M2
In this talk, we will present an initial boundary value problem for a hyperbolic system under constraints, coming from mechanics. To study the solutions of such a system, we will use a viscous approach that relaxes the constraints. We will explain the asymptotic analysis, when the viscous parameter tends to zero, which leads to an interaction between the boundary condition and the constraints for the constrained system. If time permits, we will show some numerical results.
16:00
Coffee break
Coffee break
16:00  16:30
Room: Salle Kampé de Fériet  Bâtiment M2
Poster session

An Zhang
(
CEREMADE
)
Samia Zermani
(
Institut aux Etudes d'Ingénieurs el Manar
)
Judith Berendsen
(
Institute for Computational and Applied Mathematics
)
Ahmed Ait Hammou Oulhaj
(
Laboratoire Paul Painlevé
)
Andrea Bondesan
(
Laboratoire MAP5
)
Maxime Herda
(
Institut Camille Jordan
)
Poster session
An Zhang
(
CEREMADE
)
Samia Zermani
(
Institut aux Etudes d'Ingénieurs el Manar
)
Judith Berendsen
(
Institute for Computational and Applied Mathematics
)
Ahmed Ait Hammou Oulhaj
(
Laboratoire Paul Painlevé
)
Andrea Bondesan
(
Laboratoire MAP5
)
Maxime Herda
(
Institut Camille Jordan
)
16:00  18:00
Room: Salle Kampé de Fériet  Bâtiment M2
19:30
Gala dinner
Gala dinner
19:30  22:30
Room: Restaurant Le Compostelle  Lille
Friday, 17 June 2016
08:45
Coffee
Coffee
08:45  09:30
Room: Salle Kampé de Fériet  Bâtiment M2
09:30
Stability results of dissipative systems via the frequency domain approach

Serge Nicaise
(
Laboratoire de Mathématiques et de leurs Applications de Valenciennes
)
Stability results of dissipative systems via the frequency domain approach
Serge Nicaise
(
Laboratoire de Mathématiques et de leurs Applications de Valenciennes
)
09:30  10:15
Room: Salle de Réunion  Bâtiment M2
The frequency domain approach goes back to J. Prüss [Trans. Amer. Math. Soc. 284 (1984), 847857] and F. L. Huang [Ann. Differential Equations 1 (1985), 4356] that show that a $C_0$ semigroup $(e^{tA})_{t\geq 0}$ of contractions in a Hilbert space $H$ is exponentially stable if and only if the resolvent of $A$ is uniformly bounded on the imaginary axis. Afterwards Z. Liu and B. Rao [Z. Angew. Math. Phys. 56 (2005), 630644], C. J. K. Batty and T. Duyckaerts [J. Evol. Equ. 8 (2008), 765780], and A. Bátkai, K.J. Engel, J. Prüss and R. Schnaubelt [Math. Nachr. 279 (2006), 14251440] have given some sufficient conditions on the behavior of the resolvent of $A$ on the imaginary axis that guarantee an almost polynomial decay of the semigroup. Finally an optimal result about the polynomial decay was found by A. A. Borichev and Yu. V. Tomilov [Math. Ann. 347 (2010), 455478]. This approach is a powerful tool for the study of the decay of the semigroup associated with concrete dissipative systems since it reduces to the study of the resolvent on the imaginary axis. In our talk, we will first recall these two results and then illustrate them on two particular dissipative systems, namely a generalized telegraph equation [Z. Angew. Math. Phys. 66 (2015), 32213247] and a dispersive medium model (joint work with C. Scheid (Univ. Nice)).
10:15
Uniform asymptotic preserving scheme for hyperbolic systems in 2D

Emmanuel Franck
(
Inria Nancy Grandest
)
Uniform asymptotic preserving scheme for hyperbolic systems in 2D
Emmanuel Franck
(
Inria Nancy Grandest
)
10:15  10:50
Room: Salle de Réunion  Bâtiment M2
In this work, we are interested by the discretization of hyperbolic system with stiff source term. Firstly we consider a simple linear case : the damped wave equation which can be approximative by a diffusion equation at the limit. For this equation we propose a asymptotic preserving scheme which converge uniformly on general and unstructured 2D meshes contrary to the classical extension of the AP which are not consistent in the limit regime on unstructured meshes. After that we propose to extend this method to a nonlinear problem: the Euler equations with friction. At the end the link with the wellbalanced scheme (for EulerPoisson) will be introduced.
10:50
Coffee break
Coffee break
10:50  11:25
Room: Salle Kampé de Fériet  Bâtiment M2
11:25
Complete flux schemes for conservation laws of advectiondiffusionreaction type

Jan ten Thije Boonkkamp
(
Department of Mathematics and Computer Science
)
Complete flux schemes for conservation laws of advectiondiffusionreaction type
Jan ten Thije Boonkkamp
(
Department of Mathematics and Computer Science
)
11:25  12:10
Room: Salle de Réunion  Bâtiment M2
Complete flux schemes are recently developed numerical flux approximation schemes for conservation laws of advectiondiffusionreaction type; see e.g. [1, 2]. The basic complete flux scheme is derived from a local onedimensional boundary value problem for the entire equation, including the source term. Consequently, the integral representation of the flux contains a homogeneous and an inhomogeneous part, corresponding to the advectiondiffusion operator and the source term, respectively. Suitable quadrature rules give the numerical flux. For timedependent problems, the time derivative is considered a source term and is included in the inhomogeneous flux, resulting in an implicit semidiscretisation. The implicit system proves to have much smaller dissipation and dispersion errors than the standard semidiscrete system, especially for dominant advection. Just as for scalar equations, for coupled systems of conservation laws, the complete flux approximation is derived from a local system boundary value problem, this way incorporatin the coupling between the constituent equations in the discretization. Also in the system case, the numerical flux (vector) is the superpostion of a homogeneous and an inhomogeneous component, corresponding to the advectiondiffusion operator and the source term vector, respectively. The scheme is applied to multispecies diffusion and satisfies the mass constraint exactly. References [1] J.H.M. ten Thije Boonkkamp and M.J.H. Anthonissen, The finite volumecomplete flux scheme for advectiondiffusionreaction equations, J. Sci. Comput. 46, pp. 4770 (2011). [2] J.H.M. ten Thije Boonkkamp, J. van Dijk, L. Liu and K.S.C. Peerenboom, Extension of the complete flux scheme to systems of comservation laws, J. Sci. Comput. 53, pp. 552568 (2012).
12:10
Non linear stability of Minkowski spacetime with massive scalar field

Yue Ma
(
School of Mathematics and Statistics
)
Non linear stability of Minkowski spacetime with massive scalar field
Yue Ma
(
School of Mathematics and Statistics
)
12:10  12:45
Room: Salle de Réunion  Bâtiment M2
In this talk we will present some recent work about the system of Einstein equation coupled with a massive scalar field and the system of $f(R)$ field equation (partially published in [2]). More precisely, on the nonlinear global stability of the Minkowski spacetime within these two similar contexts. In a PDE point of view, they are equivalent to the global existence of a special class of quasilinear waveKleinGordon system with small initial data. To the author’s knowledge there is not so much choice to deal with this kind of system (for a detailed explication of the major difficulty, see for example in [1] page 2), and we apply the “hyperboloidal foliation method” introduced by the author in [1] combined with some newly developed tools such as $L^∞$ estimates on KleinGordon equations in curved spacetime and $L^∞$ estimates on wave equations based on the expression of spherical means. We also adapt some tools developed in classical framework for the analysis of Einstein equation into our hyperboloidal foliation framework, such as the estimates based on wave gauge conditions and the L$^∞$ estimates on wave equations based on integration along characteristics. References [1] P. LeFloch and Y. Ma, The hyperboloidal foliation method, World Scientific, 2015 [2] P. LeFloch and Y. Ma, The nonlinear stability of Minkowski space for selfgravitating massive field. The waveKleinGordon model, Comm. Math. Phys., published online.
12:45
Lunch
Lunch
12:45  14:40
14:40
Exponential decay of a finite volume scheme to the thermal equilibrium for driftdiffusion systems

Marianne Bessemoulin
(
Laboratoire de Mathématiques Jean Leray
)
Exponential decay of a finite volume scheme to the thermal equilibrium for driftdiffusion systems
Marianne Bessemoulin
(
Laboratoire de Mathématiques Jean Leray
)
14:40  15:25
Room: Salle de Réunion  Bâtiment M2
We are interested in the largetime behavior of a numerical scheme discretizing driftdiffusion systems for semiconductors. The considered scheme is finite volume in space, and the numerical fluxes are a generalization of the classical ScharfetterGummel scheme, which allows to consider both linear or nonlinear pressure laws. We study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniformintime $L^\infty$ estimates for numerical solutions, which will be discussed. This is a joined work with Claire ChainaisHillairet.
15:25
Entropy methods for degenerate diffusions and weighted functional inequalities

Bruno Nazaret
(
SAMM  Statistique, Analyse, Modélisation multidisciplinaire
)
Entropy methods for degenerate diffusions and weighted functional inequalities
Bruno Nazaret
(
SAMM  Statistique, Analyse, Modélisation multidisciplinaire
)
15:25  16:00
Room: Salle de Réunion  Bâtiment M2
We will present results on large time asymptotics for some fast diffusion equations with power law weights. We will show that, for such diffusions, new phenomena appear : the asymptotic rates of convergence, obtained by linearization, are not global, the underlying functional inequalities may experience symmetry breaking and the Barenblatt selfsimilar profiles is not optimal.