Applied Analysis group

Scientific report

Functional analysis, harmonic analysis, analysis on varieties, geometric analysis, calculus of variations
Functional and harmonic analysis (P. Bousquet, S. Monniaux, E. Russ)
In a book, D. Mitrea, I. Mitrea, M. Mitrea and S. Monniaux are interested in situations in analysis in which one can construct a metric compatible with a given framework. In particular, they improve the theorem of Macias and Segovia of 1979 in the sense that the constants obtained are optimal. Many applications and examples are given. In particular, they show how the open application, closed graph and Banach-Steinhaus theorems can be extended to more general settings than that of complete metric spaces.
P. Auscher, S. Monniaux and P. Portal studied non autonomous Cauchy problems with operators in divergence form, using singular integrals on tent spaces (à la Coifman, Meyer, Stein).
M. Efendiev and E. Russ introduced Hardy spaces for the conjugate Beltrami equation in doubly related domains of the complex plane, and solved in these spaces the Dirichlet problem for the conjugate Beltrami equation, with Lp data on the edge, the real part of the solution being prescribed on one portion of the edge, its imaginary part on the other portion.
P. Bousquet studied critical cases for the divergence equation, Hardy inequalities for underdetermined systems, topological singularities in Sobolev spaces (density theorem for smooth functions in variety-valued Sobolev spaces), and the distributional Jacobian of variety-valued Sobolev applications.
Poincaré inequalities (E. Russ and Y. Sire)
C. Mouhot, E. Russ and Y. Sire have obtained L2 Poincaré inequalities in which the L2 norm of the gradient is replaced by a nonlocal quantity, for general μ-measures which satisfy a usual L2 Poincaré inequality. This result, obtained in the case of Rn , was extended by E. Russ and Y. Sire to the case of a Lie group with polynomial growth.
Stokes problem (S. Monniaux)
With A. McIntosh, S. Monniaux exploits a first-order formulation for the Stokes problem to obtain stronger results, and in more general domains, than those known so far and this in a larger variety of spaces. With T. ter Elst, she studies the “Dirichlet-to-Neumann” operator associated to the Stokes problem in order to find Friedlander-like estimates of the eigenvalues of the Stokes operator with Dirichlet and Neumann edge conditions.
Calculus of variations, optimization, convex analysis (P. Bousquet, L. Brasco, E. Ernst, E. Parini, P. Sicbaldi)
P. Bousquet, L. Brasco, G. Carlier and V. Julin have studied the lipschitz regularity for some elliptic equations with very large degeneracies, equations which are not much studied and which appear naturally in optimal transportation problems with congestion effects. They studied in particular equations with degeneracies confined in balls as well as anisotropic variants of the p-laplacian. P. Bousquet also analyzed the Lavrentiev phenomenon and the validity of the Euler-Lagrange equation.
E. Parini obtained results for quasi-linear equations of the p-laplacian type, with F. Charro. With N. Saintier, he also studied isoperimetric problems, related to spectral problems. E. Parini, B. Ruf and C. Tarsi have identified the optimal constants for a higher order functional inequality. F. Hamel, E. Parini and B. Ruf supervise the thesis of G. Romani on the positivity properties of the Kirchoff-Love functional, in cotutelle with the University of Milan and in the framework of a grant from the A*MIDEX Academy of Excellence.
E. Ernst gave a characterization of domains of definition which implies the automatic continuity of any convex function defined there, a problem posed in 1965 by D. Gale, V. Klee and T. Rockafellar. In collaboration with M. Volle, M. A. Lopez and N. Dinh, he discussed the convergence of augmented Lagrangian methods in multi-criteria optimization and two Hahn-Banach type theorems applied to optimization.
Spectral inequalities, rearrangement inequalities for linear and nonlinear problems (L. Brasco, F. Hamel, N. Nadirashvili, E. Russ)
L. Brasco, with G. De Philipps, A. Pratelli, B. Ruffini and B. Velichkov, showed the stability of the isoperimetric inequalities for the first Neumann eigenvalue, the second Dirichlet eigenvalue and the first Stek- lov eigenvalue, as well as the Bhattacharya-Nadirashvili-Weitsman conjecture concerning the stability of the Faber-Krahn inequality for the first Dirichlet eigenvalue. L. Brasco and G. Franzina have shown some isoperimetric in-equalities for the eigenvalues of the p-laplacian and its anisotropic versions, and have also given a new proof of the simplicity of the first Dirichlet eigenvalue of a nonlinear operator of the p-laplacian type.
L. Brasco and E. Parini, with E. Lindgren and M. Squassina, have studied the eigenvalue problem for the fractional p-laplacian, a non-local version of the classical p-laplacian, which is nothing else than the Euler-Lagrange equation for a Sobolev-Slobodeckii semi-norm. They studied the differentiability, for the super-quadratic case, of weak solutions of this nonlinear, nonlocal equation.
F. Hamel, N. Nadirashvili and E. Russ have shown Faber-Krahn type inequalities for the first eigenvalue of elliptic operators of order 2, nonsymmetric in general, on bounded domains of Rn with Dirichlet condition and under different geometric, integral or pointwise constraints on the parameters. They have introduced a new symmetrization method, different from the Schwarz method used for symmetric operators, and have shown new differential point inequalities. F. Hamel and E. Russ showed “Talenti-like” comparison results for elliptic problems with nonlinear dependence H(x,u,∇u) and at most quadratic on the terms in ∇u.
Analysis on varieties (N. Nadirashvili, P. Sicbaldi)
N. Nadirashvili studied metrics on Riemannian surfaces of given conformal type and unit area that maximize the nth eigenvalue of the Laplacian. He proved the existence of extremal metrics and characterized them in terms of the harmonic application of a given surface in the k-dimensional sphere. He gave a constructive proof of a critical metric which is regular outside a finite number of conic singularities and maximizes the first eigenvalue in the conformal class of the underlying metric. He proved the existence of an application associating to a point of the surface a family of eigenvectors corresponding to the maximized eigenvalue. He also gave new bounds on the number of negative eigenvalues of the Schrödinger operator.
In collaboration with E. Delay, J. Lamboley and F. Morabito, P. Sicbaldi studied the extremal domains, under volume constraint, for the first Laplace-Beltrami eigenvalue in Riemannian varieties, obtaining in particular results on the existence and localization of these domains, which depend on the curvature of the variety. He also generalized the study to the case of varieties with edge, where the curvature of the edge plays an important role. Finally, he studied the existence of non-trivial solutions to overdetermined elliptic problems in homogeneous varieties.
KAM theory, networks and Hamiltonian PDEs (Y. Sire)
R. de la Llave and Y. Sire studied the existence and qualitative properties of quasi-periodic solutions admitting hyperbolic directions. They have treated in particular the case of discrete networks and the case of ill-posed PDEs.

Partial Differential Equations, Control and Inverse Problems
Partial analyticity for hyperbolic or elliptic systems (N. Nadirashvili)
N. Nadirashvili and his collaborators have generalized results of S. Bernstein, H. Levy, I. Petrovsky and Morrey concerning solutions of nonlinear elliptic systems, analytic with respect to a group of variables, under assumptions of partial analytic dependence. He obtained several applications concerning the analyticity of solutions for 2D Euler and water waves without surface tension. For quasilinear wave equations he revisited a result of Alinhac and Métivier on the propagation of the partial analyticity of solutions, obtaining a more general and simpler proof.
Control and Inverse Problems (A. Benabdallah, F. Boyer, L. Cardoulis, M. Cristofol, Y. Dermenjian, M. Morancey, O. Poisson)
Within the Applied Analysis team, the Control and Inverse Problems group, which has been active for more than ten years, is composed of half a dozen permanent members who are joined by guests, post-doctoral fellows or PhD students. A monthly working group gathers its members.
The control theme is mainly studied by A. Benabdallah, F. Boyer (who moved to Toulouse since 2015), Y. Dermenjian and M. Morancey, and the inverse problem theme by L. Cardoulis, M. Cristofol and O. Poisson.
A. Benabdallah works on the observability of parabolic or elliptic operators with dis- continuous coefficients. With Y. Dermenjian and J. Le Rousseau she proved a Carleman inequality in the case of partially stratified operators, then with Y. Dermenjian and L. Thévenet she extended this inequality to the case of partially anisotropic elliptic operators. Moreover, she continued her research on the control of coupled parabolic equations. Together with F. Ammar Khodja, M. González-Burgos and L. de Teresa, she formulated necessary and sufficient conditions for controllability of classes of parabolic systems. Also with the same collaborators, she has highlighted completely new and counter-intuitive behaviors, qualified as hyperbolic phenomena (appearance of a strictly positive minimal control time, influence of the geometry). An article summarizing the results has been published, a book is being written and, with M. Morancey, an ANR project on the study of these hyperbolic phenomena in the control of parabolic equations has been submitted. On this subject, she is supervising the thesis of El Hadji Samb. With F. Boyer, M. González-Burgos and G. Olive, she obtained the first result of control by the edge of parabolic systems in space dimension > 1. With M. Cristofol, P. Gaitan and L. De Teresa, she proved an observability inequality for a parabolic system of 3 equations with space-dependent coefficients with observation of only one component.
M. Morancey studied the controllability of degenerate parabolic equations, in particular of Gru- shin type. On this problem, in collaboration with K. Beauchard and L. Miller, he characterized the minimal time required for controllability at zero. By considering in addition a singular potential, he proved the approximate controllability for adequate transmission conditions.
F. Boyer has contributed to the study of the exact or approximate controllability of equations or polar systems and/or their discretization. Among the outstanding results, we can note the identification of geometrical conditions on the control domain for the approximate controllability of coupled parabolic systems (in collaboration with G. Olive). He also continued the development of discrete Carleman estimates and their applications.
L. Cardoulis, who has just moved to the team in 2015, has worked on an inverse problem for the Schrödinger operator in an unbounded band with a stability result for two coefficients, one of which is time dependent, with P. Gaitan. The study of the reconstruction of the curvature of an unbounded waveguide has been done with M. Cristofol.
M. Cristofol treated the case of the reconstruction of coefficients for nonlinear parabolic operators with or without memory term and addressed the case of linear and nonlinear parabolic systems. In addition to the collaborations already mentioned, he used techniques from Carleman’s inequalities with P. Gaitan, K. Niinimaki, O. Poisson, L. Roques and M. Yamamoto. He has developed with L. Roques, in dimension one, a new approach based on minimal point-like observations in space for parabolic inverse problems. He worked on inverse problems of reconstruction of coefficients related to the Maxwell operator with M. Bellassoued, L. Beilina, K. Niinimaki and E. Soccorsi. He studied the case of an infinite hyperbolic waveguide with S. Li and E. Soccorsi and is currently interested in the generalization of this result to the case of time-dependent coefficient reconstruction with the numerical simulations of L. Beilina. He studied with M. Bellassoued the problem of final overdetermination for the reconstruction of a source in a parabolic system. He works, with L. Roques, on the reconstruction of the drift term in the general formulation of a diffusion problem and on the reconstruction of the volatility in a Black and Scholes type problem with M. Bellassoued, R. Brummelhuis and E. Soccorsi.
O. Poisson is working in collaboration with H. Isozaki on several inverse problem projects, including a first project on the spectral analysis of a Maxwell operator on a lattice and another on the reconstruction of a moving object for the wave operator. He has published a work which is an extension of results of Calderon’s problem for the heat equation with moving inclusion.
Homogenization, high frequency oscillations, singular perturbations (M. Bostan, O. Guès, A. Sili)
M. Bostan worked on the homogenization of transport equations. The multiscale analysis was based on the use of ergodic mean operators. This led, in particular, to a two-scale approach, with fast variable not necessarily periodic. These techniques have also allowed the treatment of parabolic problems with strongly anisotropic scattering and a theoretical study of the spectral properties of the averaged scattering matrix field. Some interesting results have been obtained in a nonlinear framework (Vlasov-Poisson system): strong convergence for not necessarily well prepared data, study of invariants, Hamiltonian structure of the limit model. This is the content of a thesis financed by the region, field of strategic activities. The perspectives aim at models with curved 3D confinement field, and the consideration of more realistic scaling laws (quasi-neutrality). Numerical simulation is also among the objectives.
A. Sili has continued his research work in this theme with various collaborators, leading to eight publications and the defense of Charef Hamid’s thesis (end of 2012).
O. Guès worked in collaboration with J.-F. Coulombel (univ. de Nantes) and M. Williams (Univ. North Carolina, USA) on the propagation and reflection of high frequency waves against a shock wave in multi-D (completing a previous work of M. Williams) under strong stability assumptions, then against an edge for weakly stable boundary conditions, highlighting the amplification of the wave, for semi-linear hyperbolic systems. These results were obtained thanks, among other things, to the precise study of a singular pseudodifferential calculation with small parameters, introduced in 2002 by M. Williams.
O. Guès, in collaboration with G. Métivier, M. Williams and K. Zumbrun, has been interested in the convergence and boundary layer problems posed by the low viscosity perturbation of a multidimensional hyperbolic system in the vicinity of a fixed edge and under a Neumann (or mixed Dirichlet-Neumann) condition, obtaining convergence in some favorable cases.
T. Auphan, in the framework of his thesis work, was interested in the theoretical problem of approaching a multidimensional symmetric hyperbolic quasilinear mixed problem of order 1, with dissipative boundary conditions by a fictitious domain volume penalization: he showed that there exist penalizations generating no spurious boundary layer at the interface, at any order, when the penalization parameter tends to 0, and he proved the convergence.
Mean field limits and kinetic models (M. Hauray, A. Nouri)
M. Hauray has worked on mean field limits and propagation for different models: 3D Vlasov equations with moderately singular interaction force, 1D Vlasov-Poisson equation with or without noise, swarm models (Vicsek), 2D Navier-Stokes equation approximated by vortex systems, 3D Landau equation with moderately soft potential. He has also been interested in stability problems of stationary profiles for plasmas in the quasi-neutral limit.
A. Nouri’s research activities are focused on applied mathematical problems modelled by kinetic equations:
– Tokamak core plasma where a Vlasov equation for the ion distribution function is coupled to the quasi-neutrality equation ;
– some quantum kinetic equations, such as the one modeling anyons;
– evolution of Bose-Einstein condensates, modeled by a Schrödinger equation of Gross-Pitaevskii type, in the middle of a gas of excitations, described by a quantum kinetic equation.
She collaborates with P. Ghendrih (IRFM of CEA Cadarache), P.-E. Jabin (University of Maryland) and C. Bardos on the study of tokamak core plasmas. With L. Arkeryd from Chalmers, Göteborg, she studies quantum kinetic equations, and with C. Schmeiser from Vienna she is interested in a chemotactic problem.
Fluid mechanics (T. Gallouët, S. Monniaux, N. Nadirashvili)
T. Gallouët continued his work on fluid mechanics problems, which led to 4 publications on the existence of solutions for elliptic problems, on the existence of solutions for compressible Stokes equations, and on the continuity in time of the entropic solution of an evolution problem (hyperbolic or degenerate parabolic).
S. Monniaux has been interested more particularly in the (linear) Stokes (or Stokes-Coriolis) system with different edge conditions in (bounded or not) lipschitzian domains. The good understanding of the properties of the linear problem allows to obtain, thanks to classical fixed point theorems, global solutions to the Navier-Stokes equations for small initial conditions. Contrary to the case of regular domains, the Stokes operator has very different properties from the vector Laplacian. She obtained elementary proofs of the existence of traces on the ∂Ω edge of functions in H1(Ω), and of integrable square u-vector fields whose divergence and rotationel are integrable square and which have a normal trace or zero tangential trace on the ∂Ω edge (one published conference proceedings and one report on these topics).
N. Nadirashvili proved that the only three-dimensional Beltrami flow of finite energy is trivial.
Propagation phenomena for reaction-diffusion type evolution PDEs (G. Chapuisat and F. Hamel)
Wave propagation phenomena are one of the most important aspects of reaction-diffusion models. W. Ding, J. Garnier, T. Giletti and F. Hamel, with H. Berestycki, M. El Smaily, J.-S. Guo, R. Huang, G. Nadin, J. Nolen, J.-M. Roquejoffre, L. Roques, L. Rossi, L. Ryzhik, Y. Sire, X. Zhao and A. Zla- toš, have studied the existence and qualitative properties of pulsating fronts in periodic media or generalized transition fronts, a notion introduced by H. Berestycki and F. Hamel unifying the known notions and constituting the natural mathematical framework for the study of space-time dynamics in complex heterogeneous media. Examples of propagation with multiple or infinite velocities have been highlighted. The thesis of H. Guo, under the supervision of F. Hamel, deals with the stability of transition fronts. Multiple research directions concern in particular problems with non-local dispersion or competition, work which is being done in the framework of the ANR NONLOCAL project.
G. Chapuisat in collaboration with H. Berestycki and J. Bouhours has studied the propagation of bistable fronts as a function of the geometry of the domain where they propagate.
Qualitative properties for local and non-local elliptic PDEs (T. Gallouët, F. Hamel, N. Na- dirashvili, P. Sicbaldi and Y. Sire)
F. Hamel, N. Nadirashvili and Y. Sire have constructed the first examples of positive solutions of semi-linear elliptic equations in convex domains with Dirichlet condition at the edge and having non-convex level sets, thus answering in the negative a conjecture of P.-L. Lions dating from 1981. M. Efendiev and F. Hamel have in particular used the dynamical systems approach to M. Efendiev and F. Hamel have used the dynamical systems approach to obtain results on the asymptotic behavior of solutions of elliptic equations in unbounded greens. Finally, one-dimensional symmetry results have been shown by D. Bonheure, F. Hamel, X. Ros-Oton, Y. Sire and E. Valdinoci for equations with non-local dispersion or for elliptic equations of order 4. The study of harmonic applications, dispersive effects and regularity theory has been analyzed by Y. Sire and his collaborators.
T. Gallouët has published two papers on existence results for elliptic problems.
P. Sicbaldi studied overdetermined elliptic problems in unbounded domains of Euclidean space. A very deep and surprising connection with minimal surfaces and surfaces with constant mean curvature allowed to find counterexamples to a Berestycki-Caffarelli-Nirenberg conjecture of 1997, and to reformulate the open questions for a possible classification. In particular, in the plane he obtained a general classification result for solutions on domains diffeomorphic to the half-plane. Work in collaboration with F. Schlenk, A. Ros and D. Ruiz.

Numerical analysis
Controllability of discretized parabolic problems (F. Boyer, F. Hubert)
Controllability results have been obtained for semi-discrete parabolic problems (in space) but also completely discretized in time and space. Time error estimates for the computation of the control on these problems have also been obtained. An essential tool underlying these results is the proof of discrete Carleman inequalities (F. Boyer, F. Hubert, collaboration with J. Le Rousseau).
Numerical methods for elliptic and parabolic equations (Ph. Angot, F. Boyer, T. Gal- louët, O. Guès, R. Herbin, F. Hubert, J. Liandrat)
Many works have been done for the development of the method called Discrete-Duality-Finite-Volume (DDFV), for example, for the development of the method for heterogeneous anisotropic operators (F. Hubert, N. Hartung, collaboration with B. Andreianov and Y. Coudière), in particular in 3 space dimensions, and for the Stokes problem (F. Boyer, F. Nabet, collaboration with S. Krell)
New discrete Schwarz methods have been proposed for Ventcell transmission conditions in the case of isotropic convection-diffusion operators (F. Hubert, in collaboration with L. Halpern) as well as in the case of anisotropic operators (F. Hubert in collaboration with L. Halpern, M. Gander, S. Krell) using DDFV methods.
A new numerical method, gathering different methods (non-conformal finite elements, gradient schemes, mimetic methods, gradient schemes, MPFA schemes…), called “Gradient Dis- cretization Method” (GDM) has been introduced for the discretization of elliptic and parabolic PDE. The interest of this method is to give a theoretical framework allowing the analysis of many schemes for linear and nonlinear PDEs, including for example a complete analysis of mass lumping. A book on MDM should be published soon (T. Gallouët, R. Herbin, collaboration with J. Dro- niou, R. Eymard, C. Guichard). Some of these schemes have been analyzed and implemented numerically for example for image processing or for flows in anisotropic heterogeneous porous media (R. Herbin, in collaboration with R. Eymard, K. Mikula, A. Handlovicova, O. Stavsovà, R. Masson, C. Guichard)
Methods coupling wavelets and dummy domains for the solution of elliptic problems in dimension 2 have been analyzed (J. Liandrat, in collaboration with P. Yin).
New methods have been developed for flow problems in porous media and for the analysis of interface and boundary layer conditions, possibly coupled with a free medium (Ph. Angot, C. Zaza, in collaboration with G. Carbou and V. Péron).
An asymptotic preserving method has been developed for nonlinear anisotropic elliptic problems appearing in tokamak modeling (Ph. Angot, T. Auphan, O. Guès).
In the context of tokamak modeling, a contribution to the penalization method for dealing with Robin-type boundary conditions has been made (J. Liandrat, in collaboration with G. Chiavassa and B. Bensiali).
High order numerical schemes have been developed for convection-diffusion equations. Their interest is to preserve the positivity of the approximated solutions on general meshes for convection and “admissible” for diffusion (R. Herbin, in collaboration with F. Babik, J.-C. Latché, B. Piar).
Nonlinear subdivision schemes (J. Liandrat)
Nonlinear subdivision schemes, obtained by nonlinear perturbation of linear schemes, have been introduced and analyzed (J. Liandrat, in collaboration with S. Amat and J. Ruiz). New schemes derived from stochastic prediction methods have been defined and analyzed (J. Liandrat, in collaboration with J. Baccou and X. Si).
Incompressible fluid mechanics (Ph. Angot, F. Boyer, T. Gallouët, R. Herbin)
Different methods for solving the incompressible Navier-Stokes equations have been introduced, splitting method, penalization method or a very original method of “vector” pressure correction (Ph. Angot, R. Cheaytou, collaborations with J.-P. Caltagirone, P. Fabrie, P. Minev). These methods are designed to be robust to large ratios of density, viscosity, permeability or open boundary conditions.
The stability of the Crank-Nicholson scheme with pressure correction has been shown (F. Boyer, F. Dar- dahlon, in collaboration with J.-C. Latché and C. Lapuerta).
Convergence results of the MAC scheme for incompressible Navier-Stokes equations (with possibly variable density) have been obtained (R. Herbin, T. Gallouët, K. Mallem, in collaboration with J.-C. Latché).
A numerical method for the solution of the 3-component Cahn-Hillard equation has been developed and analyzed (F. Boyer, S. Minjeaud)
Hyperbolic equations and systems (Ph. Angot, O. Guès, R. Herbin, M. Tournus)
A preserving asymptotic scheme has been developed for a hyperbolic system with a relaxation term (M. Tournus, in collaboration with N. Seguin).
In the context of tokamak modeling, a penalization method has been introduced and analyzed to deal with the boundary conditions problem (Ph. Angot, T. Auphan, O. Guès).
An error appearing in some papers on road traffic modeling has been highlighted and corrected (R. Herbin, in collaboration with L. Leclercq).
Transport equations (F. Boyer, T. Gallouët, R. Herbin)
The demonstration of the convergence of the upwind scheme for linear transport equations with an irregular transport field (DiPerna-Lions hypothesis) has been obtained, for the problem without boundary condition and with boundary condition (F. Boyer, A. Fettah, T. Gallouët, R. Herbin).
In the framework of a collaboration with the CEA, the development and the study of a numerical scheme for a system of transport equations have been carried out (D. Fournier, R. Herbin, in collaboration with R. LeTellier).
Compressible fluid mechanics (T. Gallouët, R. Herbin)
A proof of convergence of numerical schemes for stationary compressible Stokes equations has been given. It was then generalized for stationary or semi-stationary compressible Navier-Stokes equations. In the case of evolutionary Navier-Stokes equations, the adaptation of the so-called “strong-weak uniqueness” method has allowed to give error estimates between approximate solution and exact solution when the latter is sufficiently regular (T. Gallouët, R. Herbin, collaborations with R. Eymard, J.-C. Latché, D. Maltese, A. Novotny).
A series of works has been devoted to the study of new numerical schemes for the Euler or Navier-Stokes compressible type equations (barotropic or not, with possibly the addition of diffusion terms). These schemes do not use a Riemann solver and can be used at any Mach. The time discretizations can be implicit (with resolution by pressure correction) or partially implicit with a time step limitation depending only on the material velocities. Spatial discretizations use staggered or co-located meshes. An important originality of these methods consists in being able to discretize non-conservative equations (such as the internal energy equation) but keeping the correct velocities of the discontinuities. The good behavior of these methods is shown by numerical tests and by convergence demonstrations (consistency “in the sense of Lax”, R. Her- bin, W. Kheriji, J.-C. Latché, T.T. Nguyen, C. Zaza). A generalization to quasi-second order schemes has also been developed (R. Herbin, N. Therme, in collaboration with L. Gastaldo, J.-C. Latché).
Numerical methods have been developed and tested for the resolution of combustion models in multiphase media (D. Grapsas, R. Herbin, N. Therme, in collaboration with J.-C. Latché).
Stochastic PDE (J. Charrier, T. Gallouët)
Results have been obtained on the quantification of uncertainties for advection- diffusion problems (J. Charrier).
A series of works has been carried out on the convergence of numerical methods for the approximation of hyperbolic equations (in several space dimensions) with a stochastic source term of the “multiplicative noise” type (J. Charrier, T. Gallouët, C. Bauzet in the framework of a post-doctoral position).

Modeling and interactions with other disciplines
Interactions with physics (M. Bostan, Y. Dermenjian, C. Gomez, M. Hauray, J. Liandrat, J. Oli- vier, A. Nouri, K. Saikouk, M. Tournus)
M. Bostan has worked on the analysis of PDEs modeling tokamak plasmas: behavior in the presence of an intense stationary magnetic field (particles of disparate masses, treatment of col- lisions) or of a magnetic field strongly oscillating in time. A PhD thesis has been devoted to collisional gyro-kinetic models (relaxation, Fokker-Planck, Fokker-Planck-Landau). A second thesis is being finalized on the Vlasov-Poisson system in the finite Larmor radius regime. In parallel, he has obtained convergence results for parabolic problems with strongly anisotropic diffusion (thesis in progress, application to tokamak plasmas) and for population dynamics (swarming).
Y. Dermenjian studied guided waves of a biperiodic medium in an unbounded 3D domain, in collaboration with F. Bentosela, C. Bourrely and E. Soccorsi from CPT.
C. Gomez was interested in the phenomenon of wave propagation in random media with long-range correlations: with O. Pinaud, they provided a proof of the paraxial wave approximation with fractional white noise; with L. Ryzhik, they studied the radiative transport equation for which the collision kernel is singular (regularizing effect, scattering limits and peak forward).
In collaboration with C. Negulescu and R. Adami, M. Hauray worked on simplified models for the environment-induced quantum decoherence.
M. Hauray studied the Kac energy exchange model in dimension 1 and showed a uniform contractivity (in number of particles) in an adapted Wasserstein metric.
M. Hauray and A. Nouri studied the well-posedness of a quasi-neutral gyro-kinetic Vlasov equation in dimension 2: a model used in fusion plasmas, for example in the GYSELA code developed at CEA.
J. Liandrat works on the modeling of tokamaks. In particular, numerical simulations of particle tracking in a flow in the vicinity of walls, using interpolation techniques.
In particular, numerical simulations of particle tracking in a flow near walls, using interpolation techniques based on subdivision schemes have been carried out (in collaboration with P. Ghendrih and G. Ciraolo (CEA) and in the framework of the thesis of B. Bensiali).
In collaboration with P.-E. Jabin, A. Nouri has shown the well-posedness, local in time, in an analytical framework of a Vlasov equation coupled to the quasi-neutrality equation modeling the evolution of a tokamak plasma in the direction parallel to the magnetic field lines. With C. Bardos she showed that this system is ill-posed in the Hadamard sense in any Sobolev space.
A. Nouri works with L. Arkeryd on quantum kinetic equations. They have solved the Cauchy problem for an equation modeling anyons. They have also studied the interaction of Bose-Einstein condensates with a quasiparticle gas in a near equilibrium setting and determined the asymptotic behavior of such a system.
In collaboration with K. Martens, F. Puosi and E. Agoritsas (LiPhy – Grenoble), J. Olivier explored and tested the foundations of the mechanics of soft glassy materials. They have developed theoretical, numerical and physical tools to understand, compare and measure various behavioral scenarios in order to validate or invalidate certain theories.
K. Saikouk collaborates with J. Léchelle of the LLCC Laboratory of CEA Cadarache on the modeling and numerical simulation of nuclear fuel sintering. From a numerical point of view, this translates into the study of the evolution of interfaces between grains under mechanical and chemical effects.
M. Tournus is working with I. Aronson (Argonne National Laboratory, IL) and L. Berlyand (Penn State University, PA) on a model of an isolated microswimmer, in particular on the role of the flagellum in navigation and in collisions with solid walls.
Invasion models in ecology (F. Hamel, M. Henry)
J. Garnier, T. Giletti and F. Hamel, with O. Bonnefon, J. Coville, E. Klein and L. Roques, have studied the internal dynamics of biological invasion fronts for reaction-diffusion equations, which are well adapted to the description of certain ecological phenomena of colonization and dispersal of plants or animals. In particular, they have shown the positive role of the Allee effect on the preservation of diversity during a colonization. H. Berestycki, F. Hamel and L. Roques are supervising the thesis of M.-E. Gil on selection-mutation models in population genetics, in the framework of the ANR NONLOCAL project.
M. Henry is interested in traveling waves associated with reaction-diffusion systems, in the formation of interfaces and spatio-temporal patterns and in the dynamics of these interfaces.
Models in oncology (A. Benabdallah, G. Chapuisat, C. Gomez, F. Hamel, F. Hubert, J. Olivier, M. Tournus)
A. Benabdallah and F. Hubert have been working since 2007 on the modeling of the metastatic process. In collaboration with D. Barbolosi, they have supervised two theses (F. Verga and S. Benzekry) on the modeling of the impact of anti-cancer drugs on this progression using transport type equations. M. Henry and A. Benabdallah studied a viscosity approximation of these models. Still in collaboration with D. Barbolosi’s group, at the Center of Oncobiology and Oncopharmacology (CRO2), F. Hubert proposed a preclinical validation of the metastasis model (thesis of N. Hartung co-directed by G. Chapuisat and F. Hubert). This project was supported by the ANR (ANR MEMOREX) and the 2009-2013 Cancer Plan.
N. Hartung and F. Hubert collaborated with the team of Pr J. J. Grob’s team on the identification of efficacy indicators of anti-BRAF treatments in metastatic melanoma.
In an AMIDEX NOVUSBIO project led by E. Francescini of the Mechanics and Acoustics Laboratory, G. Chapuisat, N. Hartung and F. Hubert have started a work on the modeling of tumor growth in mice based on observations from SPECT and ultrasound imaging. N. Har- tung and F. Hubert proposed an algorithm, based on DDFV, to detect the contours of observed tumors.
Since 2013, F. Hubert and C. Gomez have been working, in collaboration with S. Honoré’s group (CRO2), on the dynamic instabilities of microtubules that result in alternating phases of poly- merization and depolymerization. They have obtained the support of A*MIDEX and the 2013-2017 Cancer Plan (PHARMATOTUBULE project). A. Barlukova in her thesis funded by LABEX ARCHIMEDE and supervised by F. Hubert and S. Honoré, uses deterministic models, discretized by finite vo- lumes techniques to account for the complexity of these dynamics and the actions of anti-microtubule chemotherapies. C. Gomez uses a stochastic approach to model this phenomenon. F. Hubert in collaboration with M. Tournus and D. White, post-doctoral fellow recruited on the project, proposed a new approach to depolymerization based on fragmentation equations.
In another context, this same fragmentation equation has been studied by M. Tournus in collaboration with M. Doumic (INRIA Roquencourt) and M. Escobedo (UPV). They are interested in the estimation (inverse problem) of the kernel and the fragmentation rate. A collaboration is in progress with W.F. Xue (School of Bioscience, UK) to apply their method to real data of proteins involved in neurodegenerative diseases.
In collaboration with O. Theodoly from the Adhesion and Diffusion laboratory in Marseille, F. Hubert and J. Olivier proposed at CEMRACS 2015 a study of cell migration in confined environments.
In the framework of C. Carrère’s thesis, A. Benabdallah and G. Chapuisat are working on the modeling of the growth of a heterogeneous tumor composed of chemotherapy resistant or sensitive cells from M. Carré’s in vitro experiments (CRO2). The optimization of chemotherapy leads to a complex optimal control problem.
Under the supervision of G. Chapuisat and F. Hamel, B. Contri studies medical models of spatial growth under the effect of periodic treatments.

Academic influence and attractiveness

Networks and institutional contracts
Members of the team are involved in many national and international projects.
– Projects led by members of the team
– ANR Blanc ANR-09-BLAN-0217-01 “MEtastases MOdeling and Reseach in EXperimental PharmacoKinetics” ( 2012-2013), coordinator : F. Hubert, I2M members : A. Benabdallah, G. Chapuisat,
– INSERM cancer plan 2009-2013, coordinator: F. Hubert, I2M members: A. Benabdallah, G. Chapuisat, N. Hartung,
– ANRNONLOCAL (ANR-14-CE25-0013) “Défi de tous les savoirs”,2014-2018. Coordinator: F. Hamel, I2M members: J. Brasseur, C. Carrère, G. Chapuisat, B. Contri, W. Ding, M.-E. Gil, H. Guo, F. Hamel, N. Nadirashvili and Y. Sire,
– INSERM PharMathTubules project, collaboration between CRO2 and I2M laboratories, 2013-2017, coordinator : F. Hubert, I2M members : A. Barlukova, C. Gomez, R. Tesson, D. White,
– PHC Tassili-2011no24296TE/11MDU834(2011-2014), coordinator: A.Benabdallah, I2M members: M. Cristofol, F. Boyer, Y. Dermenjian, P. Gaitan, F. Hubert and O. Poisson.
– Projects in which members of the team are involved
– ANR Blanc PREFERED (2008-2012), coordinator : J.-M. Roquejoffre (Toulouse), I2M members: F. Hamel, N. Nadirashvili, E. Russ and Y. Sire,
– ANR Blanc VFSitCom (2008-2012), coordinator: J.Droniou, I2M members:F.Boyer,R.Her-
bin and F. Hubert,
– ANR EMAQS (2012-2016) coordinator: K. Beauchard, I2M member: M. Morancey,
– ANR Geometrya (2013-2017), coordinator : H. Pajot, I2M member : P. Sicbaldi,
– ANR Harmonic Analysis at its Boundaries, coordinator : P. Auscher, I2M member : S. Monniaux,
– ANR INFAMIE, coordinator : R. Danchin, I2M member: S. Monniaux,
– ANR-08-BLANC-0335 CAGE (2009-2012), coordinator: F. Pacard (Créteil), I2M member: P. Sicbaldi,
– AMIDEX PHARMATOTUBULE, collaboration between CRO2 and I2M (2013-2015), coordinator: S. Honoré, I2M members: A. Barlukova, C. Gomez, F. Hubert, R. Tesson and D. White,
– GDR MOMAS (Mathematical Modeling and Numerical Simulations related to nuclear waste management problems) (2007-2012), coordinator: A. Ern (ENPC), I2M members: F. Boyer, T. Gallouët, G. Henry, R. Herbin and F. Hubert,
– GDR EGRIN (Modeling and numerical simulations Gravity flows and Natural Risks) (2015-2019), coordinator: C. Lucas, I2M members: Ph. Angot, T. Gallouët and R. Herbin,
– GDR MANU (Mathematics for nuclear energy) (2015-2019), coordinator: C. Cancès, I2M members: T. Gallouët, R. Herbin and F. Hubert,
– GDR CATIA (Control and Analysis of PDEs, Theory, Interactions and Applications) (2014-2018), coordinator: K. Beauchard, I2M members: A. Benabdallah, F. Boyer, M. Cristofol, Y. Dermenjian, P. Gaitan, F. Hubert, M. Morancey and O. Poisson,
– ERC ReaDi (ERC Grant Agreement n.321186), 2013-2017, coordinator: H.Berestycki, I2M members: G. Chapuisat and F. Hamel,
– GDRE Geometric Analysis (France-Spain), coordinators: P. Romon (Marne-la-Vallée) and J. Perez (University of Granada, Spain), I2M member: P. Sicbaldi,
– GDRE CONEDP (2009-2017), coordinator: F. Alabau, I2M members: A. Benabdallah, F. Boyer, M. Cristofol, Y. Dermenjian, P. Gaitan, F. Hubert, M. Morancey and O. Poisson,
– Analysis and control of PDEs with origin in physics and other sciences (2010-2014) (MTM2010- 15592), funded by the Spanish Ministry of Science and Innovation, coordinator: E.F. Cara, I2M member: A. Benabdallah,
– Franco-Brazilian research project SURFACES, ANR-11-IS01-0002 (2012-2015), coordinator: L. Hauswirth (Marne-la-Vallée), I2M member: P. Sicbaldi,
– GDRI Euro-maghrebian of mathematics and their interactions (2014-2018), coordinator: G. Lebeau, I2M members: A. Benabdallah, Y. Dermenjian and A. Sili,
– GDRI ReaDiNet (Korea-France-Japan-Taiwan,2014-2018), coordinator: D.Hilhorst, I2M members: G. Chapuisat, F. Hamel and M. Henry,
– LIA LAISLA (French-Mexican, 2009-2016), coordinators: H. Short and J. Seade, I2M members of the applied analysis team: A. Benabdallah.

Scientific responsibilities
Many members of the Applied Analysis team are involved in the collective tasks of organizing the research. In particular:
– A. Benabdallah is in charge of the AA team since September 2015 and a member of the I2M office.
– R. Herbin is the director of the I2M laboratory since July 2015 and was the AA team leader from 2012 to 2015.
– F. Hamel is director of the LabEx Archimède since 2014 (ANR-11-LABX-0033, this Labex gathers 5 units in mathematics and computer science). He was in charge of the ex-LATP at the ex-University of Aix- Marseille III until 2011.
– O. Guès was co-director of the mathematics department from 2012 to 2014.
– J. Liandrat was Director of Research (VP Scientific Council) at Centrale Marseille from 2010 to March 2016.
– M. Cristofol is a member of the research commission of the IUT of Aix-Marseille.
– F. Boyer and A. Nouri have been members of the UFR sciences board since 2012, F. Boyer was a member of the department board office from 2012 to 2015, and M. Bostan has been a member of the mathematics department board since November 2012 and the office since September 2015.

Expertise
Many members of the team have performed expert activities for various research bodies. Among them, we can mention :
– F. Hamel is or has been an expert, depending on the case, for the ANR, BIRS (Canada), ECOS-Sud (Chile), the ERC, Fondecyt (Chile), NSERC (Canada), RGC (Hong-Kong), for the research investment grant at the Université Pierre et Marie Curie, for international bilateral PICS projects in the fields of EDP and Scientific Computing. He was a member of the national commission in charge of the evaluation of the applications to the P.E.S. in mathematics in 2011 and 2012 and he was referent of the mathematics-computing disciplinary field for the COS of the University of Aix-Marseille in 2015.
– G.Chapuis is a member of the “Mathematics, Bioinformatics, Artificial Intelligence” Specialized Scientific Commission of INRA since 2015.
– R. Herbin was a member of the HCERES committee for the expertise of the IMB (Institut de Mathématiques de Bordeaux) in 2015, and of the AERES committee for the expertise of IRSTEA (ex-Cemagref) in 2013, and a member of several project evaluation committees (Germany, Spain, Czech Republic). She is also a member of the teaching commission of the SMAI.
– F. Hubert was a member of the ANR committee in 2012 and 2013. She is in charge of the scientific calculation option of the Aggregation of Mathematics, she has been a member of the scientific council of the GDR METICE since 2016 and was elected in 2015, vice-president of the SMAI in charge of communication and public actions.
– Cristofol has been an appointed member of CNU 26 since November 2015.
– M. Bostan has been a member of the AMU thesis commission since June 2013, of the board of the of the AMU-Entreprise development council and is a corresponding member of the Pôle des Recherches Intersectoral and Interdisciplinary Research Cluster (PR2I) – Energy, since September 2013.
Many members of the team participate each year in selection committees for positions AMU and outside AMU.

Editorial Boards
Members of the Applied Analysis team serve on editorial boards of journals including: Computational and Applied Mathematics, International Journal of Finite Volumes and ESAIM Proc., SIAM Journal On Uncertainty Quantification, Boletin de la Sociedad Matemática Mexicana, Kinetic and related Models, Taiwanese Journal of Mathematics, Tamkang Journal of Mathematics (since 2013)…

Organization of scientific events
Members of the team are involved in the organization of many international conferences including the following:
– In the context of the FVCA6 conference, R. Herbin and F. Hubert participated in the organization of a 3D performance evaluation on approximations of diffusion problems on general meshes (3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on Gene- ral Grids, R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klofkorn and G. Manzini, Proceedings of Finite Volumes for Complex Applications VI, Praha, p.895 -930, 2011).
– Conference “Kinetic equations” in November 2014 at CIRM, organized by M. Bostan, M. Hauray and A. Nouri.
– Organized several international conferences at CIRM including: Current challenges of mathematics in cancer medicine and biology: modeling and mathematical analysis, Control, New challenges of mathematics in oncology and cancer biology, etc.
– Organization of the international colloquium of LE2MI, GDRI Euro-Maghreb of mathematics and their interactions, which is an international research grouping associated with the CNRS, which includes more than thirty-five laboratories from the Maghreb and France.
– Conference “Fronts and Nonlinear PDEs” in honor of H. Berestycki, Ecole Normale Supé- rieure, Paris, June 20-24, 2011 (about 200 participants) of which F. Hamel was one of the organizers.
– Conference “France-Taiwan Joint Conference on Nonlinear Partial Differential Equations”, Tai- pei, Taiwan, October 21-25, 2013, organizers: H. Berestycki, J.-S. Guo, F. Hamel, C.-S. Lin, B. Perthame and H.-T. Yau.
– Organization of Canum 2014 (http ://smai.emath.fr/canum2014/).
– Conference on Waveguides 2016, Porquerolles, May 17-19, 2016, organizers A.-S. Bonnet- Bendhia, Ph. Briet, M. Cristofol and E. Soccorsi.
– Conference “Reaction-Diffusion Equations and Applications”, Renmin University, Beijing, China, May 26-29, 2016, organizers: S. Cantrell, F. Hamel, Y. Lou, F. Lutscher and E. Yanagida.
Thematic days and mini-colloques are organized at a rate of two or three per year:
– Around kinetic problems (June 2011, talks by N. Crouseilles, T. Goudon, J. Barré).
– Analysis day (November 2011, mini-course by L. Saint-Raymond and A. Chambolle).
– Computation of Variations Day (December 2011, lectures by F. Santambrogio, G. de Philippis, Y. Sire and A. Blanchet).
– Optimal transport and applications in analysis and DPE and numerical schemes Asymptotic Preserving for Multiscale Kinetic Equations (May 2012, mini-course by F. Bolley and M. Lemou).
– Microscopic and macroscopic modeling of crowd movements and Multiscale modeling for hydrology and erosion models (October 2012, mini-course by B. Maury and S. Cordier).
– Recent progress in Nonlinear Analysis (December 2012, mini-course by B. Ruf and T. Weth).
– Hyperbolic models for fluids and numerical schemes (December 2013, two-day colloquium day symposium, lectures by A. Novotny, D. Doyen, S. Gavrilyuk, J. Sainte-Marie, A. Beccantini, M. H. Vignal, C. Berthon, S. Minjeaud, N. Seguin, and posters by T. Auphan, R. Cheaytou, D. Maltese, F. Nabet, N. Therme, C. Zaza).
– Colloquium Inverse problems and control in the framework of the GOMS working group (November 2014, presentations by M. Bellassoued, E. Bonnetier, D. Dos Santos Ferreira, J. Garnier, H. Isozaki, M. Mo- rancey, A. Munch, M. Yamamoto).
– Thematic days on stochastic conservation laws: theory, numerical analysis and applications (June 2015, talks by S. Boyaval, J. Charrier, A. Debussche, G. Vallet, J. Vovelle, P. Wittbold and A. Zimmermann).
– Optimization and control (December 2014, presentations by M.Caponigro, C.Laurent, P.Lissy, F.Rossi).
– Stability of periodic solutions for nonlinear equations (March2015, talks by S.Benzoni, F. Chardard, P. Noble, M. Rodrigues).
– Colloquium Inverse problems and associated domains in the framework of the GOMS working group (december 2015, presentations by G. Alberti, L. Oksanen, F. Triki, M. Yamamoto)
– Modeling: Mathematics and Realities (March 2016, presentations by A. Barberousse, J. Bouhours, C. Carrère, P. Gabriel, F. Givors, R. Tesson).
Thematic meetings of applied analysis, http ://champion.univ-tln.fr/NTM/NTM2016.html, now common with the AA teams of the mathematics laboratories of Nice and Toulon, are organized every year. For example, in June 2013, there were talks by T. Auphan, J. Charrier, D. Clamond, A. Dragoul, M. Ersoy, R. Eymard, D. Esslé, I. Lucardedi, S. Minjeaud, E. Parini, M. Ribot, J. Shneider and F.Sueur.
Moreover, in 2015, F. Hubert and J. Olivier supervised a project at CEMRACS, funded by the Pharmatotubules project.

Awards
– N. Nadirashvili was awarded the Gay-Lussac Humboldt Prize (Académie des Sciences) in 2013,
– J. Garnier received the 2012 thesis prize of the University of Aix-Marseille (thesis directors: F. Hamel and L. Roques),
– F. Hamel was a junior member of the Institut Universitaire de France from 2009 to 2014,
– F. Hamel was included in the list of “Highly Cited Researchers” (Thomson Reuters) in 2014,
– G. Romani is a laureate of the “Académie d’Excellence Collège Doctoral” program of A*MIDEX (thesis 2014-2017, co-directed by F. Hamel, E. Parini and B. Ruf in co-supervision with the University of Milan),
– F. Boyer is a junior member of the Institut Universitaire de France since 2016.

Invited lectures in international conferences (since 2011)
Team members have given more than 100 invited talks, of which the top 2 per person are: