Section: New Results

Mathematical and numerical analysis

Participants : Nicolas Besse, Mihai Bostan, Nicolas Crouseilles, Sever Hirstoaga, Simon Labrunie, Sandrine Marchal, Thomas Respaud, Jean Roche, Eric Sonnendrücker.

We have established several existence and uniqueness results for collisionless kinetic models, the Vlasov-Poisson and Vlasov-Maxwell equations of plasma physics but also the Nordström-Vlasov equations used in astrophysics. We also investigated different asymptotic regimes for the Vlasov-Maxwell equations. Finally, this section includes recent results concerning the convergence of the numerical solution towards the solution of the model (Maxwell or Vlasov).

Asymptotic regimes and existence results for the Vlasov-Poisson and Vlasov-Maxwell equations

In [20] we study a special type of solution for the one dimensional Vlasov-Maxwell equations. We assume that initially the particle density is constant on its support in the phase space and we are looking for solutions with particle density having the same property at any time t>0 . More precisely, for each x the support of the density is assumed to be an interval [p_-, p ₊ ] with end-points varying in space and time. We analyze here the case of weak and strong solutions for the effective equations verified by the end-points and the electric field (water-bag model) in the relativistic setting.

We study the existence of weak solutions for the stationary Nordström-Vlasov equations in a bounded domain. The proof follows by fixed point method. The asymptotic behavior for large light speed is analyzed as well. We justify the convergence towards the stationary Vlasov-Poisson model for stellar dynamics [19] , [48] .

The subject matter of [18] concerns the existence of permanent regimes (i.e., stationary or time periodic solutions) for the Vlasov-Maxwell system in a bounded domain. We are looking for equilibrium configurations by imposing specular boundary conditions. The main difficulty is the treatment of such boundary conditions. Our analysis relies on perturbative techniques, based on uniform a priori estimates.

We investigate the well posedness of stationary Vlasov-Boltzmann equations both in the simpler case of linear problems with a space varying force field, and, the non-linear Vlasov-Poisson-Boltzmann system. For the former we obtain existence-uniqueness results for arbitrarily large integrable boundary data and justify further a priori estimates. For the later the boundary data needs to satisfy an entropy condition guaranteeing classical statistical equilibrium at the boundary. This stationary problem relates to the existence of phase transitions associated with slab geometries [22] .

The subject matter of [50] concerns the asymptotic regimes for transport equations with advection fields having components of very disparate orders of magnitude. Such models arise in the magnetic confinement context, where charged particles move under the action of strong magnetic fields. According to the different possible orderings between the typical physical scales (Larmor radius, Debye length, cyclotronic frequency, plasma frequency) we distinguish several regimes: guiding-center approximation, finite Larmor radius regime, etc. The main purpose is to derive the limit models: we justify rigorously the convergence towards these limit models and we investigate the well-posedness of them.

One of the main applications in plasma physics is the energy production through thermo-nuclear fusion. Magnetic confinement controlled fusion requires the confinement of the plasma within a bounded domain using a strong magnetic field. Several models exist for describing the evolution of strongly magnetized plasmas. In [49] we provide a rigorous derivation of the guiding-center approximation in the general three dimensional setting under the action of large stationary inhomogeneous magnetic fields. The first order corrections are computed as well: electric cross field drift, magnetic gradient drift, magnetic curvature drift, etc. The mathematical analysis relies on averaging techniques and ergodicity.

On the other hand, in order to derive a drift-kinetic model, we consider in [54] a new scaling of the Vlasov equation under the hypothesis of large external nonstationary and inhomogeneous electromagnetic field and under the condition of low-Mach number, i.e. when the kinetic energy of the fluid motion is very small in comparison to the thermal energy. To this end, we first make the dimensionless cyclotron period appear in the scaled Vlasov equation. Then we decompose the particle velocity into the mean velocity and its random part and we deduce a system of two equations giving the evolution of the new distribution function and the mean velocity (of the fluid motion). Afterward, an asymptotic analysis is made for this model and a formal derivation of the drift-kinetic model (in a five dimensional phase space) is thus obtained.

We consider the equation H(Du) = H(0), xR^N . More precisely we investigate under which hypotheses the constant functions are the only bounded solutions. In arbitrary space dimension we prove that this happens when convexity and coercivity occur. In one space dimension we show that the above property holds true for Hamiltonians in a larger class. These results apply when studying the long time behaviour of solutions for time-dependent Hamilton-Jacobi equations [23] .

We began [56] the analysis of the qualitative properties of the stationary solutions of the Vlasov-Poisson system in a model 2D singular domain (a polygon with a re-entrant corner). These functions satisfy the system:

(2)

where is a given function, is the potential created by the particles, _e is an external potential, M is the total mass of the particles, and is adjusted to satisfy the mass constraint. The qualitative properties of the solution have been established, with an emphasis on the behaviour of the potential and the density in the neighbourhood of the re-entrant corner. Asymptotics (as a function of the total mass M ) have been obtained in the case of a Maxwellian distribution, i.e. when (s) = exp(-s) . This is the first step towards the analysis of singularities of the time-dependent Vlasov–Poisson and Vlasov–Maxwell systems, though considerable difficulties have to be overcome.

We studied in [57] the existence and uniqueness of solutions to the Vlasov–Poisson system with an initial data of bounded variation. Unlike the works of Cooper–Klimas, Glassey–Schaeffer–Strauss, Guo, ... we do not assume that the initial data (and hence the solution) are bounded, continuous or compactly supported. We were able to prove local existence and uniqueness in dimension 1+1, with an explicit lower bound of the existence time in function of the data. Generalization to higher dimensions is under progress.

Convergence studies of numerical methods

The subject matter of [21] concerns the numerical approximation of reduced Vlasov-Maxwell models by semi-Lagrangian schemes. Such reduced systems have been introduced recently in the literature for studying the laser-plasma interaction. We recall the main existence and uniqueness results on these topics, we present the semi-Lagrangian scheme and finally we establish the convergence of this scheme.

In the paper [58] , we introduced a new class of forward Semi-Lagrangian schemes for the Vlasov-Poisson system based on a Cauchy Kovalevsky (CK) procedure for the numerical solution of the characteristic curves. Exact conservation properties of the first moments of the distribution function for the schemes were derived and a convergence study was performed that applies as well for the CK scheme as for a more classical Verlet scheme. The convergence in L1 norm of the schemes was proved and error estimates were obtained.

On a different topic, we analysed numerically the so-called Fourier–Singular Complement Method for the time-dependent Maxwell equations in an axisymmetric domain [52] . This work completes a series of articles on the numerical solution of the equations of electromagnetism in this type of domain : see [62] for Maxwell's equations in the case of axially symmetric data and [71] for Poisson's equation with arbitrary data. The method relies on a continuous approximation of the electromagnetic field, unlike, e.g., edge element methods. This has many advantages in the case of model coupling, e.g. if the Maxwell solver is embedded in a Vlasov–Maxwell code, either PIC or Eulerian. The symmetry of rotation is exploited by using finite elements in a meridian section of the domain only, and a spectral method in the azimuthal dimension. The analysis also incorporates the approach of [26] , which allows one to handle both: - noisy or approximate data which fail to satisfy the charge conservation equation, as may happen in a Vlasov–Maxwell code - domains with geometrical singularities (non-convex edges and/or vertices) which cause the electromagnetic field to be less regular than in a smooth or convex domain.

Domain decomposition for the solution of nonlinear equations

This is a joint work with Noureddine Alaa, Professor at the Marrakech Cadi Ayyad University. Strongly degenerate parabolic problems have received considerable attentions, and various forms of this problems have been proposed in the literature, especially in the area of reaction-diffusion equations with cross-diffusion, such problems arise from biological, chemical and physical systems. Various methods have been proposed in the mathematical literature to study the existence, uniqueness and compute numerical approximation of solutions for quasi-linear partial differential equation problems.

In this work we give a result of existence of weak solutions for some quasi-linear parabolic and periodic problem and present a method to compute a numerical solution. The algorithm is based on the Schwarz overlapping domain decomposition method, combined with finite element method. In a first step a super-solution is computed. In a second step a weak solution of the nonlinear problem is computed using a Newton method, see [37] . New numerical analysis and simulation results are published in  [13] , [30] .

Mathematical study of water-bag models

The multi-water-bag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations while keeping its kinetic character. Therefore finding water-bag-like weak solutions of the gyrokinetic equations leads to the birth of the gyro-water-bag model.

The paper [17] addresses the derivation of the nonlinear gyro-water-bag model, its quasilinear approximation and their numerical approximations by Runge-Kutta semi-Lagrangian methods and Runge-Kutta discontinuous Galerkin schemes respectively.

In [32] the water-bag concept is used in a gyrokinetic context to study finite Larmor radius effects with the possibility of using the full Larmor radius distribution instead of an averaged Larmor radius. The resulting model is used to study the ion temperature gradient (ITG) instability.

In [16] we derive different multi-water-bag (MWB) models, namely the Poisson-MWB, the quasineutral-MWB and the electromagnetic-MWB models. Then we prove some existence and uniqueness results for classical solutions of these different models.