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Waterbags and related models

We are interested in the mathematical analysis of physical models which consists in showing that they are well-posed, i.e. they admit a unique solution, in a well-suited functional framework. The notion of solution can change according to the nature of the non-linearities in the model. We have to study the particular properties of the solution such as its regularity or its long-time asymptotic behavior. Moreover spectral analysis of the models is very useful since it supplies a lot of information about the behavior of the system such as instability growth rates and spatial structure of the unstable global eigenmodes.
  For example, we have obtained a set of results about the derivation and the mathematical analysis of waterbag-type models in different physical configurations (Besse 2011a, 2011b, 2012; Bardos & Besse 2013, 2015). Waterbag models (Figure 1) come from the exact geometric reduction of the Vlasov equation thanks to geometric Liouville invariants. Their mathematical structures are closely related to hyperbolic systems of conservation laws, in finite or infinite dimension with non-local fluxes.

Figure 1: Schematic representation of phase-space with waterbags. In each of the region with a given color, the phase-space distribution function is approximated by a constant and is conserved during motion as a consequence of Liouville theorem. From Besse & Coulette (2016).

Figure 2: Toroidal geometry (left panel) and the structure of the magnetic field lines (right panels). From Besse & Coulette (2016).

  We have also made the mathematical analysis of the Vlasov-Dirac-Benney model, and show its intrinsic links with plasma physics, fluid mechanics and quantum mechanics (Bardos & Besse 2013, 2015). More precisely, in Bardos & Besse (2016), we have studied the semi-classical limit of an infinite dimensional system of coupled nonlinear Schrödinger equations towards weak solutions of the Vlasov-Dirac-Benney equation, for initial data with analytical regularity in space.
  Regarding to spectral analysis, we have performed a numerical study of the eigenvalue problem for the gyrowaterbag model in cylindrical geometry (Coulette & Besse 2013a, 2013b; Morel et al., 2014). An asymptotic and spectral analysis of the gyrowaterbag integrodifferential operator in toroidal geometry (Figure 2; Besse 2016, 2017; Besse & Coulette 2016) has led to the development of a parallel numerical code whose results are very promising and in good agreement with those of a quasilinear code and standard gyrokinetic-Vlasov code (Coulette & Besse 2017).
  In addition we often have to build and justify new reduced models, starting from original base ones, in order to take into account their own properties. This modeling effort is necessary to reduce the computational cost of original base models while it allows to retain the essential physics of the studied phenomenon.
  For example, using an Eulerian variational principle of least action (Euler-Poincaré), asymptotic perturbation methods and Lagrangian averaging techniques, we have obtained a system of PDEs called « Lagrangian averaged gyrowaterbag continuum » (LAGWBC) designed to accurately capture the dynamics of the turbulent plasma flow at length scales larger than a typical length scale while averaging the motion at scales smaller than this typical length scale. In the isotropic setting, we have shown that this system has unique strong solutions locally in time. Regarding to the original gyrowaterbag continuum, the LAGWBC equations show some additional properties and several advantages from the mathematical (regularizing terms, well-posedness) and physical (modeling of small scales, anisotropic tensor of turbulent fluctuations, dispersion nonlinear terms) viewpoints, which make this model a good candidate for describing accurately gyrokinetic turbulence in magnetically confined plasma (Besse 2016).

We also aim at performing a numerical analysis of the physical models. In other words, we design efficient numerical schemes for solving nonlinear PDEs (kinetic and multi-fluid transport equations, elliptic and waves equations). We also make the mathematical analysis of these numerical schemes, i.e. we prove in a well-suited mathematical framework, the convergence of the approximated numerical solution of the discrete problem towards the exact solution of the continuous problem and obtain a priori error estimates. For instance, we constructed and analyzed several discontinuous-Galerkin numerical schemes which do not increase a priori the total energy for the gyrokinetic-waterbag equations (Besse 2017).

Finally, we are currently developing very high-order schemes for both the 3D incompressible Euler equation in bounded domains and the Vlasov-Poisson equation in the whole space or with periodic boundary conditions. These schemes are based on a new Lagrangian formulation of these equations and make a crucial use of the invariants preserved by the flow. Moreover these numerical schemes give constructive proofs for showing high regularity of the flow launched by initial conditions with moderate smoothness.


  1. C. Bardos, N. Besse, The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits, Kinetic and Related Models, 6 (2013), 893-917.
  2. C. Bardos, N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-Benney equation Fields Inst. Commun. 75 (2015), 1-30.
  3. C. Bardos, N. Besse, Semi-classical limit of an infinite dimensional system of nonlinear Schrödinger equations, Bardos C., Besse N., Bull. Inst. Math. Acad. Sin. 11 (2016), 43-61
  4. N. Besse, On the Cauchy problem for the gyro-water-bag model, Math. Mod. Meth. Appl. Sci. 21 (2011a), 1839-1869.
  5. N. Besse, On the waterbag continuum, Arch. Rational Mech. Anal., 199 (2011b), 453-491.
  6. N. Besse, Global weak solution for the relativistic waterbag continuum, Math. Mod. Meth. Appl. Sci. 22 (2012), 1150001.
  7. N. Besse, Lagrangian averaged gyrokinetic-waterbag continuum, Commun. Math. Sci. 14 (2016), 593-626
  8. N. Besse, Discontinuous Galerkin finite element methods for the gyrokinetic-waterbag equations, IMA Journal of Numerical Analysis 37 (2017), 985-1040
  9. N. Besse, D. Coulette, Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry, J. Math. Phys. 57 (2016), 081518
  10. D. Coulette, N. Besse, Numerical comparisons of gyrokinetic multi-water-bag models, J. Comput. Phys., 248 (2013a), 1-32.
  11. D. Coulette, N. Besse, Multi-water-bag models of ion temperature gradient instability in cylindrical geometry, Phys. Plasmas, 20 (2013b), 052107.
  12. P. Morel, F. Dreydemy Ghiro, V. Berionni, D. Coulette, N. Besse, O. Gurcan, A Multi-water-bag model of drift-kinetic electron, Eur. Phys. J. D, 68 (2014), 220.
  13. D. Coulette, N. Besse, Numerical resolution of the global-eigenvalue problem for the gyrokinetic-waterbag model in toroidal geometry, J. Plasma Phys. 83 (2017), 905830207