On the Landau approximation in plasma physics

By Radjesvarane Alexandre, Cédric Villani. 2004

161 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4c78d55883ecb2c8c8c48a0e6c037474ab8bc189

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory (2002, methodology): “However, all limits of the type Boltzmann → Landau can be put into a unified formalism, first sketched in our work [446], then extended and made more precise in Alexandre and Villani [12].”
Some A priori estimates for the homogeneous Landau equation with soft potentials (2013, methodology): “This equation is obtained as a continuous approximation of the Boltzmann equation when grazing collisions prevail, see for instance [1, 2, 9, 15, 23] for a detailed study of the limiting process, and references therein on this subject.”
Smoothing Effects for Classical Solutions of the Full Landau Equation (2009, unknown): “[11], [2] and [9]).”
On the Boltzmann equation for long‐range interactions (2002, background): “2) Our second main result, discussed in our companion paper [7], is the first rigorous mathematical justification of the Landau approximation in plasma physics.”
Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation (2009, unknown): “On the other hand, in order to estimate Kk1 (v∗), write {˜̂ Fk(η +, v∗) − ˜̂Fk(η, v∗) }˜̂ hk(η, v∗) = {˜̂ Fk(η +, v∗) − ˜̂Fk(η, v∗) }{˜̂ hk(η, v∗) − ˜̂hk(η+, v∗) } −η− · ( ∇˜̂Fk ) (η+, v∗) ˜̂ hk(η+, v∗) − ∫ 1 0 {( ∇˜̂Fk ) (η+ + τ (η − η+), v∗) − ( ∇˜̂Fk ) (η+, v∗))dτ } · (η − η+)˜̂hk(η+, v∗).”
Global Existence of Renormalized Solutions to Entropy-Dissipating Reaction–Diffusion Systems (2015, background): “Wewould like to emphasize that renormalized solutions have foundwidespread use in the theory of PDEs, also, in particular, for elliptic and parabolic equations; see for example [1,2,7,31,39].”
Cercignani's Conjecture is Sometimes True and Always Almost True (2003, background): “The links between Boltzmann’s and Landau’s entropy production functionals have been studied in detail (see for instance [ 1 , 43]).”
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation (2005, background): “See [2,31] and the references included for mathematical and physical background about this model, which is of great importance in plasma physics.”
Global Existence for Quadratic Systems of Reaction-Diffusion (2007, background): “[16, 1, 2] for Landau’s equation of plasma physics and Boltzmann’s equation without angular cutoff).”
Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators (2006, background): “When all collisions become concentrated on θ = 0, one obtains by the so-called grazing collision limit asymptotic (see for instance [6, 15, 16, 18, 29, 5]) the Landau operator QL(f, f)(v) = ∇v · ( ∫”