Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions

Arsénio, Diogo; Saint‐Raymond, Laure

doi:10.1016/j.crma.2013.04.025

Cited by 10 publications

(7 citation statements)

References 14 publications

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“…In the case of long-range microscopic interactions giving rise to a collision crosssection with a singularity for grazing collisions, treated in the third part, we start by proving the existence of renormalized solutions with a defect measure in the spirit of the construction by Alexandre and Villani [1]. This result, which is important independently of the study of hydrodynamic limits, has been addressed in the note [7]. The study of hydrodynamic limits follows then essentially the lines of [4] (combined with the results of the conditional part).…”

Section: Prefacementioning

confidence: 97%

“…Following the strategy by Alexandre and Villani [1], and renormalizing the Vlasov-Boltzmann equation by concave functions, we thus get some global renormalized solutions involving a defect measure (which is formally 0 because of the conservation of mass). This construction has been sketched in [7]. It will be detailed and used to obtain fully rigorous convergence results in Part 3.…”

Section: Coupling the Boltzmann Equation With Maxwell's Equationsmentioning

confidence: 99%

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From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics

Arsénio¹,

Saint‐Raymond²

2016

Preprint

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Part 2. Conditional convergence results Chapter 4. Two typical regimes iii iv CONTENTS 4.1. Renormalized solutions 4.1.1. The Vlasov-Boltzmann equation 4.1.2. Coupling the Boltzmann equation with Maxwell's equations 4.1.3. The setting of our conditional study 4.1.4. Macroscopic conservation laws 4.2. The incompressible quasi-static Navier-Stokes-Fourier-Maxwell-Poisson system 4.3. The two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with (solenoidal) Ohm's law 4.3.1. Weak interactions 4.3.2. Strong interactions 4.4. Outline of proofs Chapter 5. Weak compactness and relaxation estimates 5.1. Controls from the relative entropy bound 5.2. Controls from the entropy dissipation bound 5.3. Relaxation towards thermodynamic equilibrium 5.3.1. Infinitesimal Maxwellians 5.3.2. Bulk velocity and temperature 5.4. Improved integrability in velocity Chapter 6. Lower order linear constraint equations and energy inequalities 6.1. Macroscopic constraint equations for one species 6.2. Macroscopic constraint equations for two species, weak interactions 6.3. Energy inequalities 6.4. The limiting Maxwell's equations Chapter 7. Strong compactness and hypoellipticity 7.1. Compactness with respect to v 7.1.1. Compactness of the gain term 7.1.2. Relative entropy, entropy dissipation and strong compactness 7.2. Compactness with respect to x 7.2.1. Hypoellipticity and the transfer of compactness 7.2.2. Compactness of fluctuations for one species 7.2.3. Compactness of fluctuations for two species Chapter 8. Higher order and nonlinear constraint equations 8.1. Macroscopic constraint equations for two species, weak interactions 8.1.1. Proof of Proposition 8.1 8.1.1.1. An admissible renormalization 8.1.1.2. Convergence of collision integrals 8.1.1.3. Decomposition of flux terms 8.1.1.4. Decomposition of acceleration terms 8.1.1.5. Convergence 8.2. Macroscopic constraint equations for two species, strong interactions 8.3. Energy inequalities Chapter 9. Approximate macroscopic equations 9.1. Approximate conservation of mass, momentum and energy for one species 9.1.1. Conservation defects 9.1.2. Decomposition of flux terms 9.1.3. Decomposition of acceleration terms CONTENTS v 9.2. Approximate conservation of mass, momentum and energy for two species 9.2.1. Conservation defects 9.2.2. Decomposition of flux terms 9.2.3. Decomposition of acceleration terms 9.2.4.

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Section: Prefacementioning

confidence: 97%

Section: Coupling the Boltzmann Equation With Maxwell's Equationsmentioning

confidence: 99%

From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics

Arsénio¹,

Saint‐Raymond²

2016

Preprint

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“…where f s = f s (x, v, t), f s ′ = f s ′ (x, v ′ , t), where Π w = I−ww/|w| 2 is the projection orthogonal to w, and where the plasma parameter Λ arises as a cut-off in the collision integral for impact factors greater than the Debye length (and, in principle, depends upon s, s ′ pairs). Although this is a standard kinetic model for a plasma [1], it has never been rigorously derived from a microscopic description [44,45] and global existence of (strong) solutions is an open problem [46]. Physically, alternative collision-integrals such as that of Balescu-Lenard [47,48] might give improved accuracy when large-velocity bumps or tails develop in the distribution functions [49].…”

Section: A Basic Equationsmentioning

confidence: 99%

Cascades and Dissipative Anomalies in Nearly Collisionless Plasma Turbulence

Eyink

2018

Phys. Rev. X

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We develop first-principles theory of kinetic plasma turbulence governed by the Vlasov-Maxwell-Landau equations in the limit of vanishing collision rates. Following an exact renormalization-group approach pioneered by Onsager, we demonstrate the existence of a "collisionless range" of scales (lengths and velocities) in 1-particle phase space where the ideal Vlasov-Maxwell equations are satisfied in a "coarse-grained sense". Entropy conservation may nevertheless be violated in that range by a "dissipative anomaly" due to nonlinear entropy cascade. We derive "4/5th-law" type expressions for the entropy flux, which allow us to characterize the singularities (structure-function scaling exponents) required for its non-vanishing. Conservation laws of mass, momentum and energy are not afflicted with anomalous transfers in the collisionless limit. In a subsequent limit of small gyroradii, however, anomalous contributions to inertial-range energy balance may appear due both to cascade of bulk energy and to turbulent redistribution of internal energy in phase space. In that same limit the "generalized Ohm's law" derived from the particle momentum balances reduces to an "ideal Ohm's law", but only in a coarse-grained sense that does not imply magnetic flux-freezing and that permits magnetic reconnection at all inertial-range scales. We compare our results with prior theory based on the gyrokinetic (high gyro-frequency) limit, with numerical simulations, and with spacecraft measurements of the solar wind and terrestrial magnetosphere.

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“…Among them, we would only mention Desvillettes-Dolbeault [22] for the long time asymptotics of the system, Bernis-Desvillettes [4] for the propagation of regularity of solutions, Mischler [66] for the initial boundary value problem, Bostan-Gamba-Goudon-Vasseur [9] for the stationary problem on the bounded domain, and Guo [43] for global existence of classical solutions near vacuum. Note that the existence of renormalised solutions of the much more complex Vlasov-Maxwell-Boltzmann system with a defect measure has been recently studied in Arsenio-Saint-Raymond [3].…”

Section: Literature and Backgroundmentioning

confidence: 99%

The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

Duan

Liu²

2017

J Stat Phys

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The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in [42]. It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-trivial long-term dynamics of the binary system. In the paper, we consider the problem in the setting of rarefaction waves. The analytical tool is based on the macro-micro decomposition introduced in [59] that we can be able to develop into the case for the two-component Boltzmann equations around local bi-Maxwellians. Our focus is to explore how the disparate masses and charges of particles play a role in the analysis of the approach of the complex coupling system time-asymptotically toward a nonconstant equilibrium state whose macroscopic quantities satisfy the quasineutral nonisentropic Euler system. 10 2.3. Diffusion and heat-conductivity 15 3. Quasineutral Euler equations and rarefaction waves 20 4. Preliminary estimates on two-component collision operator 23 5. Proof of the main result 27 6. A priori estimates on the fluid part 32 6.1. Estimate on zero-order energy 32 6.2. Estimate on first-order dissipation 37 6.3. Estimate on first-order energy 44 7. A priori estimates on the non-fluid part 51 7.1. Estimate on zero-order dissipation 51 7.2. Estimate on high-order energy 55 7.3. Estimate on energy with mixed derivatives 60 References 61 2010 Mathematics Subject Classification. Primary: 35Q20, 76P05; Secondary: 35B35, 35B40.

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Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions

Cited by 10 publications

References 14 publications

From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics

From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics

Cascades and Dissipative Anomalies in Nearly Collisionless Plasma Turbulence

The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

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