From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics - Volume 1

Description

The Vlasov–Maxwell–Boltzmann system is a microscopic model to describe the dynamics of charged particles subject to self-induced electromagnetic forces. At the macroscopic scale, in the incompressible viscous fluid limit the evolution of the plasma is governed by equations of Navier–Stokes–Fourier type, with some electromagnetic forcing that may take on various forms depending on the number of species and on the strength of the interactions. From the mathematical point of view, these models have very different behaviors. Their analysis therefore requires various mathematical methods which this book aims to present in a systematic, painstaking, and exhaustive way. The first part of this work is devoted to the systematic formal analysis of viscous hydrodynamic limits of the Vlasov–Maxwell–Boltzmann system, leading to a precise classification of physically relevant models for viscous incompressible plasmas, some of which have not previously been described in the literature. In the second part, the convergence results are made precise and rigorous, assuming the existence of renormalized solutions for the Vlasov–Maxwell–Boltzmann system. The analysis is based essentially on the scaled entropy inequality. Important mathematical tools are introduced, with new developments used to prove these convergence results (Chapman–Enskog-type decomposition and regularity in the v variable, hypoelliptic transfer of compactness, analysis of high frequency time oscillations, and more). The third and fourth parts (which will be published in a second volume) show how to adapt the arguments presented in the conditional case to deal with a weaker notion of solutions to the Vlasov–Maxwell–Boltzmann system, the existence of which is known. Keywords: Plasma, magneto-hydro-dynamics, fluid limits, kinetic theory, entropy method, moment method, hypoellipticity, electromagetic waves, Ohm’s law Preface......Page 6 Contents......Page 10 I Formal derivations and macroscopic weak stability......Page 14 The Vlasov–Maxwell–Boltzmann system......Page 16 The Boltzmann collision operator......Page 18 Formal macroscopic properties......Page 21 The mathematical framework......Page 25 Incompressible viscous regimes......Page 28 Scalings for the electromagnetic field......Page 30 Formal analysis of the one-species asymptotics......Page 34 Thermodynamic equilibrium......Page 35 Macroscopic constraints......Page 38 Evolution equations......Page 40 Summary......Page 46 The Vlasov–Poisson–Boltzmann system......Page 49 Formal analysis of the two-species asymptotics......Page 50 Thermodynamic equilibrium......Page 52 The case of very weak interspecies collisions......Page 59 Macroscopic hydrodynamic constraints......Page 68 Hydrodynamic evolution equations......Page 69 Macroscopic electrodynamic constraints and evolution......Page 73 Summary......Page 78 The two-species Vlasov–Poisson–Boltzmann system......Page 82 Weak stability of the limiting macroscopic systems......Page 86 The incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system......Page 87 The two-fluid incompressible Navier–Stokes–Fourier–Maxwellsystem with (solenoidal) Ohm's law......Page 90 Large global solutions in two dimensions......Page 94 Small global solutions in three dimensions......Page 96 Weak-strong stability and dissipative solutions......Page 97 The incompressible Navier–Stokes–Maxwell system......Page 98 The two-fluid incompressible Navier–Stokes–Maxwell system with Ohm's law......Page 105 The two-fluid incompressible Navier–Stokes–Maxwell system with solenoidal Ohm's law......Page 118 II Conditional convergence results......Page 132 Two typical regimes......Page 134 The Vlasov–Boltzmann equation......Page 135 Coupling the Boltzmann equation with Maxwell's equations......Page 144 The setting of our conditional study......Page 145 Macroscopic conservation laws......Page 147 The incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system......Page 153 The two-fluid incompressible Navier–Stokes–Fourier–Maxwellsystem with (solenoidal) Ohm's law......Page 158 Weak interactions......Page 160 Strong interactions......Page 164 Outline of proofs......Page 168 Weak compactness and relaxation estimates......Page 170 Controls from the relative entropy bound......Page 171 Controls from the entropy dissipation bound......Page 174 Relaxation towards thermodynamic equilibrium......Page 177 Infinitesimal Maxwellians......Page 181 Bulk velocity and temperature......Page 183 Improved integrability in velocity......Page 187 Macroscopic constraint equations for one species......Page 196 Macroscopic constraint equations for two species, weak interactions......Page 199 Energy inequalities......Page 203 The limiting Maxwell equations......Page 209 Strong compactness and hypoellipticity......Page 212 Compactness of the gain term......Page 213 Relative entropy, entropy dissipation and strong compactness......Page 216 Compactness with respect to x......Page 220 Hypoellipticity and the transfer of compactness......Page 221 Compactness of fluctuations for one species......Page 226 Compactness of fluctuations for two species......Page 234 Macroscopic constraint equations for two species, weak interactions......Page 246 An admissible renormalization......Page 248 Convergence of conservation defects......Page 250 Decomposition of flux terms......Page 255 Decomposition of acceleration terms......Page 257 Macroscopic constraint equations for two species, strong interactions......Page 260 Energy inequalities for two species, strong interaction......Page 271 Approximate macroscopic equations......Page 276 Approximate conservation of mass, momentum and energy for one species......Page 277 Conservation defects......Page 279 Decomposition of flux terms......Page 283 Decomposition of acceleration terms......Page 289 Approximate conservation of mass, momentum and energy for two species......Page 291 Conservation defects......Page 300 Decomposition of flux terms......Page 303 Decomposition of acceleration terms......Page 306 Proof of Propositions 9.5 and 9.6......Page 308 Proofs of Lemmas 9.7, 9.8, 9.9 and 9.10......Page 312 Acoustic and electromagnetic waves......Page 326 Formal filtering of oscillations......Page 327 Rigorous filtering of oscillations......Page 331 Weak convergence of fluctuations, collision integrandsand electromagnetic fields......Page 336 Constraint equations, Maxwell's system and the energy inequality......Page 337 Evolution equations......Page 338 Temporal continuity, initial data and conclusion of the proof......Page 341 The relative entropy method: old and new......Page 344 Weak convergence of fluctuations, collision integrandsand electromagnetic fields......Page 345 Constraint equations, Maxwell's system and the energy inequality......Page 347 The renormalized modulated entropy inequality......Page 348 Convergence and conclusion of the proof......Page 368 Weak convergence of fluctuations, collision integrandsand electromagnetic fields......Page 369 Constraint equations, Maxwell's system and the energy inequality......Page 372 The renormalized modulated entropy inequality......Page 373 Convergence and conclusion of the proof......Page 393 Appendix A: Cross-section for momentum and energy transfer......Page 398 Appendix B: Young inequalitites......Page 402 Appendix C: End of proof of Lemma 7.7 on hypoelliptic transferof compactness......Page 406 Bibliography......Page 412 Index......Page 416