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Lecture 3: the Vlasov Equation

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Lecture 3: the Vlasov Equation TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Lecture 3: The Vlasov equation Seung Yeal Ha Department of Mathematical Sciences Seoul National University Feb. 10th, 2015 TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Outline Two expansions Hydrodynamic limits The Vlasov equation The incompressible Euler limit TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Lecture 3 The Boltzmann equation and Vlasov equation TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT A singular perturbation problem Note that small Knudsen limit Kn ! 0 to the Boltzmann equation corresponds to a singular perturbation problem: Kn @t F + ξ · rx F = Q(F; F): Thus, formally, as long as @t F + ξ · rx F is uniformly bounded in the zero Knudsen limit, we may argue that Q(F; F) ! 0; as Kn ! 0 In other words, as Kn ! 0, F ! M in suitable sense: Thus, zero Knudsen limit, we may hope that the Boltzmann equation behaves like the Euler equations. TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Question: Can this singular perturbation limit be justified rigorously ? TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT The compressible Euler limit Consider the Boltzmann equation " := Kn: " @t F + ξ · rx F = Q(F; F): • The Hilbert expansion D. Hilbert, Begrndung der kinetischen Gastheorie, Math. Ann. 72 (1912), 562-577. Expand F as a formal power series of " = Kn: 1 X n 2 F = " Fn = F0 + "F1 + " F2 + ··· : n=0 L.H.S.: 2 3 "(t F0 + ξ · rx F0) + " (t F1 + ξ · rx F1) + " (t F2 + ξ · rx F2) + ··· R.H.S.: Q(F0; F0) + 2"Q(F1; F0) 2 +" (2Q(F2; F0) + Q(F1; F1)) + ··· TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Compare various orders in ": O(1): Q(F0; F0) = 0 =) F0 = M: " : 2Q(F1; F0) = @t F0 + ξ · rx F0 =: S0; 2 " : 2Q(F2; F0) = @t F1 + ξ · rx F1 − Q(F1; F1) =: S1; For the solvability of LM (F1) := 2Q(F1; F0) = S0, by the Fredholm alternative, ∗ S0 is orthogonal to the kernel of LM ∗ But KerLM is spanned by the collision invariants, hS0; αi = 0; α = 0; 1; ··· ; 4: : Euler equations: TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT • Some relevant references. 1. Russel E. Caflisch: The fluid dynamic limit of the nonlinear Boltzmann equation. CPAM. 33 (1980), no. 5, 651Ð666. 2. Shih-Hsien Yu: Hydrodynamic limits with Shock Waves of the Boltzmann Equation. CPAM 58 (2005), 409-443. 3. Laure Saint-Raymond: A mathematical PDE perspective on the Chapman-Enskog expansion. Bulletin of AMS, 51 (2014), 247-275. cf. 1. The compressible Navier-Stokes limit: The Chapman-Enskog expansion 2. The incompressible Euler limit: M ! 0. TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT References 1. Earle H. Kennard: Kinetic theory of gases (1938). 2. Stephen G. Brush: The kinetic theory of gases: An anthology of classic papers with historical commentary. Imperial College Press, 2003. TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT The Vlasov equation TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT If you first hear of the Vlasov equation, you might ask the following questions: • QA 1: What is the Vlasov equation? • QA 2: When do we use the Vlasov equation? • QA 3: What are the relations with other fluid equations? • ··· TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT What is the Vlasov equation ? The Vlasov equation = collisionless Boltzmann equation Physical situation: Consider an ensemble of particles moving in a mean-field force fields (e.g. a many-body particle systems in mean-field setting) Let F = F(x; ξ; t) be a one-particle distribution function of particles and we assume that the collisions between particles are secondary, in contrast collisions (interactions) between particle and fields are important. d F(x(t); ξ(t); t) = @ F + x_ (t) · r F + ξ_(t) · r F = 0; dt t x ξ or equivalently ξ @ F + · r F + E(x; t) · r F = 0: t m x ξ TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Let V = V (x; y) be the pairwise potential between particles at position x and y, respectively. In this case, the self-consistent force field ZZ E(x; t) = −∇x V (x; y)F(y; ξ∗; t)d∗dy 3 3 Z R ×R = −∇x V (x; y)ρ(y; t)dy: R3 e.g. 1. Electrostatic potential generated by a charge q: q 1 V (x; y) = : repulsive 4π jx − yj 2. Gravitational potential generated by a mass m: Gm V (x; y) = − : attractive jx − yj TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Thus, the self-consistent Vlasov equation reads as ξ @t F + · rx F + E(x; t) · rξF = 0; m Z E(x; t) = −∇x V (x; y)ρ(y; t)dy: R3 TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT When do we use the Vlasov equation ? In plasma physics, the equation was first suggested for description of plasma by Anatoly Vlasov in 1938 " A. A. Vlasov (1938). "On Vibration Properties of Electron Gas". J. Exp. Theor. Phys. (in Russian) 8 (3): 291" TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT • Plasma physics Plasma= the fourth fundamental state of gases, completely ionized gases. e.g., gas inside light bulb. Ice =) water =) vapor =) plasma: TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT The Vlasov-Maxwell system Vlasov equations for electrons and ions + Maxwell equationsfor electric and magnetic force fields. • Dynamic variables Fi = Fi (x; ξ; t); Fe = Fe(x; ξ; t): distribution functions for ion and electron; E = E(x; t); B = B(x; t): electric and magentic fields density: We set a relativistic velocity related to momentum ξ: ξ vα(ξ) = ; α = i; e: p 2 2 2 mα + jξj =c where c is the speed of light. Then it is easy to see that jvα(ξ)j < c: TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT • The Vlasov-Maxwell system v @ F + v · r F + e E + α × B · r F = 0; t α α x α α c ξ α @t E = cr × B − j; r · E = ρ, @t B = −cr × E; r · B = 0; where ρ and j are the charge and current densities: Z X ρ = C(d) eαFαdξ; charge density α Z X j = C(d) vαeαFαdξ currrent density: α cf. Small data and global existence: TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT A single species plamsa • Electron gun (laser) and plamsa sheath Only one species of charged particles, say electrons or ions. TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT The Vlasov-Poisson system Recall a Vlasov-Maxwell system for a single species. v(ξ) @ F + v(ξ) · r F + e E + × B · r F = 0; t x c ξ 1 j @ E = r × B − ; r · E = ρ, c t c 1 @ B = −∇ × E; r · B = 0: c t Consider a regime where jv(ξ)j c;@t B ≈ 0;@xi Bi ≈ 0: TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT i.e., letting c ! 1 and B = 0. Then, the Vlasov-Maxwell system becomes @t F + v(ξ) · rx F + eE · rξF = 0; r · E = ρ, r × E = 0: We set E = rx ' and obtain the Vlasov-Poisson system: 3 @t F + ξ · rx F + rx ' · rξF = 0; x; ξ 2 R ; t 2 R; Z ∆' = ρ, ρ = eFdξ: cf. Rigorous justification: Degond, Ukai ’80 in finite-time interval TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT • Astrophysics: A galaxy is a gravitationally bound system consisting of stars, stellar remnants, an interstellar medium of gas and dust, and dark matter cf. Size of galaxy: From 104 to 1014, observable universe: ≥ 1014-galaxies TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT James Hopwood Jeans: On the theory of star-streaming and the structure of the universe, Monthly Notices of the Royal Astronomical Society, 76 (1915), 70 -84. cf. Reinhard Rein TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Inner beauties of the V-P system TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Inner beauties of the V-P system • The V-P system is a Hamiltonian system. Recall that the Vlasov-Poisson (V-P) system reads as ξ 3 @t F + · rx F − rx ' · rξF = 0; x; ξ 2 R ; t 2 R; m Z ∆' = ρ, ρ = Fdξ: Then, the V-P system is equivalent to the ODE system: dx ξ dξ = ; = E = −∇ ': dt m dt x TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT Define a particle path [x(s) = x(s; x; v; t); ξ(s) = ξ(s; x; v; t)]: dx(s) ξ(s) dξ(s) = ; = E(x(s); ξ(s); s); ds m ds (x(t); ξ(t)) = (x; ξ): Then, along the particle path, we have 3 F(x(s); ξ(s); s) = F0(x; ξ); s > 0; x; ξ 2 R : Note that (x; ξ) ! (x(s); ξ(s)) is a measure preserving. TWO EXPANSIONS HYDRODYNAMIC LIMITS THE VLASOV EQUATION THE INCOMPRESSIBLE EULER LIMIT We claim: The above ODE system is a Hamilton’s ODEs We set 1 ZZ H(x; ξ; t) := jξj2 + V (x; y)f (y; ξ; t)dξdy 2m 3× 3 Z R R 1 2 = jξj + V (x; y)ρF (y; t)dy; 2m R3 | {z } =:' where potential energy is mean-field.
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