Quasineutral Limits of the Vlasov- Introduction VPFP system Poisson/Maxwell-Fokker-Planck System Euler with damping Euler without damping VMFP System F UCAI LI Home Page Department of Mathematics, Nanjing University Title Page Email: fli@nju.edu.cn ◭◭ ◮◮ ◭ ◮ ( A joint work with Ling Hsiao and Shu Wang ) Page 1 of 37 Go Back Full Screen Close Quit
1. Introduction Kinetic equations are mathematical models used to describe the “dilute parti- cle gases” at an intermediate scale between the microscopic and the macro- Introduction scopic level. They appear in a variety of sciences such as plasma, astrophysics, VPFP system aerospace engineering, nuclear engineering, particle-fluid interactions, semi- Euler with damping conductor technology, social sciences and biologies, see Euler without damping VMFP System • B. Perthame. Bull. Amer. Math. Soc. (N.S.) , 41(2):205–244 2004. Home Page • N. Bellomo, A. Palczewski, and G. Toscani. Mathematical topics in non- Title Page linear kinetic theory ,1988. ◭◭ ◮◮ • L. Arlotti, N. Bellomo, and E. De Angelis. Math. Models Methods Appl. ◭ ◮ Sci. , 12(4):567–591, 2002. Page 2 of 37 • A. Decoster, P. A. Markowich, and B. Perthame. Modeling of collisions , Go Back 1998. Full Screen • N. Bellomo and M. Pulvirenti, editors. Modeling in applied sciences, A Close kinetic theory approach , 2000. Quit
Now the study on kinetic equations is one of the most active areas in applied mathematics. The quasineutral limit problem plays an important role in the theory of kinetic equations since it gives the relations between the microscopic mathematical models and the macroscopic ones. Introduction We discuss the quasineutral limits of some kinetic equations including VPFP system • Vlasov-Poisson-Fokker-Planck (VPFP) system Euler with damping Euler without damping • Vlasov-Maxwell-Fokker-Planck (VMFP) system VMFP System We shall show that the solution of VPFP system converges to the solution of Home Page incompressible Euler equations with or without damping, and the renormalized Title Page solution of VMFP system converges to the solution of so-called electron mag- ◭◭ ◮◮ netohydrodynamics (e-MHD) equations. ◭ ◮ Page 3 of 37 Go Back Full Screen Close Quit
2. The Vlasov-Poisson-Fokker-Planck (VPFP) system The VPFP system takes the form Introduction ∂ t f ǫ + ξ · ∇ x f ǫ − ∇ x Φ ǫ · ∇ ξ f ǫ = 1 ξf ǫ + σ ∇ ξ f ǫ � � , τ div ξ (1) VPFP system Euler with damping � R N f ǫ ( t, x, ξ ) dξ = 1 − ǫ ∆Φ ǫ (2) Euler without damping VMFP System with initial datum f ǫ (0 , x, ξ ) = f ǫ Home Page 0 ( x, ξ ) ≥ 0 . (3) Title Page Here f ( t, x, ξ ) denotes the distribution function for particles, which expresses the probability of finding a particle at time t ≥ 0 in a position x ∈ [0 , 1] N ≡ T N , ◭◭ ◮◮ ◭ ◮ and with a velocity ξ ∈ R N , N = 1 , 2 , 3 . The spatially periodic electric potential Φ ǫ is coupled with f ǫ ( t, x, ξ ) through Page 4 of 37 the Poisson equation (2). Go Back Full Screen • ǫ > 0 : The vacuum electric permittivity Close • σ > 0 : The thermal diffusion coefficient Quit
• τ > 0 : The relaxation time due to collisions of the particles with the thermal bath. For more physical background and the derivation of the VPFP system, see Introduction • S. Chandrasekhar. Stochastic problems in physics and astronomy. Rev. VPFP system Modern Physics , 15:1–89, 1943. Euler with damping Euler without damping • S. Chandrasekhar. Brownian motion, dynamical friction, and stellar dy- VMFP System namics. Rev. Modern Physics , 21:383–388, 1949. Home Page Some know results of VPFP system: Title Page 1. The existence and uniqueness of smooth solutions of VPFP system: ◭◭ ◮◮ • F. Bouchut. J. Funct. Anal. ,1993.; ◭ ◮ Page 5 of 37 • G. Rein and J. Weckler. J. Differential Equations , 1992; Go Back • Jr. H. D. Victory and B. P. O’Dwyer. Indiana Univ. Math. J. , 1990. Full Screen Close Quit
2. Different kind of limits problem on VPFP system. Some results on different kind of limits problem on VPFP system under different scalings and different assumptions have been obtained: • Parabolic limit. F. Poupaud and J. Soler [ Math. Models Methods Appl. Introduction Sci. , 2000 ] consider the form VPFP system ǫ 2 ∂ t f ǫ + ǫ (( v · ∇ x ) f ǫ − ( ∇ x Φ ǫ · ∇ v ) f ǫ ) = ∆ v f ǫ + div v ( vf ǫ ) , Euler with damping Euler without damping � ∆ x Φ ǫ = R 3 f ǫ dv VMFP System Home Page They showed the above VPFP sysmtem converges to a parabolic equation as ǫ → 0 . Title Page ◭◭ ◮◮ • High-field limit. T. Goudon, J. Nieto, F. Poupaud, and J. Soler [ JDE , 2005] ◭ ◮ consider the form Page 6 of 37 ǫ ( ∂ t f ǫ + v · ∇ x f ǫ ) − ∇ x Φ ǫ · ∇ v f ǫ = ∆ v f ǫ + div v ( vf ǫ ) , Go Back � ∆ x Φ ǫ = n ǫ − D ǫ , n ǫ ( t, x ) = R 3 f ǫ dv Full Screen Close They showed the above VPFP system converges to a nonlinear systems of Quit PDEs related to Ohm’s Law as ǫ → 0 .
• Low-field limit. Using the Hilbert expansion, A. Arnold, J. A. Carrillo, I. Gamba, and C.-W. Shu [ Transport Theory Statist. Phys. , 2001] obtain the convergence of VPFP system to drift-diffusion equations. We are interested in the quasineutral limits of VPFP system (1)-(2), i.e., ∂ t f ǫ + ξ · ∇ x f ǫ − ∇ x Φ ǫ · ∇ ξ f ǫ = 1 Introduction ξf ǫ + σ ∇ ξ f ǫ � � , τ div ξ VPFP system � Euler with damping R N f ǫ ( t, x, ξ ) dξ = 1 − ǫ ∆Φ ǫ . Euler without damping VMFP System . We discuss two cases: Home Page • ǫ → 0 , τ > 0 is fixed, and σ → 0 . In this case the VPFP system Title Page converges to incompressible Euler Equations with damping. ◭◭ ◮◮ • ǫ → 0 , τ → + ∞ , and σ > 0 is fixed. In this case the VPFP system ◭ ◮ converges to incompressible Euler Equations without damping. Page 7 of 37 Remark: 1. The case ǫ, τ, → 0 is trivial because we obtain J = 0 formally. Go Back 2. The case for which τ → 0 , ǫ > 0 and σ > 0 are fixed corresponds to the Full Screen hyperbolic limit conjectured by F. Poupaud and J. Soler in the paper [ Math. Close Models Methods Appl. Sci. , 2000], but we don’t know whether or not they have Quit obtained the result.
3. The convergence to incompressible Euler Introduction VPFP system equations with damping Euler with damping Euler without damping We study the quasi-neutral limits of VPFP system for the first case, i.e. ǫ → 0 , VMFP System τ, σ > 0 are fixed. We set τ = 1 for convenience. Home Page Now the VPFP system reads Title Page ∂ t f ǫ + ξ · ∇ x f ǫ − ∇ x Φ ǫ · ∇ ξ f ǫ = div ξ ξf ǫ + σ ∇ ξ f ǫ � � , (4) ◭◭ ◮◮ � R N f ǫ ( t, x, ξ ) dξ = 1 − ǫ ∆Φ ǫ (5) ◭ ◮ Page 8 of 37 with initial datum Go Back f ǫ (0 , x, ξ ) = f ǫ 0 ( x, ξ ) ≥ 0 . (6) Full Screen Close Quit
Denote � � ρ ǫ ( t, x ) = R N f ǫ ( t, x, ξ ) dξ, J ǫ ( t, x ) = ρ ǫ u ǫ ( t, x ) = R N ξ f ǫ ( t, x, ξ ) dξ. (7) Multiplying the equation (4) by 1 and ξ respectively, and integrating the results with respect to ξ , we have Introduction ∂ t ρ ǫ + ∇ · J ǫ = 0 , VPFP system (8) Euler with damping Euler without damping � ∂ t J ǫ + ∇ x · R N ( ξ ⊗ ξ ) f ǫ dξ + ∇ Φ ǫ VMFP System + ǫ 2 ∇ ( |∇ Φ ǫ | 2 ) − ǫ ∇ · ( ∇ Φ ǫ ⊗ ∇ Φ ǫ ) + J ǫ = 0 Home Page (9) Title Page with ◭◭ ◮◮ − ǫ ∆Φ ǫ = ρ ǫ − 1 , (10) ◭ ◮ where we have used the identity Page 9 of 37 ∇ Φ ǫ (∆Φ ǫ ) = 1 2 ∇ ( |∇ Φ ǫ | 2 ) − ∇ · ( ∇ Φ ǫ ⊗ ∇ Φ ǫ ) Go Back Full Screen and the symbol “ ⊗ ” which denotes the tensor product of vectors. Close Quit
R N ( ξ ⊗ ξ ) f ǫ dξ , we rewrite the � To understand the fluid structure of the term distribution function f ǫ in the system (4)-(5) as ρ ǫ −| ξ − u ǫ | 2 � � f ǫ = + ǫG ǫ ( t, x, ξ ) ≡ M ( ρ ǫ ,u ǫ ,K ǫ ) ( ξ )+ ǫG ǫ , (11) (2 πK ǫ ) N/ 2 exp 2 K ǫ Introduction VPFP system where M ( ρ ǫ ,u ǫ ,K ǫ ) ( ξ ) is a local Maxwellian denoting the macroscopic fluid part of Euler with damping f ǫ , G ǫ is the microscopic non-fluid part of f ǫ , and K ǫ denotes the temperature. Euler without damping Then, a direct computation gives VMFP System R N ( ξ ⊗ ξ ) f ǫ dξ = J ǫ � J ǫ � + ρ ǫ K ǫ I N + O ( ǫ ) , (12) Home Page ρ ǫ Title Page where I N is an identity matrix, and ◭◭ ◮◮ 1 � 3 2 K ǫ + 1 � � ◭ ◮ R N | ξ | 2 f ǫ ( t, x, ξ ) dξ = ρ ǫ 2 | u ǫ | 2 ( t, x ) + O ( ǫ ) . 2 Page 10 of 37 Go Back Full Screen Close Quit
Recommend
More recommend
