Analisis para la ecuacion de Boltzmann Soluciones y Approximaciones

Transcription

1 Analisis para la ecuacion de Boltzmann Soluciones y Approximaciones Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin Mar del PLata, Julio 2012 Collaborators: R. Alonso, Rice U. E. Carneiro, IMPA (Rio, Brasil) C. Bardos, Paris VII, J. Haack, UT Austin D. Levermore, U Maryland S.H. Tharskabhushanam, Birmingham, UK 7/12/2012

2

3 Overview Introduction to classical kinetic equations for elastic and inelastic interactions: The Boltzmann equation for binary elastic and inelastic collision. Description of interactions, collisional frequency: hard and soft potentials, angular cross sections. Grad Cut-off assumption: integrable and non-integrable angular cross section. Collision invariants.. Energy dissipation & heat source mechanisms. Self-similar models. Analytical issues for the Boltzmann Transport equation Young s inequality and Hardy Littlewood Sobolev inequality. Gain of integrabilty estimates and regularity of the estimates. Propagation of L p estimates. Existence, regularity and stability of solutions near Maxwellian Distributions Kaniel-Shimbrot iteration. L p propagation of gradients in physical and velocity space. L p stability.

4 Spectral - Lagrangian constrained solvers for collisional problems Deterministic solvers for Dissipative models - The space homogeneous problem FFT application - Computations of Self-similar solutions Space inhomogeneous problems Time splitting algorithms Simulations of boundary value layers problems Benchmark simulations

5 Part I Introduction to classical kinetic equations for elastic and inelastic interactions: The Boltzmann equation for binary elastic and inelastic collision. Description of interactions, collisional frequency: hard and soft potentials, angular cross sections. Grad Cut-off assumption: integrable and non-integrable angular cross section. Collision invariants.. Energy dissipation & heat source mechanisms. Self-similar models.

6 elastic collisions

7 Interchange of velocities during a binary collision or interaction σ = u ref / u is the unit vector in the direction of the relative velocity with respect to an elastic collision v * u' v * γ. θ u σ v v' v * β v * u'. 1-β γ. e σ θ. u v' v Elastic collision Inelastic collision β =(1+e)/2 1- β +e = β Remark: θ 0 grazing and θ π head on collisions or interactions

8 The Boltzmann Transport equation for elastic or inelastic collisions := mass density := statistical correlation function (sort of mean field ansatz,i.e. independent of v) = for elastic interactions (e=1) i.e. enough intersitial space May be extended to multi-linear interactions (in some special cases to see later)

9 it is assumed that the restitution coefficient is only a function of the impact velocity e = e( u η ). The properties of the map z e(z) are given in the next page v' = v+ (1+e) (u. η) η and v' * = v * + (1+e) (u. η) η 2 2 The notation for pre-collision perspective uses symbols 'v, 'v * : Then, for 'e = e( 'u η ) = 1/e, the pre-collisional velocities are clearly given by 'v = v+ (1+'e) ('u. η) η and 'v * = v * + (1+'e) ('u. η) η 2 2 In addition, the Jacobian of the transformation is then given by J(e(z)) = e(z) + ze z (z) = θ z (z) =( z e(z) ) z However, for a handy weak formulation we need to write the equation in a different set of coordinates involving σ := u'/ u the unit direction of the specular (elastic) reflection of the postcollisional relative velocity, for d=3 γ σ θ

10 β=1 elastic σ 7/12/2012

11 Collisional kernel or transition probability of interactions is calculated using intramolecular potential laws: is the angular cross section satisfies In addition, sometimes we use an α-growth condition of the type which is satisfied for angular cross section function for α > d-1 (in 3-d is for α>2)

12 7/12/2012

13 7/12/2012, for the case of inelastic interactions.

14

15

16

17 Exact energy identity for a Maxwell type interaction models Then f(v,t) δ 0 as t to a singular concentrated measure (unless there is source ) Current issues of interest regarding energy dissipation: Can one tell the shape or classify possible stationary states and their asymptotics, such as self-similarity? Non-Gaussian (or Maxwellian) statistics!

18 Reviewing inelastic properties INELASTIC Boltzmann collision term: No classical H-Theorem if e = constant < 1 Yet, it dissipates total energy for e=e(z) < 1 (by Jensen's inequality): Inelasticity brings loss of micro reversibility but keeps time irreversibility!!: That is, there are stationary states and, in some particular cases we can show stability to stationary and self-similar states (Multi-linear Maxwell molecule equations of collisional type and variable hard potentials for collisions with a background thermostat) However: Existence of NESS: Non Equilibrium Statistical States (stable stationary states are non-gaussian pdf s)

19 Conservation Laws: Even further, any solution F of the Inhomogeneous Boltzmann equation formally satisfies the following the local conservation laws: when ξ(v, x, t) is any quantity that satisfies: ξ(, x, t) Span{1, v 1, v 2,, v D, v 2 }, and t ξ + v x ξ = 0 It has been known (since Boltzmann who worked out the case D = 3), that the only such quantities ξ are linear combinations of the 4 + 2D + D(D 1)/2 quantities 1, v, x vt, v 2 / 2, v x= v x T -xv T, v (x vt), x vt 2 / 2 By integrating the corresponding local conservation laws over space and time,

20 Local and global Maxwellians If ρ, u, and θ functions of (x, t) the following pdf are called local Maxwellians The family of all global Maxwellians over the spatial domain RD with positive mass, zero net momentum, and center of mass at the origin has the form with m > 0 and (a, b, c, B) Ω where Ω is the open cone in R x R x R x R D D defined by Ω = { (a, b, c, B) : (ac-b 2 ) I + B 2 > 0 } Any global Maxwellian can be written as a local Maxwellian form: with and

21 Property 1: let M 1 and M 2 be global Maxwellians with parameters given by (m 1, a 1, b 11, c 1, B 1 ) R + Ω and (m 2, a 2, b 2, c 2, B 2 ) R + Ω resp. respectively. Then M 1 M 2 for every (v, x, t) if and only if meaning Property 2 (for stability): Let the collision kernel b have the separated form b= u β b, for some β ( D, 2]. Let M be a global Maxwellian for some (m, a, b, c, B) R + Ω. Let F be any measurable function that satisfies the pointwise bounds 0 F (v, x, t) x, s + t) M(v, x, s+t), s in [0, ). Then for every [t 1, t 2 ] [0, ) one has the L 1 -bound with

22 Next we need to recall self-similarity:

23 A general form statistical transport : The space-homogenous BTE with external heating sources Important examples from mathematical physics and social sciences: The term models external heating sources: background thermostat (linear collisions), thermal bath (diffusion) shear flow (friction), dynamically scaled long time limits (self-similar solutions). v v η v * v * Inelastic Collision u = (1-β) u + β u σ, with σ the direction of elastic post-collisional relative velocity

24 Energy dissipation implies the appearance of Non-Equilibrium Stationary Statistical States

25 Part II Analytical issues for the Boltzmann Transport equation Young s inequality and Hardy Littlewood Sobolev inequality. Gain of integrabilty estimates and regularity of the estimates. Propagation of L p estimates.

26 The existence theory of the space homogeneous Boltzmann equation for variable hard potentials Φ( u )= u γ, with 0 < γ 1 and integrable b(u.σ) creation of moments estimates creation of exponentially weighted lower bound Remark: existence theory and stability for γ=0 (Maxwell type of interaction) was developed by Wild, Morgensten 40 and Bobylev 75.

27 Sharp Povzner estimates Summability of moments series (JMPA 08) creation of exponentially L - weighted estimates: in progress (with R. Alonso and C. Mouhot

28

29

30

31 7/12/2012

32 I- Sketch of proofs for Radial Symmetryzation, Young s and HLS inequalities for the collisional integrals 1- Radial symmetrization

33

34

35

36

37

38

39

40 Remark: this counterexample may not be a solution of the BTE!! We do not know whether is possible to find a counterexample in within the class of solutions of the BTE.

41 7/12/2012

42 7/12/2012

43 7/12/2012

44 7/12/2012

45 Important properties for non-integrable angular cross-section A quantitative form of a Cancellation Lemma (R. Alonso, E. Carneiro& I.M.G, 12) (originally by Alexandre, Villani, Wennberg and Desvilletes, '01) Let b as in hypothesis H2-A (non-integrable angular cross-section) and s > 2. Then where C > 0 is a constant that depends only on s. Local lower estimate: f is a solution of the Boltzmann eq. with bounded mass, energy and entropy. Then, there exists δ = δ(m 0, m 2, H(f 0 )) > 0 and a constant C =C(m 0,m 2,λ) > 0, such that for any v R n and t 0 Then, for the norms Propagation of Moments and L p s estimates:

46 Theorem 1(Propagation of Moments)(Alonso, Carneiro& I.M.G'12): Under assumptions H1-A (non-mollified potential ) and H2-A (non-integrable angular cross-section), let λ ( 2, 0] (not too soft potentials) and s > 2 (initial moments). Assume that the initial data f 0 L 1 s+ λ (R n ) L logl (R n ). Then Comments on the proof: the constrain λ ( 2, 0] is due to the cancellation lemma and The local lower estimate and the norms we use: to obtain with C 8 = C 8 (m 0, m 2, H(f 0 ), b, s, λ+2 ) and C 9 = C 9 (m 0, m 2, b, s, λ), with λ+2>0. and, by interpolation with

47 Comments: Our estimate improves the one of Desvilletes & Mouhot Asymp.Anal.'07, where also uniform propagation was also obtained in the same range λ ( 2, 0] under much stronger assumptions, namely: 1- Mollified potential Φ, 2- Integrable and bounded-by-below angular cross-section b(θ), 3- Initial datum f 0 L 1 2s L 2 q 0 for some q 0 > Their method consists of first proving polynomial bounds and then combining these with quantitative results of convergence of the solution to a Maxwellian equilibrium. Instead we use: i) Finite entropy hypothesis (which would be implied by f 0 L p, for any p > 1), ii) Diminish the number of additional moments required from s to λ. iii) Drop the smoothness on Φ and the integrability on b assumptions (non-cut-off) iv) The method for the a priori bound in our proof is direct and based on the use of an appropriate cancellation lemma and local lower estimate.

48 Theorem 2 ( Propagation of L p -norms )(Alonso, Carneiro& I.M.G'12): Assume the collision kernel satisfies H1-B (mollified potential kernel), H2-B (integrable angular cross section), λ ( n, 0] (up to very soft potentials) p (1, ) and s 0. Let f be a non-negative solution of the Boltzmann equation such that Then, In particular when λ ( 2, 0], the L p s λ /p norm of f is uniformly propagated for initial data f 0 L 1 max{2,s+ λ (2+1/p )} L p s. Theorem 3: ( Propagation of L p -norms )(Alonso, Carneiro& I.M.G'12): Assume the collision kernel satisfying H1-A (non-mollified potential) and b L a with a>1 (this is H2-B+ condition) Let s 0 and p ( (n/ λ ), ) and assume f 0 belonging to L=L 1 2+s L p s. Then, there exists λ 0 ( 2, 0) depending on f 0 L and such that

49 Comments on the propagation of L p -norms. 1- These estimates treats mollified potentials and its proof and uses the gain of integrability estimates (Alonso, I.M.Gamba, KRM'11) as done for propagation of L p -integrability in the case of hard potentials (λ ϵ (0,1] ) (as also done in Mouhot & Villani ARMA'04 for propagation of any high order Sobolev norms) 2- Here we get a unified approach to treat both hard and soft potentials as well as relaxes and simplifies considerably the methods and assumptions Lions, Wennberg and Mouhot & Villani 3- These results also show that propagation of L p integrability is a consequence of a priori uniform propagation of just a few moments calculated explicitly

50 Part III Existence, regularity and stability of solutions near Maxwellian Distributions Kaniel-Shimbrot iteration. L p propagation of gradients in physical and velocity space. L p stability.

51 7/12/2012

52

53

54 This estimate implies global in time control. This and all estimates that follow hold for L - Global Maxwellians weights of the form

55

56

57

58

59 Scattering property for Boltzmann Solutions. Recall from the classical scattering results: the advection operator A = v x generates the group e ta that acts on every function F that is defined almost everywhere by the formula When F in is locally integrable then F = e ta F in is the unique distribution solution of the initial-value problem In the Boltzmann setting we have the following scattering Theorem: Let F (v, x, t) be a global mild solution of the Cauchy problem for the Boltzmann eq. with potentials λ (-2, n-1] that also satisfying all estimates listed above. Then there exists a unique F (v, x) integrable such that and F satisfies the bound almost everywhere over R n x R n. with

60 II- Sketch of proofs for L p gradients regularity for solutions to the Boltzmann equation in L M a,b (R n ) and L p stability

61

62

63

64

65

66

67 Part IV Approximations by Spectral-Lagrangian Constrained methods Deterministic solvers for Dissipative models - The space homogeneous problem FFT application - Computations of Self-similar solutions Space inhomogeneous problems Time splitting algorithms Simulations of boundary value layers problems Benchmark simulations Collaborators: R. Alonso, Rice U. J. Haack, UT Austin S.H. Tharskabhushanam, Birmingham, UK

68 7/12/2012

69 7/12/2012

70 7/12/2012

71 7/12/2012

72 7/12/2012

73 7/12/2012

74 7/12/2012

75 7/12/2012

76 7/12/2012

77 7/12/2012

78 7/12/2012

79 7/12/2012

80

81 7/12/2012

82 Elastic collisions contours evolution Elastic collisions evolution -3D

83 Inelastic collisions contours evolution 7/12/2012 Inelastic collisions evolution - 3D

84 Discontinuous Initial Data Elastic collisions

85

86

87 Next we need to recall self-similarity:

88

89

90

91

92 7/12/2012

93

94 7/12/2012

95

96 7/12/2012

97 Bibliography and recent work related to these problems: -R.Alonso, E.Carneiro, I.M.G. ArXiv.org 08 &09, CMP 10 (weigthed Young s inequality and Hardy Sobolev s inequalities for collisional integrals with integrable (grad cut-off)angular cross section) -R. Alonso and I.M.Gamba, JSP 09&11, (Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section) -R. Alonso and I.M.Gamba, KRM 10 (Gain of integrability estimates) -R.Alonso, Canizo, I.M.Gamba, C. Mouhot, in preparation (sharper decay for moments creation estimates for variable hard potentials) -Cercignani, C.;'95(inelastic BTE derivation); -Cercignani, C.; The Boltzmann equation and its applications. Applied Mathematical Sciences, 67. Springer-Verlag, New York, Cercignani, C.; Illner, R.; Pulvirenti, M.; The mathematical theory of dilute gases. Applied Mathematical Sciences, 106. Springer-Verlag, New York, Bobylev, A.V., JSP 97 (elastic,hard spheres in 3 d: propagation of L 1 -exponential estimates ); -Bobylev, A.V., Carrillo, J.A. and Gamba,I.M; JSP'00 (inelastic Maxwell type interactions- self similarity- mean field); -A.V. Bobylev, I.M. Gamba, V.Panferov, C.Villani, JSP'04, CMP 04 (inelastic + heat sources); -A.V. Bobylev and I.M.Gamba, JSP'06 (Maxwell type interactions-inelastic/elastic + thermostat), -A.V. Bobylev, C.Cercignani and I.M.Gamba arxiv.org,06 (CMP 09); (generalized multi-linear Maxwell type interactions-inelastic/elastic: global energy dissipation) -I.M.Gamba, V.Panferov, C.Villani, arxiv.org 07, ARMA 09 (elastic n-dimensional variable hard potentials Grad cutoff: propagation of L 1 and L - exponential estimates) -C.Mouhot and C. Villani, ARMA 04(L p estimates and regularity for the space homogeneous Boltzmann equation) -C. Mouhot, CMP 06 (elastic, VHP, bounded angular cross section: creation of L 1 -exponential ) -R. Alonso and I.M.Gamba, JMPA 08 (Grad cut-off, propagation of regularity bounds-elastic d-dim VHP) -I.M.Gamba and Harsha Tarskabhushanam JCP 09(spectral-lagrangian solvers-computation of singularities) -I.M.Gamba and Harsha Tarskabhushanam JCM 09 (Shock and Boundary Structure formation by Spectral- Lagrangian methods for the Inhomogeneous Boltzmann Transport Equation). -I. Ibragimov and S. Rjasanow; Numerical Solution of the Boltzmann Equation on the Uniform Grid; Computing, 69, , (2002).

98 Thank you very much for your attention! References: at and references therein