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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

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 Lp estimates.

• Existence, regularity and stability of solutions near Maxwellian Distributions

• Kaniel-Shimbrot iteration.

• Lp propagation of gradients in physical and velocity space.

• Lp stability.

• 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

• 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.

Part I

elastic collisions

v*

v

. u

v’* u'

θ

v*

v

. σ

. . 1-β

u

v'

v’*

β

u' e

β =(1+e)/2 ↔ 1- β +e = β

Elastic collision Inelastic collision

σ = uref/|u| is the unit vector in the direction

of the relative velocity with respect to an elastic collision

Interchange of velocities during a binary collision or interaction

γ

Remark: θ ≈ 0 grazing and θ ≈ π head on collisions or interactions

v' θ

σ

γ

i.e. enough intersitial space

May be extended to multi-linear interactions (in some special cases to see later)

:= statistical correlation function (sort of mean field ansatz,i.e. independent of v)

= for elastic interactions (e=1)

:= mass density

The Boltzmann Transport equation for elastic or inelastic collisions

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

e(z) + zez(z) = θz(z) =( z e(z) )z J(e(z)) = In addition, the Jacobian of the

transformation is then given by

γ

θ

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

σ

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β=1 elastic

σ

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)

is the angular cross section

satisfies

Collisional kernel or transition probability of interactions is calculated using intramolecular

potential laws:

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, for the case of inelastic interactions.

Exact energy identity for a Maxwell type interaction models

Then f(v,t) → δ0 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!

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)

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, v1 , v2 , · · · , vD , |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 xT -xvT, v · (x − vt) , |x − vt|2 / 2

By integrating the corresponding local conservation laws over space and time,

with m > 0 and (a, b, c, B) Ω where Ω is the open cone in R x R x R x RD∧D defined by

Ω = { (a, b, c, B) : (ac-b2) I + B2 > 0 }

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

Any global Maxwellian can be written as a local Maxwellian form:

with

and

Local and global Maxwellians

Property 1: let M1 and M2 be global Maxwellians with parameters given

by (m1, a1 , b11 , c1 , B1 ) ∈ R+ × Ω and (m2, a2, b2, c2, B2 ) ∈ R+ × Ω resp. respectively.

Then M1≤ M2 for every (v, x, t) if and only if

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 [t1, t2] ⊂ [0, ∞) one has the L1 -bound

meaning

with

Next we need to recall self-similarity:

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).

Inelastic Collision u’= (1-β) u + β |u| σ , with σ the direction of elastic post-collisional relative velocity

‘v

‘v*

v

v*

η

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

•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 Lp estimates.

Part II

creation of moments estimates

creation of exponentially weighted

lower bound

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)

Remark: existence theory and stability for γ=0 (Maxwell type of interaction) was developed

by Wild, Morgensten’40 and Bobylev’75.

Sharp Povzner estimates

Summability of moments

series

creation of exponentially L∞-weighted estimates:

in progress (with R. Alonso and C. Mouhot (JMPA’08)

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I- Sketch of proofs for Radial Symmetryzation, Young’s and HLS

inequalities for the collisional integrals

1- Radial symmetrization

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.

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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.

Then, for the norms

Propagation of Moments and Lps estimates:

Local lower estimate: f is a solution of the Boltzmann eq. with bounded mass, energy and

entropy. Then, there exists δ = δ(m0 , m2 , H(f0)) > 0 and a constant

C =C(m0,m2,λ) > 0, such that for any v ∈ Rn and t ≥ 0

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 f0 ∈ L1s+|λ|(R

n) ∩ L logL (Rn). 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 C8 = C8(m0 , m2 , H(f0), b, s, λ+2 ) and C9 = C9 (m0 , m2 , b, s, λ), with λ+2>0.

and, by interpolation with

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 f0 ∈ L12s ∩ L2

q0 for some q0 > 0.

4- 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 f0 ∈ Lp, 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.

Theorem 2 ( Propagation of Lp-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 Lp

s−|λ|/p norm of f is uniformly propagated

for initial data f0 ∈ L1max{2,s+|λ|(2+1/p )} ∩ Lp

s .

Theorem 3: ( Propagation of Lp-norms )(Alonso, Carneiro& I.M.G'12):

Assume the collision kernel satisfying H1-A (non-mollified potential) and

b∈ La with a>1 (this is H2-B+ condition)

Let s ≥ 0 and p ∈ ( (n/|λ|) , ∞) and assume f0 belonging to L=L12+s ∩ Lp

s. Then, there exists λ0 ∈ (−2, 0) depending on ||f0||L and such that

Comments on the propagation of Lp-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 Lp -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 Lp integrability is a

consequence of a priori uniform propagation of just a few moments calculated

explicitly

Part III

• Existence, regularity and stability of solutions near Maxwellian Distributions

• Kaniel-Shimbrot iteration.

• Lp propagation of gradients in physical and velocity space.

• Lp stability.

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This estimate implies global in time control.

This and all estimates that follow hold for L∞ - Global Maxwellians weights of the form

Scattering property for Boltzmann Solutions.

Recall from the classical scattering results:

the advection operator A = −v · ∇ x generates the group etA that acts on every function F

that is defined almost everywhere by the formula

When Fin is locally integrable then F = etA Fin 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 Rn x Rn .

with

II- Sketch of proofs for Lp gradients regularity for solutions to the

Boltzmann equation in L∞ Ma,b

(Rn ) and Lp stability

Part IV

Approximations by

Spectral-Lagrangian Constrained methods

Collaborators:

R. Alonso, Rice U.

J. Haack, UT Austin

S.H. Tharskabhushanam, Birmingham, UK

• 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

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Elastic collisions contours evolution

Elastic collisions evolution -3D

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Inelastic collisions contours evolution

Inelastic collisions evolution - 3D

Discontinuous Initial Data – Elastic collisions

Next we need to recall self-similarity:

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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, 1988.

-Cercignani, C.; Illner, R.; Pulvirenti, M.; The mathematical theory of dilute gases. Applied

Mathematical Sciences, 106. Springer-Verlag, New York, 1994.

-Bobylev, A.V. , JSP 97 (elastic,hard spheres in 3 d: propagation of L1-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 cut-

off: propagation of L1 and L∞-exponential estimates)

-C.Mouhot and C. Villani, ARMA’04(Lp

estimates and regularity for the space homogeneous Boltzmann equation)

-C. Mouhot, CMP’06 (elastic, VHP, bounded angular cross section: creation of L1-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, 163-186, (2002).

Thank you very much for your attention!

References: at www.ma.utexas.edu/users/gamba/research and references therein