Boltzmann equation for soft potentials with Boltzmann equation for soft potentials with integrable angular cross sectionintegrable angular cross section

The Cauchy problemThe Cauchy problem

Irene M. Gamba

The University of Texas at Austin

Mathematics and ICES

IPAM April 2009- KTWSII

In collaboration with Ricardo Alonso

‘v

‘v*

v

v*

C = number of particle in the box a = diameter of the spheresN=space dimension η

elastic collisioninelastic collisionη the impact direction

i.e. enough intersitial space

May be extended to multi-linear interactions

d-1

Consider the Cauchy Boltzmann problem (Maxwell, Boltzmann 1860s-80s);

Grad 1950s; Cercignani 60s; Kaniel Shimbrot 80’s, Di Perna-Lions late 80’s) Find a function f (t, x, v) ≥ 0 that solves the equation (written in strong form)

σ is the impact direction: ′u=u - 2(u· σ) σ (specular reflection condition)

Assumption on the model: The collision kernel B(u, û · σ) satisfies

(i) B(u, û · σ) = |u|λ b(û · σ) with -n < λ ≤ 1 ; we call soft potentials: -n < λ < 0

(ii) Grad’s assumption: b(û · σ) L∈ 1(S n−1), that is

with

Conservative interaction(elastic)

Grad’s assumption allows to split the collision operator in a gain and a loss part,

Q( f, g) = Q+( f, g) − Q−( f, g) = Gain - Loss

But not pointwise bounds are assumed on b(û · σ)

The loss operator has the following structure

Q−( f, g) = f R(g), with R(g), called the collision frequency, given by

The loss bilinear form is a convolution.

We shall see also the gain is a weighted convolution

λ

|u|λ

|u|λ

More generally, the exchange of velocities in center of mass-relative velocity frame

Energy dissipation parameter or restitution parameters

Recall: Q+(v) operator in weak (Maxwell) form, and then it can easily be extended to dissipative (inelastic) collisions

Same the collision kernel form

With the Grad Cut-off assumption:

with

Q−( f, g) = fAnd convolution structure in the loss term: λ

β=1 elastic interactionelastic interaction

ω is the scattering direction with respect to an elastic collision: ω= u′ /|u| were u′ and u satisfyThe relation of specular reflection: u′ = u -2(u· σ) σ

cos (u· ω) = π – 2 cos (u· σ).

Outline

•In Alonso’s lecture:• Average angular estimates (for the inelastic case as well)Average angular estimates (for the inelastic case as well)

• weighted Young’s inequalities for 1 ≤ p , q , r ≤ ∞ weighted Young’s inequalities for 1 ≤ p , q , r ≤ ∞ (with exact constants) (with exact constants) 0 ≤0 ≤ λλ = 1 = 1

• Sharp constantsSharp constants for for Maxwell typeMaxwell type interaction for (p, q , r) = interaction for (p, q , r) = (1,2, 2) and (2,1,2) λλ = 0= 0 •Hardy Littlewood Sobolev inequalities , for 1 < p , q , r < ∞ Hardy Littlewood Sobolev inequalities , for 1 < p , q , r < ∞ (with exact constants) (with exact constants) -n ≤-n ≤ λλ < 0 < 0

In this lectureExistence, uniqueness and regularity estimates for the near vacuum and near (different) Existence, uniqueness and regularity estimates for the near vacuum and near (different) Maxwellian solutions for the space inhomogeneous problem Maxwellian solutions for the space inhomogeneous problem (using Kaniel-Shimbrot iteration type solutions) elastic interactions for soft potential (using Kaniel-Shimbrot iteration type solutions) elastic interactions for soft potential and the above estimates.and the above estimates.

LLpp stability estimates in the soft potential case, for 1 < p < ∞ stability estimates in the soft potential case, for 1 < p < ∞

Average angular estimates & weighted Young’s inequalities & Average angular estimates & weighted Young’s inequalities & Hardy Littlewood Sobolev inequalities & sharp constantsHardy Littlewood Sobolev inequalities & sharp constants

R. Alonso and E. Carneiro’08, and R. Alonso and E. Carneiro, IG, 09 (ArXiv.org):by means of radial symmertrization techniques radial symmertrization techniques

Denoting by

Bobylev’s variables and operatorBobylev’s variables and operator

Bobylev’s operator on Maxwell type interactions λ=0is the well know identity for the Fourier transform of the Q+

Translation and reflection operators

is invariant under rotations

Young’s for variable hard potentials and Maxwell type interactions Young’s for variable hard potentials and Maxwell type interactions

Hardy-Littlewood-Sobolev type inequality for soft potentialsHardy-Littlewood-Sobolev type inequality for soft potentials

0 ≤0 ≤ λλ = 1 = 1

-n <-n < λλ < 0 < 0

for

Inequalities with Maxwellian weightsInequalities with Maxwellian weights

As an application of these ideas one can also show Young type estimates for the non-symmetric Boltzmann collision operator with Maxwellian weights.

For any a > 0 define the global Maxwellian as

Consider the Cauchy Boltzmann problem:

B(u, û · σ) = |u|−λ b(û · σ) with 0 ≤ λ < n-1 with the Grad’s assumption:

Q−( f, g) = f

Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross sectionwith integrable angular cross section (Ricardo Alonso & I.M.G., 09 submitted)

with

(1)

Definition: A distributional (mild) solution in [0; T] of BTE initial value problem is a function f ϵ W1;1(0; T;L∞(R2n)) that solves (1) a.e. in (0; T] x R2n such that

, satisfies

Consider the space with the norm

Notation and spaces:

Kaniel & Shinbrot iteration ’78 (DP-L -11yrs)

Kaniel-Shinbrot:(also Illner & Shinbrot ’84) define the sequences {ln(t)} and {un(t)} as the mild solutions to the system

and

which relies in choosing a initial pair of functions (l0, u0) satisfying so called the beginning condition in [0, T]:

where the pair (l1, u1) solves the system with initial state (l0, u0).

Theorem: Let {ln(t)} and {un(t)} the sequences defined by the mild solutions of the linear system above, such that the beginning condition is satisfied in [0, T], then

(i) The sequences {ln(t)} and {un(t)} are well defined for n ≥ 1. In addition, {ln(t)}, {un(t)} areincreasing and decreasing sequences respectively, and

l#n (t) ≤ u#

n (t) a.e. in 0 ≤ t ≤ T.

(ii) If 0 ≤ ln(0) = f0 = un(0) for n ≥ 1, then

lim ln(t) = lim un(t) = f(t) a.e. in [0; T]: n∞ n∞

In addition the limit f (t) ∈ C(0, T; M#α,β) is the unique distributional solution of the Boltzmann

equation in [0, T] and fulfills

0 ≤ l#0(t) ≤ f #(t) ≤ u#

0(t) a.e. in [0, T].

Lemma : Assume −1 ≤ λ < n − 1. Then, for any 0 ≤ s ≤ t ≤ T and functions f #, g# that lie in L∞(0, T;M#

α,β), then the following inequality holds

with

Distributional solutions for small initial data: (near vacuum)

Theorem: Let B(u, û · σ) = |u|−λ b(û · σ) with -1 ≤ λ < n-1 with the Grad’s assumption Then, the Boltzmann equation has a unique global distributional solution if

. Moreover for any T ≥ 0 ,

As a consequence, one concludes that the distributional solution f is controlled by a traveling Maxwellian, and that

Hard and soft potentials case for small initial data

It behaves like the heat equation, asmass spreads as t grows

## ##

##

Theorem: Let B(u, û · σ) = |u|−λ b(û · σ) with -n < λ ≤ 0 with the Grad’s assumption In addition, assume that f0 is ε–close to the local Maxwellian distribution M(x, v) = C Mα,β(x − v, v) , with 0 < α, 0 < β. • Then, for sufficiently small ε the Boltzmann equation has a unique solution satisfying

C1(t) Mα1,β1 (x − (t + 1)v, v ) ≤ f ( t, x-vt , v) ≤ C2(t) Mα2,β2 (x − (t + 1)v, v)

for some positive functions 0 < C1(t) ≤ C ≤ C2(t) < ∞, and parameters 0 < α2 ≤ α ≤ α1

and 0 < β2 ≤ β ≤ β1.

• Moreover, the case α = 0 (infinite mass) is permitted as long as α 1 = α 2 = 0. (this last part extends the result of Mishler & Perthame ’97 to soft potentials)

Distributional solutions near local Maxwellians :Distributional solutions near local Maxwellians : Ricardo Ricardo Alonso, IMG’08Alonso, IMG’08 Previous work by Toscani ’88, Goudon’97, Mischler – Perthame ‘97

Distributional solutions near local Maxwellians :Distributional solutions near local Maxwellians : Ricardo Ricardo Alonso, IMG’08Alonso, IMG’08

Define the distance between two Maxwellian distributions Mi = CiMαi,βi

for i = 1, 2 as d(M1, M2) := |C2 − C1| + |α2 − α1| + |β2 − β1|.

Second, we say that f is ε–close to the Maxwellian distribution M = C Mα,β if there exist Maxwellian distributions Mi (i = 1, 2) such that d(Mi, M) <ε for some small ε > 0, and M1 ≤ f ≤ M2.Also define

and notice that for -n < λ ≤ 0

Sketch of proof:

Following the Kaniel-Shinbrot procedure, one obtains the following non-linear system of inequations

which can be solved in C1(t) and C2(t) for an initial data for t0 ≥ 1 that satisfy an

admissible beginning condition.

Sketch of proof:1-1-

So choose C1(t) and C2(t) such that

(Remark: Mischler &Perthame for λ=0 and ϕ 1 = ϕ2 )

which clearly implies for any t0 , t ≥1or

Therefore

which has a solution of the form

For

2-2-

3-3- Therefore, C2(t) will be uniformly bounded for t ≥ 1 as long as

which can be obtained done by taking d (M1, M2) ≤ ϵ and

Then, for sufficiently small ϵ the Boltzmann equation has a unique solution satisfying

C1(t) Mα1,β1 (x − (t + 1)v, v ) ≤ f ( t, x-vt , v) ≤ C2(t) Mα2,β2 (x − (t + 1)v, v)

for some positive functions 0 < C1(t) ≤ C ≤ C2(t) < ∞, and parameters 0 < α2 ≤ α ≤ α1 and 0 < β2 ≤ β ≤ β1. i.e. the distributional solution f is controlled by a traveling Maxwellian, and so it spreads its mass as t ∞,

In particular, the ‘beginning condition’ follows since, with source and absorption coefficient fixed, a simple comparison arguments of ODE’s shows that The evolution equation for C1(t) with an initial state C1(0)=C1 , with f0 ≥ C1Mαi,βi

, implies . Similarly arguments work for C2(t).

Classical solutions Classical solutions (Different approach from Guo’03, our methods follow some of the those by Boudin & Desvilletes ‘00, plus new ones ) (Different approach from Guo’03, our methods follow some of the those by Boudin & Desvilletes ‘00, plus new ones )

Definition. A classical solution in [0, T] of problem our is a function such that

,

Theorem (Application of HLS inequality to Q+ for soft potentials) : Let the collision kernel satisfying assumptions λ < n and the Grad cut-off, then for 1 < p < ∞

where γ = n/(n−λ) and Ci = C(n, λ, p, ||b||L1(Sn−1) ) with i = 1, 2,3.

The constants can be explicitly computed and are proportional to

with parameter 1 < q = q(n, λ, p) < ∞, (the singularity at s = 1 is removed by symmetrazing b(s) when f = g )

Theorem (space regularity, globally in time ) Fix 0 ≤ T ≤ ∞ and assume the collision kernel satisfies B(u, û · σ) = |u|−λ b(û · σ) with -1 ≤ λ < n-1 with the Grad’s assumption.

Also, assume that f0 satisfies the smallness assumption or is near to a local Maxwellian. In addition, assume that ∇f0 ∈ Lp(R2n) for some 1 < p < ∞. Then, there is a unique classical solution f to the problem in the interval [0, T] satisfying the estimates of these theorems, and

for all t [0, ∈ T],

with constant

Proof: set

: ∫

with

›

x x

for a fix h > 0 and x ∈ S n−1 and the corresp. translation operator and transformingx∗ → x∗ + hx in the collision operator.

›

x

and integrate: ∫

And estimate

By Gronwall inequality

with a = n/(n−λ)

Multiply by

Using HLS estimates on Q(f,g)

and, since the distributional solution f(t; x; v) is controlled by a traveling Maxwellian, then

(by similar arguments)

Then, as h 0 tox

x globally in time

Velocity regularity Velocity regularity (local in time)

Proof : Take for a fix h > 0 and ˆv ∈ S n−1 and the corresp. translation operator and transformingv∗ → v∗ + hˆv in the collision operator.

and : ∫

Theorem Let f be a classical solution in [0, T] with f0 satisfying the condition of the smallness assumption or is near to a local Maxwellian and ∇x f0 L∈ p(R2n) for some 1 < p < ∞. In addition assume that ∇v f0 ∈ Lp(R2n). Then, f satisfies the estimate

multiply by

apply HLS on Q

with

then

Which is solved by

Just set

Then, by the regularity estimate with 0 < λ < n-1

(Bernoulli Eq. )

x

x

Then, as h 0 to

LLpp and M and Mα,βα,β stability stability

Set

Now, since f and g are controlled by traveling Maxwellians one has

and 0 < λ < n-1with a = n/(n−λ)

Theorem Let f and g distributional solutions of problem associated to the initial datumf0 and g0 respectively. Assume that these datum satisfies the condition of theorems for small data or near Maxwellians solutions (0 < λ < n-1) . Then, there exist C > 0 independent of time such that

Moreover, for f0 and g0 sufficiently small in Mα,β

Remark: The result of Ha 06 for L1 stabiltity requires b(û · σ) bounded as a function of the scattering angle. Our result is for integrable b(û · σ) …. but p >1

References and preprints http://rene.ma.utexas.edu/users/gamba/publications-web.htm

Thank you for your attention!Thank you for your attention!