Boltzmann equation for soft potentials with
integrable angular cross section
The Cauchy problem
Irene M. Gamba
The University of Texas at Austin
Mathematics and ICES
IPAM April 2009- KTWSII
In collaboration with Ricardo Alonso
‘v
C = number of particle in the box
a = diameter of the spheres
N=space dimension
η the impact direction
η
‘v*
d-1
i.e. enough intersitial space
May be extended to multi-linear interactions
v
v*
elastic col
inelastic
collision
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)
with
Conservative interaction
(elastic)
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) ∈ L1(S n−1), that is
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
|u|λ
|u|λ
The loss bilinear form is a convolution.
We shall see also the
gain is a weighted convolution
λ
Recall: Q+(v) operator in weak (Maxwell) form, and then it can easily be extended to
dissipative (inelastic) collisions
ω is the scattering direction with respect to an elastic collision: ω= u′ /|u| were u′ and u satisfy
The relation of specular reflection: u′ = u -2(u· σ) σ

cos (u· ω) = π – 2 cos (u· σ).
More generally, the exchange of velocities in center of mass-relative velocity frame
Energy dissipation parameter or restitution parameters
with
β=1 elastic interaction
Same the collision kernel form
With the Grad Cut-off assumption:
And convolution structure in the loss term:
Q−( f, g) = f
λ
Outline
•In Alonso’s lecture:
• Average angular estimates (for the inelastic case as well)
• weighted Young’s inequalities for 1 ≤ p , q , r ≤ ∞ (with exact constants) 0 ≤ λ = 1
• Sharp constants for Maxwell type interaction for (p, q , r) = (1,2, 2) and (2,1,2)
λ= 0
•Hardy Littlewood Sobolev inequalities , for 1 < p , q , r < ∞ (with exact constants) -n ≤ λ < 0
In this lecture
Existence, uniqueness and regularity estimates for the near vacuum and near (different)
Maxwellian solutions for the space inhomogeneous problem
(using Kaniel-Shimbrot iteration type solutions) elastic interactions for soft potential
and the above estimates.
Lp stability estimates in the soft potential case, for 1 < p < ∞
Average angular estimates & weighted Young’s inequalities &
Hardy 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
Bobylev’s variables and operator
is invariant under rotations
Denoting by
Translation and reflection operators
Bobylev’s operator on Maxwell type interactions λ=0
is the well know identity for the Fourier transform of the Q+
for
Young’s for variable hard potentials and Maxwell type interactions
0≤ λ=1
Hardy-Littlewood-Sobolev type inequality for soft potentials
-n < λ < 0
Inequalities 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
Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials
with integrable angular cross section (Ricardo Alonso & I.M.G., 09 submitted)
Consider the Cauchy Boltzmann problem:
(1)
B(u, û · σ) = |u|−λ b(û · σ) with 0 ≤ λ < n-1 with the Grad’s assumption:
with
Q−( f, g) = f
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
Kaniel & Shinbrot iteration ’78
(DP-L -11yrs)
Notation and spaces:
Consider the space
with the norm
Kaniel-Shinbrot:(also Illner & Shinbrot ’84)
define the sequences {ln(t)} and {un(t)} as the mild solutions to the system
which relies in choosing a initial pair of functions (l0, u0) satisfying so called
the beginning condition in [0, T]:
and
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)} are
increasing and decreasing sequences respectively, and
l#n (t) ≤ u#n (t)
(ii) If 0 ≤ ln(0) = f0 = un(0) for n ≥ 1,
lim
n∞
ln(t) = lim
n∞
a.e. in 0 ≤ t ≤ T.
then
un(t)
=
f(t)
a.e. in [0; T]:
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].
Hard and soft potentials case for small initial data
Lemma : Assume −1 ≤ λ < n − 1.
Then, for any 0 ≤ s ≤ t ≤ T and functions
#
L∞(0, T;M α,β), then the following inequality holds
f #, g# that lie in
#
#
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
It behaves like the heat equation, as
mass spreads as t grows
Distributional solutions near local Maxwellians : Ricardo Alonso, IMG’08
Previous work by Toscani ’88, Goudon’97, Mischler – Perthame ‘97
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 : Ricardo Alonso, IMG’08
Sketch of proof:
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
Also define
and notice that for -n < λ ≤ 0
for some small ε > 0, and
M1 ≤ f ≤ M2.
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-
So choose C1(t) and C2(t) such that
(Remark: Mischler &Perthame for λ=0 and
ϕ 1 = ϕ2 )
which clearly implies
2-
Therefore
which has a solution of the form
For
or
for any t0 , t ≥1
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
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 C 1(t) with an initial state C1(0)=C1 ,
with f0 ≥ C1Mαi,βi , implies
. Similarly arguments work for C2(t).
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 ∞,
Classical solutions
(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 ∇fx0 ∈ 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],
x
x
with constant
set
›
with
for a fix h > 0 and x ∈ S n−1 and the
corresp. translation operator and transforming
x∗ → x∗ + hx in the collision operator.
›
Proof:
:∫
and integrate: ∫
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
with a = n/(n−λ)
And estimate
(by similar arguments)
By Gronwall inequality
Then, as h  0 to
x
x
globally in time
Velocity regularity (local in time)
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 ∈Lp(R2n) for some 1 < p < ∞. In
addition assume that ∇v f0 ∈ Lp(R2n). Then, f satisfies the estimate
for a fix h > 0 and ˆv ∈ S n−1 and the
corresp. translation operator and transforming
v∗ → v∗ + hˆv in the collision operator.
Proof : Take
multiply by
and :
apply HLS on Q
∫
Just set
then
(Bernoulli Eq. )
with
x
Which is solved by
Then, by the regularity estimate
with 0 < λ < n-1
x
Then, as h  0 to
Lp and Mα,β stability
Set
Now, since f and g are controlled by traveling Maxwellians one has
with a = n/(n−λ)
and 0 < λ < n-1
Theorem Let f and g distributional solutions of problem associated to the initial datum
f0 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
Thank you for your attention!
References and preprints http://rene.ma.utexas.edu/users/gamba/publications-web.htm