1. The University of Texas at Austin
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 KTWSII
In collaboration with Ricardo Alonso

2. elastic collision ‘v v inelastic collision v* ‘v*
C = number of particle in the box a = diameter of the spheres
N=space dimension
inelastic collision
η the impact direction η v*
‘v*
d-1
i.e. enough intersitial space
May be extended to multi-linear interactions

3. Conservative interaction
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
B(u, û · σ) = |u|λ b(û · σ) with -n < λ ≤ 1 ; we call soft potentials: -n < λ < 0
Grad’s assumption: b(û · σ) ∈ L1(S n−1), that is

4. Q( f, g) = Q+( f, g) − Q−( f, g) = Gain - Loss
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 λ
The loss bilinear form is a convolution.
We shall see also the gain is a weighted convolution

5. dissipative (inelastic) collisions
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: λ
Q−( f, g) = f
And convolution structure in the loss term:

6. 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 < ∞

7. 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+

8. Young’s for variable hard potentials and Maxwell type interactions
0 ≤ λ = 1
Hardy-Littlewood-Sobolev type inequality for soft potentials
-n < λ < 0

9. 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

10. 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

11. 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).

12. increasing and decreasing sequences respectively, and
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) 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].

13. Hard and soft potentials case for small initial data
Lemma : Assume −1 ≤ λ < n − 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
It behaves like the heat equation, as
mass spreads as t grows

14. 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)

15. 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 for some small ε > 0, and M1 ≤ f ≤ M2.
Also define
and notice that for -n < λ ≤ 0

16. which can be solved in C1(t) and C2(t)
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 )

17. for any t0 , t ≥1 Therefore or which clearly implies 2-
which has a solution of the form
For

18. 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 C1(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 ∞,

19. 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 )

20. 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 x x x
Proof: 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. › ›
: ∫

21. Multiply by
and integrate: ∫
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
globally in time x

22. 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
Proof : Take
for a fix h > 0 and ˆv ∈ S n−1 and the
corresp. translation operator and transforming
v∗ → v∗ + hˆv in the collision operator.
multiply by
and : ∫
apply HLS on Q

23. 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

24. 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

25. 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

26. Thank you for your attention!
References and preprints