the boltzmann equation, besov spaces, and · 2012. 7. 25. · v. sohinger, r. strain (upenn)...

The Boltzmann equation, Besov spaces, andoptimal time decay rates in Rn

Vedran Sohinger (University of Pennsylvania)Robert Strain (University of Pennsylvania)

Conference in honor of Michael Taylor’s 65th birthdayUniversity of North Carolina, Chapel Hill

July 16-20, 2012

V. Sohinger, R. Strain (UPenn) Boltzmann equation Besov decay rates

The Boltzmann equation

We are considering the Boltzmann equation on Rn:

∂tF + v · ∇xF = Q(F ,F ), F (0, x , v) = F0(x , v). (1)

Here, x ∈ Rn, v ∈ Rn, t ≥ 0.F = F (t , x , v) ≥ 0 physically represents a probabilitydensity of particles in position and velocity.We note that the linear part of the equation is given by atransport equation.The collision operator Q(F ,F ) models binary collisionsbetween particles. It is quadratic in F and local in (t , x).One can also consider (??) for x ∈ Ω ⊆ Rn or x ∈ Tn.

History of the equation and Physical interpretation

The Boltzmann equation was derived from rarefied gasdynamics by Maxwell in 1860 and by Boltzmann in 1872.The dilute gas is modeled by a large number of particleswhich obey Newton’s laws of classical mechanics.They travel in a straight line (→ transport equation) untilthey collide by a binary collision (→ collision operator Q).The equation (??) is obtained in an appropriate limit as thenumber of particles goes to infinity and when the size ofeach particle goes to zero; Boltzmann-Grad limit.

Conservation laws

Given two particles which collide, we denote their incomingvelocities by v , v∗ and we denote their outgoing velocities byv ′, v ′∗. The following conservation laws then hold:

(Conservation of momentum) v + v∗ = v ′ + v ′∗.(Conservation of energy) |v |2 + |v∗|2 = |v ′|2 + |v ′∗|2.

It is useful to parametrize the set of solutions (v ′, v ′∗) to theabove conservation laws by σ ∈ Sn−1. More precisely, we canwrite:

v ′ =v + v∗

2+|v − v∗|

v ′∗ =v + v∗

2− |v − v∗|

for some σ ∈ Sn−1.

Collision operator

We can now define the Boltzmann collision operator Q.

Q(F ,G)(v)def=

dv∗∫

Sn−1dσ B(v − v∗, σ)[F ′∗G

′ − F∗G] (2)

Here, G = G(v),F∗ = F (v∗),G′ = G(v ′),F ′∗ = F (v ′∗).The Boltzmann collision kernel is given by:

B(v − v∗, σ)def= |v − v∗|γb(cos θ), cos θ =

v − v∗|v − v∗|

· σ, γ > −n.

We are working in the non cut-off Boltzmann regime, in whichthe angular function t 7→ b(t) is not locally integrable:

sinn−2 θ · b(cos θ) ∼ θ−1−2s, ∀θ ∈ (0,π

2], s ∈ (0,1), γ+2s > −n

The non cut-off Boltzmann equation 1

The lack of local integrability was avoided by modifying thecollision operator Q by introducing a cut-off in the angularsingularity (Grad 1963)→ different behavior of the equation.The first global stability results for the non cut-off Boltzmannequation on Tn were proved by Gressman and Strain (JAMS2011), and on Rn by Strain (preprint 2010). These worksrigorously justify stability results for Maxwellian equilibria in theoriginal model proposed by Ludwig Boltzmann, and henceresolve a problem which had been open for over 140 years.There is also related work on Rn by using different methodsand obtaining different results by Alexandre, Morimoto, Ukai,Xu, Yang. Further results for the non cut-off Boltzmannequation by Lions, Desvilettes, Chen-Desvilettes-He.

The key idea in the work of Gressman and Strain is to definethe right weighted fractional seminorm:

|f |2Ns,γdef=

dv 〈v〉γ+2s+1∫

d(v ,v ′)≤1dv ′|f (v)− f (v ′)|2

d(v , v ′)n+2s

where d(v , v ′) :=√|v − v ′|2 + 1

4(|v |2 − |v ′|2)2 represents the

distance between two points on the paraboloid (v , 12 |v |

2). If onedefines |f |2Hα`

∫Rn dv 〈v〉`|(I −∆v )

α2 f (v)|2, |f |L2

def= |f |H0

`, and

|f |Ns,γ ∼ |f |L2γ+2s

+ |f |Ns,γ , it is possible to show:

|f |2L2γ+2s

+ |f |2Hsγ. |f |2Ns,γ . |f |2Hs

γ+2s.

Hence, Ns,γ is an anisotropic space which lies in betweenstrictly different standard isotropic spaces.

The key idea in the work of Gressman and Strain is to definethe right weighted fractional seminorm:

|f |2Ns,γdef=

dv 〈v〉γ+2s+1∫

d(v ,v ′)≤1dv ′|f (v)− f (v ′)|2

d(v , v ′)n+2s

where d(v , v ′) :=√|v − v ′|2 + 1

4(|v |2 − |v ′|2)2 represents the

distance between two points on the paraboloid (v , 12 |v |

2). If onedefines |f |2Hα`

∫Rn dv 〈v〉`|(I −∆v )

α2 f (v)|2, |f |L2

def= |f |H0

`, and

|f |Ns,γ ∼ |f |L2γ+2s

+ |f |Ns,γ , it is possible to show:

|f |2L2γ+2s

+ |f |2Hsγ. |f |2Ns,γ . |f |2Hs

γ+2s.

Hence, Ns,γ is an anisotropic space which lies in betweenstrictly different standard isotropic spaces.

Linearization 1

We linearize the Boltzmann equation near the Maxwellian

equilibrium µ(v)def= (2π)−

n2 e−

|v|22 in the following way:

F (t , x , v) = µ(v) +√µ(v)f (t , x , v).

The perturbation f then solves:

∂t f + v · ∇x f + L(f ) = Γ(f , f ), f (0, x , v) = f0(x , v). (3)

Here, the linearized Boltzmann operator L is given by:

L(g)def= −µ−

12Q(µ,

√µg)− µ−

12Q(√µg, µ)

and the bilinear operator Γ is given by:

Γ(g,h)def= µ−

12Q(√µg,√µh).

Linearization 1

We linearize the Boltzmann equation near the Maxwellian

equilibrium µ(v)def= (2π)−

n2 e−

|v|22 in the following way:

F (t , x , v) = µ(v) +√µ(v)f (t , x , v).

The perturbation f then solves:

∂t f + v · ∇x f + L(f ) = Γ(f , f ), f (0, x , v) = f0(x , v). (3)

Here, the linearized Boltzmann operator L is given by:

L(g)def= −µ−

12Q(µ,

√µg)− µ−

12Q(√µg, µ)

and the bilinear operator Γ is given by:

Γ(g,h)def= µ−

12Q(√µg,√µh).

Linearization 2

It is well-known that the linearized operator L has ann + 2-dimensional null-space N(L) consisting of elements of L2

vwhich are exponentially decreasing.

P := Orthogonal projection onto N(L) = Macroscopicprojection.I− P = Microscopic projection.

One then has the following coercivity bound, for some λ > 0(proved by Gressman and Strain):

〈g,Lg〉L2v≥ λ|(I− P)g|2Ns,γ .

Hence, only the microscopic components are part of thedissipation.

Linearization 2

It is well-known that the linearized operator L has ann + 2-dimensional null-space N(L) consisting of elements of L2

vwhich are exponentially decreasing.

P := Orthogonal projection onto N(L) = Macroscopicprojection.I− P = Microscopic projection.

One then has the following coercivity bound, for some λ > 0(proved by Gressman and Strain):

〈g,Lg〉L2v≥ λ|(I− P)g|2Ns,γ .

Hence, only the microscopic components are part of thedissipation.

Global existence

We take for our initial data to be the following perturbation ofthe Maxwellian:

F0 = µ+√µf0.

and we assume that the following quantity

εK ,`def=

∑|α|+|β|≤K

‖w(`, |β|)∂αβ f0‖2L2x L2

v 1. (4)

for K , ` sufficiently large, and w a polynomial weight in v , alldepending on n and ∂αβ

def= ∂α1

x1· · · ∂αn

xn ∂β1v1· · · ∂βn

vn .Work of Gressman and Strain→ global solution to theBoltzmann equation.

Statement of our problem

We work on Rn for n ≥ 3 and we are interested in whathappens when we take the perturbation of the initial data, i.e.f0, to lie in an a homogeneous mixed Besov space given by(semi)norms of the type:

‖f‖B%,∞p L2v

def= sup

j∈Z(2%j‖∆j f‖Lp

x L2v)

where ∆j denotes the Littlewood-Paley projection in x . We notethat this is not the standard definition of mixed-norm spaces.Question: Given initial data in the above mixed Besov space,what can we say about decay rates to the Maxwellianequlibrium?Sobolev embedding:Lp(Rn) ⊆ B−%,∞2 (Rn), % = n

p −n2 , p ∈ [1,2]. Hence, this

question also covers the case of an L1x -type perturbation.

‖f‖B%,∞p L2v

def= sup

x L2v)

‖f‖B%,∞p L2v

def= sup

x L2v)

Statement of our results 1

For all of the results, we assume that the previously definedquantity εK ,` is sufficiently small, for K = K (n).

Theorem 1: (S.-Strain (2012), optimal decay results)

Fix % ∈ (0, n2 ], and k ∈ 0,1, . . . ,K − 1. Suppose that

‖f0‖B−%,∞2 L2v<∞. Then, for t ≥ 0:∑

k≤|α|≤K

‖∂αf (t)‖2L2x L2

v. (1 + t)−(k+%).

Furthermore, if 2 ≤ r ≤ ∞ and k < K − 1− n2 + n

r , then, fort ≥ 0: ∑

|α|=k

‖∂αf (t)‖2Lrx L2

v. (1 + t)−k−%− n

2+nr .

Statement of our results 2

Under certain assumptions, we are able to extend the range of%.

Theorem 2: (S.-Strain (2012), faster decay results)Suppose that we are working in the setting of hard potentials,i.e. γ + 2s ≥ 0, and % ∈ (0, n+2

2 ] ,‖f0‖B−%,∞2 L2v<∞. Recalling the

macroscopic projection P, we assume that:

‖Pf0‖B−%,∞2 L2v

+ ‖(I− P)f0‖B−%+1,∞2 L2

v<∞.

Then, for all t ≥ 0:∑|α|+|β|≤K

‖w(`, |β|)∂αβ f (t)‖2L2x L2

v. (1 + t)−%

for the polynomial weight w defined earlier.

Comments on the results

In Theorem 1, we don’t need to assume that the‖f0‖B−%,∞2 L2

vis small; we avoid the difficulty of keeping an

L1-type quantity uniformly small along the time evolution.The decay results are said to be optimal since they are thesame as those obtained for the linear equation:

∂t f + (v · ∇x + L)f = 0, f (0, x , v) = f0(x , v).

Same decay results as for the cut-off Boltzmann equation:Ukai and Asano (1982) and Ukai (1986).There are analogous decay results for the (linear) heatequation with initial data in B−%,∞2 (Rn) ∀% > 0.The restriction ρ ∈ (0, n+2

2 ] in Theorem 2 is also obtained inthe study of decay properties of Leray-Hopf weak solutionsto the Navier-Stokes equations with initial data in(L2 ∩ B−%,∞2 )n by Wiegner (1987).

∂t f + (v · ∇x + L)f = 0, f (0, x , v) = f0(x , v).

Main ideas of the proof 1

Main ideas of the proof of Theorem 1: We use a wide classof nonlinear energy estimates, motivated by the previous workof Guo-Strain, Strain, Duan. One estimate is, fork ∈ 0,1, . . . ,K:

∑|α|=k

‖∂αf‖2L2x L2

∑|α|=k

‖(I− P)∂αf (t)‖2L2x Ns,γ

. εK ,`∑

mink+1,K≤|α|≤K

‖∂αf‖2L2x Ns,γ .

These types of estimates require us to use interpolation andembedding results for spaces of the type Lp

xHv , where Hv is aseparable Hilbert space in the v -variable. This requires the usevector-valued variants of the Calderón-Zygmund Theory.

Main ideas of the proof 2

Main ideas of the proof of Theorem 2: In the proof ofTheorem 2, we use an improved estimate for solutions to thelinear problem evolving for purely microscopic initial data, i.e.Pf0 = 0.In order to study this case, we use an approach based on thework of Ellis and Pinsky (JMPA 1975) by which we can give aprecise description of the spectrum of the operatorB(ξ)

def= 2πiv · ξ + L, when |ξ| 1.

The main point is that, for |ξ| sufficiently small, B(ξ) has n + 2eigenvalues whose eigenprojections Pj(ξ) satisfy:

Pj(ξ) = P(0)j (ξ) + |ξ| · P(1)

j (ξ).

where P(0)j (ξ)f0 = 0 and P(1)

j (ξ) are uniformly bounded on L2v .

The extra factor of |ξ| is then helpful in obtaining the betterdecay bound.

the boltzmann equation, besov spaces, and · 2012. 7. 25. · v. sohinger, r. strain (upenn)...

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