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Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Non Isotropic Cauchy Theory for the Boltzmann Nordheim Equation for Bosons.

Amit Einav, University of Cambridge1

Nonlocal Nonlinear Partial Differential Equations and Applications Anacapri, Italy

17th of September, 2015

1Joint Work with Marc Briant

The Author was supported by EPSRC grant EP/L002302/1

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Table of Contents

1 The Spatially Homogeneous Boltzmann-Nordheim equation for Bosons 2 Bose Einstein Condensation 3 Known Results 4 A Local in Time Cauchy Theory for the Boltzmann-Nordheim Equation 5 Strategy of the Proof 6 Global Existence for the Boltzmann-Nordheim Equation 7 Final Remarks

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

The Spatially Homogeneous Boltzmann Equation

One of the most influential equation in the kinetic theory of gases is the so-called Boltzmann equation, describing the time evolution of the probability density of a particle in a classical dilute gas. In its spatially homogeneous form it reads as

• ∂tf (v) = QB(f )(v)

t > 0, v ∈ Rd f |t=0 = f0, with QB(f )(v) =

• Rd ×Sd−1 B(v, v∗, σ)
• f (v ′)f (v ′

∗) − f (v)f (v∗)

• dv∗dσ,

where dσ is the uniform probability measure on the sphere, B is the collision kernel, containing all the physical information about the interactions between the particles, and v ′ = v + v∗ 2 + |v − v∗| σ 2 , v ′

∗ = v + v∗

2 − |v − v∗| σ 2 .

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

The Collision Kernel

Most normal physical situations correspond to the case where B(v, v∗, σ) = Φ (|v − v∗|) b (cos θ) , with cos θ =

• v−v∗

|v−v∗|, σ

• and Φ(z) = CΦzγ.

The power γ represents the ’hardness’ of the potential. In what will follow we will assume that γ ∈ [0, 1], a regime containing the interesting cases of the Maxwell Molecules (γ = 0) and Hard Spheres (γ = 1). In general, the angular part of the collision kernel, b, satisfies b(cos θ) sind−2(θ) ∼

θ→0+ b0θ−(1+ν),

for some ν ∈ (−∞, 2). Removing the singularity, i.e. requiring that lb = π b(cos θ) sind−2(θ)dθ < ∞ corresponds to the so-called Grad’s angular cut off condition, which we will also assume in what follows. In fact, we will require that b∞ = bL∞ < ∞.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

The Spatially Homogeneous Boltzmann-Nordheim Equation

The Boltzmann equation arises in classical mechanics, and bears no consideration to possible quantum statistical effects. This was corrected in 1928 by Nordheim, who suggested a new equation that takes into account the fact that in the quantum statistics the probability to end up in a state may also depend on the number of particles already occupying that state. His modification to the original equation reads as

• ∂tf (v) = QBN(f )(v)

t > 0, v ∈ Rd f |t=0 = f0, (1) where QBN(f )(v) =

• Rd ×Sd−1 B(v, v∗, σ)
• f (v ′)f (v ′

∗) (1 + αf (v)) (1 + αf (v∗))

−f (v)f (v∗)

• 1 + αf (v ′)

1 + αf (v ′

∗)

• dvdv∗dσ,

with α = 1 for bosons (the probability to occupy the same state as another particle is increased) and α = −1 for fermions (the probability to

• ccupy the same state as another particle is decreased).

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Conserved Quantities and the Entropy

Much like the classic Boltzmann equation, the Boltzmann-Nordheim equation for both bosons and fermions conserves mass, momentum and energy:

• Rd

  1 v |v|2   f (t, v)dv =

• Rd

  1 v |v|2   f0(v)dv =   M0 u M2   . The Boltzmann-Nordheim equation also admits an entropy functional. In the bosonic gas case it is given by S(f ) =

• Rd ((1 + f (v)) log(1 + f (v)) − f (v) log f (v)) dv.

Under the Boltzmann-Nordheim flow the entropy increases with time, yet as a difference of two terms it gives no control over possible blow-ups.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

The Bose Einstein Condensation

It has been shown that for a given mass M0, momentum u, and energy M2 there exists a unique maximiser to the entropy S which is of the form FBE(v) = m0δ(v − u) + 1 e

β 2 ((v−u)2−µ) − 1

, with

m0 ≥ 0 and m0 or µ must be zero. β ∈ (0, ∞] is the inverse of the equilibrium temperature. µ ∈ (−∞, 0] is the chemical potential.

This suggests that any solution to the Boltzmann-Nordheim equation should converge to an appropriate FBE as time goes to infinity. The phenomena of the appearance of a delta function at the average momentum u corresponds to the physical aggregation of all the particles at the same velocity. This is known as the Bose-Einstein Condensation.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

The Bose Einstein Condensation Cont.

The Condensation phenomena can be expressed in a simple, physical, condition depending on the mass and energy of the gas. In the case when d = 3 one has that m0 = 0 if and only if M0 ≤ ζ(3/2) (ζ(5/2))

3 5

4π 3 3

5

M

3 5

2 ,

where ζ is the Riemann Zeta function. This can be recast in terms of a critical temperature condition.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Known Results

Up till now, most results for the Boltzmann-Nordheim equation for bosons have been obtained in the isotropic setting.

• X. Lu: Global in time Cauchy theory for isotropic initial data with

bounded mass and energy in L1 1 + |v|2 . Lu also developed the theory for distributional solution and has shown long time convergence towards equilibrium. Escobedo and Vel´ azques: Local in time Cauchy Theory in L∞ 1 + |v|6+0 . Additionally, Escobedo and Vel´ azques gave conditions under which, in the isotropic setting, a blow up (implying possible condensation) in finite time must occur. The main goal of the presented work is to develop a robust Local in time Cauchy Theory for the Boltzmann-Nordheim equation in a general framework.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Local in Time Cauchy Theory for the Boltzmann-Nordheim Equation

• Notations

In what follows we will use the following notations: Lp

s,v =

• f ∈ Lp

v (R) | f Lp

s,v = (1 + |v|s) f (v)Lp v (R) < ∞

• Mα =
• Rd |v|α f (v)dv

b∞ = bL∞ lb =

• Sd−1 b (cos θ) dσ

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Local in Time Cauchy Theory for the Boltzmann-Nordheim Equation

• Main Result

Theorem (Briant & E. 2015) Let f0 ≥ 0 be in L1

2,v ∩ L∞ s,v where d ≥ 3 and d − 1 < s. Then, there exists

T0 > 0 for which there exists a unique non-negative solution to the Boltzmann-Nordheim equation on [0, T0) × Rd, f ∈ L∞

loc

• [0, T0), L1

2,v ∩ L∞ v

• ,

that preserves mass and energy. Moreover, this solution satisfies For any 0 ≤ s′ < ¯ s, where ¯ s = min

• s,

d 1+γ

• s − d + 1 + γ + 2(1+γ)

d

• ,

we have that f ∈ L∞

loc

• [0, T0), L1

2,v ∩ L∞ s′,v

• ,

T0 = ∞

• r

lim

T→T −

f L∞

[0,T]×Rd = ∞,

f preserves the momentum of f0, for all α > 0 and for all 0 < T < T0, Mα(t) ∈ L∞

loc ([T, T0)) .

T0 only depends on d, s, CΦ, b∞, lb, γ, f0L1

2,v and f0L∞ s,v .

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Local in Time Cauchy Theory for the Boltzmann-Nordheim Equation

• Key Observations

Due to Grad’s angular cut off assumption we can write the collision

• perator QBN in the following way:

QBN(f )(v) = Q+

BN(f )(v) − f (v)Q− BN(f )(v),

with Q+

BN(f )(v) = CΦ

• Rd ×Sd−1 |v − v∗|γ b (cos θ) f (v ′)f (v ′

∗)

(1 + f (v) + f (v∗)) dv∗dσ, Q−

BN(f )(v) = CΦ

• Rd ×Sd−1 |v − v∗|γ b (cos θ) f (v∗)
• 1 + f (v ′) + f (v ′

∗)

• dv∗dσ.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Local in Time Cauchy Theory for the Boltzmann-Nordheim Equation

• Key Observations Cont.

The gain and loss terms of the Boltzmann-Nordheim collision operator control, and are controlled, by the gain and loss terms of the classic Boltzman collision operator up to a factor of f L∞

v :

Q+/−

B

(f ) ≤ Q+/−

BN (f ) ≤

• 1 + 2 f L∞

v

• Q+/−

B

(f ). As for small periods of time we don’t expect any blow up we conclude that the Cauchy Theory for the Boltzmann-Nordheim equation should follow from similar arguments to those given for the classic Boltzmann equation together with L∞ estimations. The main theorem presented is shown by appropriate variants of the Mischler-Wennberg method (existence and uniqueness), entangled with techniques developed by Arkeryd (L∞ control).

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Constructing a Solution

The construction of a local solution to the Boltzmann-Nordheim equation is done by an explicit Euler Scheme. The first step is to define the truncated collision kernel, Qn, by replacing the collision kernel B(v, v∗, σ) in QBN with Bn(v, v∗, σ) = (|v − v∗| ∧ n)γ b (cos θ) . The next step would be to create a scheme of the form

• f (0)

n

= f0 f (k+1)

n

(v) = f (k)

n

• 1 − ∆nQ−

n (f (k) n

)

• + ∆nQ+

n (f (k) n

), (2) for an appropriate time step ∆n and the natural truncated gain and loss

• perators Q−

n and Q+ n . While the mass and energy are easy to control in

the above scheme, the L∞ control, especially for the gain operator, is not trivial.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Carleman’s Representation

In order to deal with the L∞ norm for the gain operator (truncated or not) one recalls the Carleman representation for the classic Boltzmann equation Q+

B (f )(v) =

• Rd ×Sd−1 B(v − v∗, σ)f (v ′)f (v ′

∗)dv∗dσ

=

• Rd

dv ′ |v − v∗|    

• Evv′

B

• 2v − v ′

∗ − v ′, v′

∗−v′

|v′

∗−v′|

• |v ′

∗ − v ′|d−2

f (v ′)f (v ′

∗)dE(v ′ ∗)

    , where Evv′ is the hyperplane that passes through v and is perpendicular to v − v ′, and dE is the Lebesgue measure restricted to it.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Carleman’s Representation Cont.

One can show that in our setting Q+

BN(f )(v) ≤ CΦb∞

• 1 + 2 f L∞

v

• sup

v,v′

• Evv′

f (v ′

∗)dE(v ′ ∗)

• Rd f (v ′)
• v − v ′

−d+1+γ dv ′

• L∞

v

, for all v ∈ Rd. The integration of the solution over the hyperplane Evv′ plays a pivotal part in the L∞ estimation. The condition f0 ∈ L∞

s,v for high enough s will

guarantee that sup

v,v′

• Evv′

f0(v ′

∗)dE(v ′ ∗) < ∞.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Constructing an Approximate Sequence

Denoting by CL = CΦlbM0, K∞ = 2 f0L∞

v

min (1, CL), E∞ = sup

v,v′

• Evv′

f0(v ′

∗)dE(v ′ ∗)

• + C+EM0(1 + 2K∞)
• Sd−1
• d + γ − 1K∞ + M0
• C∞ = CΦCγlb(M0 + M2)K∞ + C+E∞ (1 + 2K∞)
• Sd−1
• 1 + γ K∞ + M0
• .

with C+ and C+E explicit constants that depend only on d, CΦ and b∞,

• ne can finally define the time of existence of a solution

T0 = min

• 1 ; K∞

2C∞ min (1, CL)

• ,

and the time step for the scheme ∆n = min

• 1,

1 2CΦlbnγM0 [1 + 2K∞]

• .

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Constructing an Approximate Sequence Cont.

With the mentioned notations one shows that the scheme (2), with k ∈

• 0, 1, . . .
• T0

∆n

• satisfies:

(i) f (k)

n

≥ 0; (ii)

• f (k)

n

• L1

v

= M0,

• |v|2 f (k)

n

• L1

v

= M2 and

• Rd vf (k)

n

dv = M1; (iii) sup

v∈Rd

 f (k)

n

(v) + CL∆n

k−1

• j=0

(nγ ∧ (1 + |v|γ)) f (j)

n (v)

  ≤ K∞ and for almost every (v, v′),

• Evv′

f (k)

n

(v′

∗)dE(v′ ∗) ≤ E∞.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Constructing an Approximate Sequence Cont.

Defining fn(t, v) = f (k)

n

(v) (t, v) ∈ [k∆n, (k + 1)∆n) × Rd, we have constructed a uniformly in time bounded sequence, with the same mass, moment and energy as f0. At this point we can pass to weak and weak-∗ limits and find a function f ∈ L1 [0, T0) × Rd ∩ L∞ [0, T0) × Rd that solves the Boltzmann-Nordheim equation with initial datum f0, and has the same mass, moment and energy (the latter is not as simple to prove as the mass and momentum and requires a use of a Povzner type inequality).

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Uniqueness

Assuming that f and g are two bounded solutions to the Boltzmann-Nordheim equation that conserve the mass, momentum, energy and have the same initial data f0, one can show the following inequalities: d dt f − gL1

v ≤ CT

• f − gL1

2,v + f − gL∞ v

• d

dt f − gL1

2,v ≤ CT

• M2+γ(t) f − gL1

v + f − gL1 2,v

+(1 + M2+γ(t)) f − gL∞

v

• (3)

f − gL∞

v

≤ CT t

• f − gL1

2,v (u) + f − gL∞ v (u)

• du.

where CT is a constant depending on T < T0, d, the collision kernel, the appropriate norms of f0 and the L∞

t,v bounds on f and g.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Uniqueness Cont.

The possible blow up rate of M2+γ plays an important role in the mentioned inequalities. Fortunately, one finds that if M2+γ is unbounded in (0, T] then there exists CT, like before, with M2+γ(t) ≤ CT t . The above follows from a refined Povzner-like inequality, which is also the main tool to show the instantaneous creation of moments of all order to the solution. Armed with the new information about M2+γ one can show that (3) implies the existence of a constant Cn,T such that for any t ∈ [0, T] and any n ∈ N max

• f − gL1

v , f − gL1 2,v , f − gL∞ v

• ≤ Cn.Ttn.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Strategy of the Proof - Uniqueness Cont.

Following the spirit of Nagumo’s fixed point theorem one defines X(t) = f − gL1

2,v

tn0 for an appropriate large n0 and shows that for any t ∈ [0, t0], for a given t0 small enough, one has that X(t) ≤ K ntn n! sup

u∈[0,t0]

X(u) for all n ∈ N. This is enough to show the desired uniqueness.

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Global in Time Theory for the Boltzmann-Nordheim Equation

Theorem (Briant & E. 2015) Let f0 ≥ 0 be in L1

2,v ∩ L∞ s,v when d ≥ 3 and d − 1 < s.

Then there exist explicit constant C > 0, depending only on s, d and the collision kernel, such that if C

• M0(f0)

f0L∞

v

1

d

1 M0(f0)

• 1 + 6 f0L∞

v

• ×

 f0L∞

s,v +

3 f0L∞

v

M0(f0)

• γ

d−1

1 + 6 f0L∞

v

3 f0L∞

v

+ M0(f0)

• 

 ≤ 1 the solution f to (1) associated to f0 is defined on [0, +∞) and satisfies ∀t ∈ [0, +∞), f (t, ·)L∞

v

≤ 3 f0L∞

v ,

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Future Venues of Interest

Understanding the phenomena of the Bose Einstein Condensation better. Finding a Global Existence condition that corresponds to the critical temperature criterion. Finding an explicit rate of convergence to equilibrium in the case where no condensation occurs. And much more....

Homogen. Boltzmann- Nordheim Equations Bose Einstein Condensa- tion Known Results Local Cauchy Theory for Boltzmann- Nordheim Equation Strategy of the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks

Thank You!