Why do Solutions of the

Maxwell–Boltzmann Equation Tend to be Gaussians?

A Simple Answer

Dirk Hundertmark, Young-Ran Lee

Received: December 20, 2016

Communicated by Heinz Siedentop

Abstract. The Maxwell-Boltzmann functional equation has recently attraction renewed interest since besides its importance in Boltzmann’s kinetic theory of gases it also characterizes maximizers of certain bilinear estimates for solutions of the free Schrödinger equation. In this note we give a short and simple proof that, under some mild growth restrictions, any measurable complex-valued solution of the Maxwell-Boltzmann equation is a Gaussian. This covers most, if not all, of the applications.

2010 Mathematics Subject Classification: 39B22, 39B32

Keywords and Phrases: Maxwell–Boltzmann functional equation. Gaussian maximizers

1 Introduction

The Maxwell-Boltzmann functional equation for a measurable function f : ℝ^{d} ℂ states that

f x f y H x^{2} y^{2}, x y for almost all x, y ℝ^{d}  (1)

for some measurable function H : ℝ_{}ℝ^{d} ℂ. This equation has attracted a lot of attention in kinetic theory, since it determines the collision invariant of the Boltzmann equation. In this case, it is natural to also assume that f in non-negative and integrable. In recent years the Maxwell–Boltzmann equation has regained attention for complex-valued functions f, since it determines in some cases the maximizers of the Strichartz inequality in low dimensions d 2 [4, 5] and of the Ozawa and Tsutsumi bilinear inequality [7] and other bilinear estimates [3, 8], which are space–time inequalities for solutions of the free Schrödinger equation. In [2] a class of bilinear estimates was proved, which includes the ones of [3, 7, 8] for solutions of the Schrödinger equation. It was also noticed in [2] that any maximizer f of their bilinear inequality obeys the Maxwell–Boltzmann equation and, furthermore, that f L^{1} e^{γx^{2}} dx for some suitable γ 0. They conclude, with the help of known results [6, 9, 10] on solutions of the Maxwell–Boltzmann equation, that any maximizer of their bilinear inequality must be a Gaussian. However, the known approaches to show that solutions f of the Maxwell-Boltzmann equation are Gaussians need that f is non-negative and f 0. This is the case in [10], where Villani uses an argument due to Desvillettes to reduce the proof for f L^{1} and f 0 to f C^{2} and f 0. In [6] Lions uses the estimates he established in his work to prove that any solution of Maxwell– Boltzmann is smooth, this is elegant but technical and still needs that f 0. In [9] the proof needs f 0 and f L_{2}^{1} ℝ^{d} , that is, x 1 x ^{2} f x is integrable, and, upon closer inspection, it seems to us that it also needs that the Fourier transform of f does not vanish. Using a completely different method, another approach in [1] also shows that solutions f of the Maxwell–Boltzmann functional equation are Gaussians if f 0 is measurable and finite and f 0 on a set of positive measure. Our note is intended to give a short proof that complex valued-functions, which obey the Maxwell-Boltzmann equation together with a mild growth condition, are necessarily Gaussians with a rotationally symmetric covariance. More importantly, we believe that the proof we give is very simple. It does not require any technical tools besides some simple linear algebra and the chain rule. The proof we give below is inspired by the proof in [5], where Gaussians were shown to be the only maximizers in the sharp Stichartz inequality in low dimensions.

2 The Maxwell–Boltzmann equation and Gaussians

Theorem 1. Let f L^{1} ℝ^{d}, e^{γx^{2}} dx for some γ 0 obey the Maxwell– Boltzmann equation

f x f y H x^{2} y^{2}, x y for almost all x, y ℝ^{d}  (2)

for some measurable function H : ℝ_{}ℝ^{d} ℂ. Then there exist a, A ℂ with Re a γ and b ℂ^{d} such that

f x Ae^{ax^{2} b x} for almost every x ℝ^{d}.  (3)

Proof. First, we assume in addition that f L^{1} ℝ^{d} . We will relax this in Step 4 below. Step 1: Assume that f C^{2} ℝ^{d} L^{1} ℝ^{d} , i.e., it is integrable and twice continuously differentiable. Assume, furthermore, that f never vanishes. Then there exist a, A ℂ with Re a 0 and b ℂ^{d} such that

f x Ae^{ax^{2} b x} for x ℝ^{d}.

To prove this, we will need suitable rotations in two-dimensional subspaces of ℝ^{2d}. From our assumption f x f y H x^{2} y^{2}, x y one easily deduces that the product f x f y is invariant under a large class of rotations of ℝ^{2d}, namely the rotations of ℝ^{2d} which leave ℝ^{d} ℝ^{d} x, y x y invariant. To exploit this, we will construct a convenient basis in ℝ^{2d}, in which these rotation have a simple expression. Let e_{j}, j 1 …, d be the standard basis for ℝ^{d}, that is, e_{j} has only zero entries, except for the j^{th} slot, in which it has a one, and define the vectors α_{j}, j 1, …, d by

α_{j} : \frac{1}{\overline{2}} \begin{matrix}e_{j} \\ e_{j} \end{matrix} .

so that α_{j} are unit vectors. Then the equation x y c ℝ^{d} is equivalent to

the d equations c_{j} x_{j} y_{j} \overline{2} α_{j}, z _{ℝ^{2d}}, j 1, …, d with z \begin{matrix}x \\ y \end{matrix} and

, _{ℝ^{2d}} the standard scalar product in ℝ^{2d}. To construct suitable rotations in the orthogonal complement of span α_{1}, …, α_{d} ℝ^{2d}, one has to find a basis for this orthogonal complement. This can be done either in a systematic manner or by simply guessing that the vectors

β_{j} : \frac{1}{\overline{2}} \begin{matrix}e_{j} \\ e_{j} \end{matrix}

will do the job: It is easy to check that α_{1}, …, α_{d}, β_{1}, …, β_{d} form an orthonormal basis of ℝ^{2d}. Fix j, k 1, …, d with j k. We define the rotation by an angle φ in the plane spanned by β_{j} and β_{k} by the matrix

R_{j,k} φ _{l 1}^{d} α_{l}α_{l}^{t} _{\begin{matrix}m 1,...,d \\ m j,k \end{matrix}} β_{m}β_{m}^{t}

cos φ β_{j}β_{j}^{t} sin φ β_{j}β_{k}^{t} sin φ β_{k}β_{j}^{t} cos φ β_{k}β_{k}^{t} .

(4)

In the following, we will suppress in our notation that the matrix depends on j and k. This rotation keeps all the directions α_{l}, l 1, …, d and β_{m}, m j, k invariant. Since the function

F z : f x f y H x^{2} y^{2}, x y , z x, y ℝ^{2d}

is invariant under such a rotation, we have

F R φ z const

for all fixed z ℝ^{2d} and all φ ℝ. Thus, since F is twice continuously differentiable, by assumption,

0 \frac{d}{dφ}F R φ z _{φ 0}  ∇_{2d}F z , \frac{d}{dφ}R φ _{φ 0} z _{ℝ^{2d}} .  (5)

Here ∇_{2d} is the gradient in ℝ^{2d}. One easily calculates

\frac{d}{dφ}R φ _{φ 0} β_{j}β_{k}^{t} β_{k}β_{j}^{t}.

In the splitting ℝ^{2d} ℝ^{d} ℝ^{d}, the matrix β_{j}β_{k}^{t} has a simple block structure,

β_{j}β_{k}^{t} \frac{1}{2} \begin{matrix}e_{j} \\ e_{j} \end{matrix} e_{k}^{t} e_{k}^{t} \frac{1}{2} \begin{matrix}e_{j}e_{k}^{t} & e_{j}e_{k}^{t} \\ e_{j}e_{k}^{t} & e_{j}e_{k}^{t} \end{matrix} .

So

\frac{d}{dφ}R φ _{φ 0} \frac{1}{2} \begin{matrix}e_{j}e_{k}^{t} e_{k}e_{j}^{t} & e_{j}e_{k}^{t} e_{k}e_{j}^{t} \\ e_{j}e_{k}^{t} e_{k}e_{j}^{t} & e_{j}e_{k}^{t} e_{k}e_{j}^{t} \end{matrix}

and thus, since e_{k}^{t}x e_{k}, x _{ℝ^{d}} x_{k}, one has

\frac{d}{dφ}R φ _{φ 0} z  \begin{matrix}x_{k} y_{k} e_{j} x_{j} y_{j} e_{k} \\ x_{k} y_{k} e_{j} x_{j} y_{j} e_{k} \end{matrix}  for z  \begin{matrix}x \\ y \end{matrix} .

Hence, writing

∇_{2d}F z f x f y \begin{matrix}∇q x \\ ∇q y \end{matrix}

with q ln f, which is well-defined, since, by assumption, f never vanishes, we get from (5) the differential equation

\begin{aligned}0 & \begin{matrix}∇q x \\ ∇q y \end{matrix} , \begin{matrix}x_{k} y_{k} e_{j} x_{j} y_{j} e_{k} \\ x_{k} y_{k} e_{j} x_{j} y_{j} e_{k} \end{matrix} _{ℝ^{2d}} \\ & x_{k} y_{k} _{j}q x x_{j} y_{j} _{k}q x x_{k} y_{k} _{j}q y x_{j} y_{j} _{k}q y \end{aligned}

for all j k and all x, y ℝ^{d}. Differentiating this with respect to y_{j} yields

0 _{k}q x x_{k} y_{k} _{j}^{2}q y _{k}q y x_{j} y_{j} _{j} _{k}q y  (6)

and differentiating this again with respect to x_{j}, we arrive at

j _{k}q x _{j} _{k}q y 0

for all x, y ℝ^{d}, which setting x y shows

j _{k}q x 0 for all j k,  (7)

whereas differentiating (6) with respect to x_{k} gives

_{k}^{2}q x _{j}^{2}q y for all j k  (8)

and for all x, y. The two equations (7) and (8) show that there exists a constant a ℂ such that

∇ _{j}q x 2ae_{j}

for all j 1, …, d. Integrating this gives

jq x 2ax_{j} b_{j}

for some constants b_{j} ℂ, i.e.,

∇q x 2ax b

and integrating this yields ln f x q x ax^{2} b x c. That is,

f x Ae^{ax^{2} b x} for x ℝ^{d}

for some constants a, A ℂ and b ℂ^{d} and in order that this is in L^{1} ℝ^{d} , we need to have Re a 0. For the second step let

g x \frac{1}{π^{d 2}}e^{x^{2}}

a centered L^{1}-normalized Gaussian and for ε 0

g_{ε} x ε^{d} g x ε

its scaled version, which serves as an approximation of the delta-distributions as ε 0. Step 2: Assume that f L^{1} ℝ^{d} and the convolution g_{ε} f never vanishes for all small enough ε 0. Then there exist a, A ℂ with Re a 0 and b ℂ^{d} such that

f x Ae^{ax^{2} b x} .

Indeed, G_{ε} x, y g_{ε} x g_{ε} y is a centered Gaussian in ℝ^{2d}, in particular, invariant under all rotations of ℝ^{2d}. Let H x, y H x^{2} y^{2}, x y and set { \atop H_{ε}} G_{ε} H, the convolution now on ℝ^{2d}. From the Maxwell-Boltzmann equation for f one gets

g_{ε} f x g_{ε} f y G_{ε} H x, y for all x, y ℝ^{d}

A simple calculation, using that G_{ε} is invariant under all rotations of ℝ^{2d}, shows that { \atop H_{ε}} inherits all rational invariances of H, that is, it is invariant under all rotations of ℝ^{2d} which leave ℝ^{2} ℝ^{d} x, y x y invariant. Clearly g_{ε} f is infinitely often differentiable and, by assumption it does not vanish for all small enough ε 0. So Step 1 applies to g_{ε} f and shows that there exist a_{ε}, A_{ε} ℂ with Re a_{ε} 0 and b_{ε} ℂ^{d} such that

g_{ε} f x A_{ε}e^{a_{ε}x^{2} b_{ε} x}

for all x ℝ^{d}. Taking the limit ε 0 shows that f is the L^{1}-limit of Gaussians, hence it must be a Gaussian.

To finish the proof for f L^{1} ℝ^{d} it is enough to show Step 3: Let f L^{1} obey the Maxwell–Boltzmann equation and f not be the zero function. Then for any small enough ε 0 the convolution g_{ε} f never vanishes. Indeed, as above one sees

ℝ^{d} ℝ^{d} x, y g_{ε} f x g_{ε} f y

is invariant under all rotations of ℝ^{2d} which leave ℝ^{d} ℝ^{d} x, y x y invariant. So taking the modulus and applying the same argument again, shows that for any δ 0

ℝ^{d} ℝ^{d} x, y g_{δ} g_{ε} f x g_{δ} g_{ε} f y

is also invariant under all rotations of ℝ^{2d} which leave ℝ^{d} ℝ^{d} x, y x y invariant. Since f is not identically zero and g_{ε} f converges to f in L^{1}, we see that g_{ε} f is not identically zero for all small enough ε 0. But in this case g_{ε} f is non negative and positive on some set of positive measure, thus g_{δ} g_{ε} f x 0

for all δ 0 and all x ℝ^{d}, i.e., it is infinitely often differentiable and it never vanishes. By Step 1, we see that g_{ε} f is a Gaussian and thus, if it vanishes somewhere it must vanish everywhere. So if g_{ε} f vanishes somewhere for all small ε, it is identically zero for all small ε, and taking the limit ε 0, we see that f must be equal to zero almost everywhere, in contradiction to our assumption. So g_{ε} f never vanishes for all small enough ε 0. This finished the proof in case f L^{1} ℝ^{d} obeys the Maxwell–Boltzmann equation. The last step is to relax the integrability assumption on f, which is easy: Step 4: If f L^{1} ℝ^{d}, e^{γx^{2}} dx obeys the Maxwell–Boltzmann equation, then there exist a, A ℂ with Re a γ and b ℂ^{d} such that

f x Ae^{ax^{2} b x}

for almost all x ℝ^{d}. Indeed, let f_{γ} x e^{γx^{2}} f x . Then f_{γ} L^{1} ℝ^{d} and it also obeys the Maxwell–Boltzmann equation. So by the above there exist a_{0}, A ℂ with Re a_{0} 0 and b ℂ^{d} such that

f_{γ} x Ae^{a_{0}x^{2} b x} for almost all x ℝ^{d}.

Then cleary

f x Ae^{ax^{2} b x}

with a a_{0} γ and Re a γ.

Solutions of the Maxwell–Boltzmann Equation 1273

Acknowledgements Y.-R. Lee thanks the Department of Mathematics at KIT and D. Hundertmark thanks the Department of Mathematics at Sogang University for their warm hospitality. D. Hundertmark thanks the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 ‘Wave Phenomena’ and the Alfried Krupp von Bohlen und Halbach Foundation for financial support. Y.- R. Lee thanks the National Research Foundation of Korea (NRF) for financial support funded by the Ministry of Education (No. 2014R1A1A2058848).

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D. Hundertmark Institute for Analysis Karlsruhe Institute of Technology (KIT) Englerstraße 2 76131 Karlsruhe Germany dirk.hundertmark@kit.edu Y.-R. Lee Department of Mathematics Sogang University 35 Baekbeom-ro, Mapo-gu Seoul 04107 South Korea younglee@sogang.ac.kr 1274

Documenta Mathematica 22 (2017)