Grossly Determined Solutions for a Boltzmann-like Equation

In gas dynamics, the connection between the continuum physics model offered by the Navier-Stokes equations and the heat equation and the molecular model offered by the kinetic theory of gases has been understood for some time, especially through the work of Chapman and Enskog, but it has never been established rigorously. This paper established a precise bridge between this two models for a simple linear Boltzman-like equation. Specifically a special class of solutions, the grossly determined solutions, of this kinetic model are shown to exist and satisfy closed form balance equations representing a class of continuum model solutions.


Introduction
The Maxwell-Boltzmann (or Boltzmann) equation models the dynamics of a dilute gas: where C(F, F ) is the collisions operator. The unknown F (t, x, v) is the molecular density function of the gas. We require F (t, x, v) : R × R 3 × R 3 → R to be a non-negative integrable function with respect to v. Define n(t, x) := V F (t, x, v) dv where V = R 3 represents "velocity" space. Then, F (t, x, v)/n(t, x) is a probability distribution with respect to v. Specifically, we interpret this distribution as the probability of seeing a molecule of velocity v (in R 3 ) at position x (the point x in R 3 ) at time t.
The collisions operator C(F, F ) is normally a bilinear integral operator which acts only on the velocity variables v. Different models of intermolecular interaction (often called the encounter problem) yield different forms of C(F, F ), but there is a commonality to all collisions operators in the full theory. Specifically, collisions operators are required to satisfy the properties of conservation of mass, momentum and energy.
In Fundamentals of Maxwell's Kinetic Theory of a Simple Monotonic Gas [17], C. Truesdell and R. G. Muncaster write a text designed to put the Maxwell-Boltzmann equation on both firm mathematical and historical ground. In the epilogue of the text, the authors discuss what they term the main open problems of kinetic theory. Specifically, they discuss the need for a more detailed existence and uniqueness theory, the impact of the Boltzmann H-theorem on the "trend to equilibrium" of a gas, and they discuss a concept of their own invention -grossly determined solutions. In the 35 years since their writing, a great deal has been accomplished in regards to existence theory for both the homogeneous [2,6,14,15] and the inhomogeneous Boltzmann equation [1,7,13] under varying assumptions about the collisions operator. Implications of the H-theorem also continue to be a great source of scholarly interest. In [3], Cercignani directly addresses the problem of existence as stated in [17] and, in doing so, reframes Truesdell and Muncaster's question about the H-theorem leading to great productivity (see [5,18]). Until now, the main problem on grossly determined solutions has not received much attention.
In contrast to the Maxwell-Boltzmann equation, the Navier-Stokes equations model the dynamics of a gas via physical fields of the gas: where ν (the bulk viscosity), µ (the shear viscosity), and f (the force) are given or defined via balance laws and constitutive equations. Here, the unknowns are the velocity and density fields, v and ρ. In [17,Ch. XXIII], C. Truesdell and R. G. Muncaster remark that -no matter which model of gas flow you begin with -the ultimate goal is the same: determine the density, velocity and temperature fields of the gas. They then note that many of the known exact solutions of Boltzmann's equation -such as those solutions derived from Hilbert's iteration (see [4, pg. 316] or [17, Ch. XXII]), or the Chapman and Enskog procedure (see [10, pg. 86]) -shared the property that the solution class could be represented as being dependent on one (or more) of the gas's physical properties. This led them to define the concept of a grossly determined solution: a solution which is determined at any given instant by the gross conditions (mass density, velocity, temperature) of the gas at that time. In their epilogue, the authors suggest that these concepts may lead to a new way forward: (1) In general, can we determine a set of conservation laws that define the gross field properties? (2) Can we use these conservation laws to determine the class of grossly determined solutions to the problem? (3) If one could find the class of general solutions, can we show that the general solutions evolve asymptotically in time to the class grossly determined solutions?
In addition to finding a new, richer class of solutions to the Maxwell-Boltzmann equation, the class of grossly determine solutions would now be in terms akin to the solutions of the Navier-Stokes equations. In spirit, this type of research is already being done. For example, relaxations and generalization of the Chapman-Enskog procedure to the Navier-Stokes equations [16] or the Burnett equations [9,12] are attempting to accomplish the same goal as grossly determined solutions. However, to date, no one has explicitly explored Truesdell and Muncaster's conjecture. The goal of this paper is to prove that grossly determined solutions exist for a linearized form of the Boltzmann equation, demonstrating steps (1) and (2) above. In a forthcoming paper, step (3) will be established. The following theorem is the main result of this paper.
where f (t, x, v) is the molecular density function of the gas and φ is the probability density function φ(v) := e −v 2 / √ π. Let ρ(t, x) represent the density function of the gas: where the Fourier transformρ(t, ξ) has support within (− √ π, 0) ∪ (0, √ π). Letρ 0 (ξ) denote the Fourier transform of the density function at t = 0. Then a solution to equation (1.2) is given by where the Fourier transform of f iŝ Section 2 of this paper gives an extremely brief introduction of the Maxwell-Boltzmann equation and the role of balance laws in the kinetic theory. In Section 3, we will justify why the partial integro-differential equation (1.2) is an appropriate proxy for the full onedimensional Boltzmann equation. Section 4 derives the class of grossly determined solutions stated in Theorem 1. .) For an intuition of the structure of C(F, F ), consider two particles P and Q and let v and v and v * and v * be the pre-and post-collision velocities of the particles P and Q, respectively. Let F (t, x, v) be the molecular density function for the gas. For notational convenience, let We have introduced new unknowns v and v * into our problem. These can be derived from the Encounter Problem [17, Ch. VI], the modeling of the interaction of two particles in otherwise empty space. 1 In this framework, under appropriate assumptions, the encounter problem is akin to solving a two-body problem. Thus, we can interpret v and v * as v = V (v, v * , s 1 , s 2 ) and v * = V * (v, v * , s 1 , s 2 ) where S = R 2 is a parameter space representing the spatial trajectories of the molecules P and Q.
The net increase in the density of molecules of velocity v by collisions is modeled as being proportional to the difference F (v )F (v * ) − F (v)F (v * ). To ensure that this difference is itself a molecular density function, we modify by an appropriate weight function w. This results in the collisions operator While the derivation of the collisions operator and its properties are rife with motivational and simplifying assumptions, we will take the viewpoint that the following conservation properties are axiomatic.
The quantities 1, v i and |v| 2 are called the summational invariants. The summational invariant conditions are derived from using C(F, F ) and the assumption that the total mass, momentum and energy before a collision are equal to those same quantities after a collision.
Equipped with the above conservation properties, the collisions operator has another additional characteristic.

Balance Equations / Conservation
Laws derived from the Boltzmann Equation. In the classical theory, the summational invariants of the collisions operator are used to derive the balance equations associated with continuum fluid dynamics. Here, the Boltzmann equation is converted into a system of PDEs that are dependent upon the gross field properties of the gas.
Recall that F (t, x, v) is a non-normalized, probability distribution with respect to v. From this, we establish the gross (physical) properties of density, momentum (velocity) and energy. Let m be the molecular mass. Then Now, beginning with the Boltzmann Equation we use the moments to derive the field equations.
We include the proof of the continuity equation to motivate some of the computations in the following chapter. The others are unimportant to this paper and are omitted.
To derive the continuity equation, multiply the Boltzmann Equation by the constant m. Integrate over the velocity space V : By derivation of the density function above and properties of the collision condition, we obtain ∂ρ ∂t The balance equations have introduced new unknown functions. The term P = [P ij ] in balance equation (2) is called the stress tensor. In traditional kinetic theory of gas texts (versus elasticity), this term is called the pressure tensor. (The pressure tensor is the negative of the stress tensor.) Similarly, one can interpret the function T = (T 1 , T 2 , T 3 ) as an energy flux vector. In the classical theory, assumptions are now made about the gas with the goal of representing these tensors back in terms of density, momentum and energy (i.e. constitutive relations). In other words, the system of PDEs that comprise the balance laws are now a closed system in terms of the density, momentum and energy functions. The ultimate goal of this exercise is that we now hope that this new system of PDEs in the gross fields alone are solvable via classical PDE methods.

Derivation of a 1D Approximation of the Boltzmann Equation
3.1. Approximating the Collisions Operator. We begin by simplifying the Maxwell-Boltzmann equation via imposing the condition that the state spaces be one-dimensional.
We seek to replace C with a term C that simplifies the equation, but still retains some of the basic characteristics of the full collisions operator. In Truesdell and Muncaster's text [17,Ch. VII], alternative forms of the collisions operator are explored. We first note that the collisions operator can be written more generally as a symmetric bilinear operator: Or, more simply denoted, where G and H are any functions such that the integral is finite. Note that if we let F = G = H, then the above simplifies to equation (2.1), the original collisions operator. Akin to the traditional linearization technique (see [8]), we perturb a solution F about a Maxwellian density function. Let φ(v) be a uniform Maxwellian (normal) distribution. Note our choice of φ is independent of t and x. Define the function The function F can be interpreted as a slight deviation from the equilibrium solution φ(v). Requiring F to be a solution to the Boltzmann equation, consider the action of C on F : (by the bilinearity of C.) Since C(φ, φ) = 0 (because φ is Maxwellian) and C(φf, φ) = C(φ, φf ) (by symmetry of C), Substituting F into the rest of the one-dimensional Maxwell-Boltzmann equation leads one to consider the Boltzmann equation at first order Using equation (3.1), and the Boltzmann equation at first order becomes (3.2) We seek to further simplify this approximation. As is, with the reduction of dimensions, it will be impossible for the approximated collisions operator in (3.2) to satisfy all the properties 6 of the original C(F, F ). Minimally, we must require the approximated collisions operator to satisfy the conservation of mass condition. The expansion of 2C(φ, φf ) suggests we consider the following collisions operator.
Consider a collisions operator of the form Then C(f )(v) satisfies the conservation of mass condition required of a Maxwell-Boltzmann collisions operator. Proof.
It should be noted that by disposing of the term have removed the need to solve the associated two-body problem. In other words, while we will show that the operator C(f ) has many of the important properties of the full collisions operator, we have essentially removed any "proper" collisions from this model. Replacing the righthand side of (3.2) by C(f ) results in the equation Since φ(v) = 0 on all of R, we can simplify further and state the final form of the model we will work with for the remainder of the paper.

A 1D Approximation of the Boltzmann Equation:
Modeling Fluid Flow along the Real Line. Let x ∈ R represent the position of a molecule and let v ∈ R be the velocity of that molecule. Then the molecular density function f (t, x, v) satisfies the equation 3.3. Properties of C. For the rest of this paper, we will be working with the simplified partial integro-differential equation (PIDE) (3.5). In keeping with the traditional approach, we need to understand the right-hand side of (3.5) as a collisions operator. Define C(f ) as In order to retain the conservation of mass condition, Proposition 5, our future work will require that we work with the weighted L 2 inner product Note that in this notation Proposition 5 takes the form (2) Let C(f ) = 0. Then (3) First we will show that C is a bounded operator on F v .

4.1.
Introduction. In the full kinetic theory each solution of the Maxwell-Boltzmann equation leads immediately to a collection of fields that satisfy the five balance laws, Proposition 4. In classical gas dynamics one wishes to solve the five balance laws for the gross condition of the gas (density, momentum and energy) without any appeal to the kinetic theory. Solving the balance laws directly, however, is impossible as we have introduced additional unknown functions (the pressure tensor P and the energy flux vector T). The goal of some classical iterative solution constructions (for example, the Chapman-Enskog procedure) has been to convert these new unknowns into functions of the gross condition of the gas and thereby "close" the balance laws and create PDEs that must be solved. Our goal here is similar, but at the level of the Maxwell-Boltzmann equation rather than at the level of the balance laws. Specifically one might hope to find a class of solutions for the molecular density F , the grossly determined solutions (GDS), that are completely determined by their own gross fields. For this class, then, P and T are functions of the gross fields and then the balance laws become a well defined system of PDEs that we can identify with classical gas dynamics. We endeavor to accomplish this goal for ∂f where φ(w) is the probability density function φ(v) := 1 √ π e −v 2 (i.e. R φ(v)dv = 1). That is, we will search for a set of grossly determined solutions for our simplified problem that represent a "classical" theory of gas dynamics embedded in our "kinetic" theory of gases.

Derivation of the Continuity Equation.
By construction, we can define only one gross field. The mass-density is For simplicity we let m = 1 and define the density function ρ(t, x): As a result of the one gross field, we do not expect to be able to derive more than one balance law.
Proposition 7. The associated continuity equation is ∂ρ ∂t where Proof. By the definition of ρ(t, x) we see that Multiply the last equation by the the probability density function φ(v) and integrate over the velocity field V = R. This results in the continuity equation: x, v) dv plays the role of mass flux and this results in the balance law ∂ρ ∂t + ∂T ∂x = 0.
As we had in the traditional theory, a new unknown function T has been added to the system. However, if we can describe T as a function of ρ, then this will "close" the Continuity Equation in ρ(t, x) and lead to the class of grossly determined solutions.

4.3.1.
Observations and Assumptions on the form of the GDS. For this problem, there is only one gross field property -mass density. In this setting, the question posited by Truesdell and Muncaster is "Could there be a special class of solutions of (4.1), each determined in some way by their own density field ρ?" Assume that a solution f is dependent on the density field ρ(t, x). That is, •)](x, v) dv is a function of ρ. Given that T is now a function of ρ, we see that the continuity equation ρ t + T x = 0 is a closed system PDE in ρ alone. Moreover, if we are able to determine G, we should be able to solve this PDE. Additionally, the gross field property can now be written for all ρ. We now look for a way to find (or approximate) G. By self-similarity conditions, since (4.1) is autonomous in x (and t), one expects solutions f to be invariant with respect to translations in x. Additionally, since the original problem is a linear PIDE, there is no harm in hoping to find solutions in which G is linear in ρ. In Hörmander's Linear Partial Differential Operators [11, pg 15], he proves an interesting representation theorem for linear maps of distributions: Lemma 8. Let U be a linear mapping of C ∞ 0 (R n ) into C ∞ (R n ) which commutes with translations and is continuous in the sense that U ψ j → 0 in C ∞ (R n ) if the sequence ψ j → 0 in C ∞ 0 (R n ). Then there exists one and only one distribution u such that U ψ = u * ψ, ψ ∈ C ∞ 0 (R n ). Again, we have the freedom to create a solution (dependent on ρ) by any means necessary. As we are already embracing an ansatz, we will assume that "G is continuous at zero". In G's current form, it is dependent on x and v. If we can show that G[ρ(t, •)](x, v) is invariant in x, then the lemma suggests we should look for grossly determined solutions f that are convolutions with ρ.
. For fixed y, assume that f (t, x + y, v) is another solution in this class. Then f (t, x + y, v) = G[ρ y (t, •)](x, v) for some different density field ρ y . What is the connection between ρ y and ρ? We have So, f (t, x + y, v) = G[ρ(t, • + y)](x, v). Redefining the variables, we let x = 0 and y = x. Then Thus, by Hörmander's lemma, f is a convolution and can be represented in the form: While in this context, K v (y) is being interpreted as the kernel in the spacial dimension, we use the notation K v to remember that this portion of the solution will also be dependent on velocity.

Solving for the Kernel
x − y) dy and substitute f into (4.1). This results in the equation To rid this equation of convolutions, we use the Fourier transform in the spacial dimension x. Defineĝ Applying the Fourier transform to the restated PIDE above yields Additionally, we can transform the gross field property. Using the convolution solution, the density becomes Under the transform, we getρ Upon the support ofρ(t, ξ), equation (4.6) requires that Last, we transform the continuity equation (4.2): Applying the Fourier transform, we obtain ∂ρ ∂t Then, the transformed continuity equation becomes ∂ρ ∂t (t, ξ) = −iξρ(t, ξ)k(ξ) (4.9) 13 Note that we have succeeded into converting the balance law into a separable PDE. Given an initial density condition ρ(0, ξ), we see that the transformed representation of ρ(t, x) iŝ ρ(t, ξ) =ρ 0 (ξ)e −iξk(ξ)t whereρ 0 (ξ) :=ρ(0, ξ). We see that understanding K v (ξ) andρ(t, ξ) requires a better understanding of k(ξ). Substituting (4.9) into the transformed PIDE (4.5) yields Again requiring thatρ(t, ξ) ≡ 0, we can simplify to This results in a representation of K v (ξ) in terms of k(ξ).
Moreover, apart from knowing k, we have an explicit form of K v in the variable v alone. Combining (4.8) and (4.10) we find a representation of k(ξ) that suppresses K v : Now, for any fixed value of ξ, k(ξ) will yield a number in C. So, for fixed ξ, let that number be k ξ = r + ai. Then Note that if we let k ξ be pure imaginary (i.e. r = 0), then the real-part integral vanishes as vφ(v)(1 + aξ) (1 + aξ) 2 + (vξ) 2 is an odd function in v. Again, we have the ability to simplify any way we deem appropriate. We are just trying to find a class of solutions in which each is dependent on its own density. So, we let r = 0. Then This last equation results in a constraint on the freedom of ξ in our class of solutions. To better understand this, let us define the function Ξ(c) as follows: Note that we now are able to represent ξ as a parametric function of c. To examine the values of ξ defined over the range of c, we begin with the following graphical observation, Figure 1. It appears that for the solution class, our transform variable is bounded. In fact, we can show that |ξ| = |Ξ(c)| ∈ (0, √ π). 15 Claim 10. Let Ξ(c) be defined as in (4.11). Then lim c→0 + Ξ(c) = √ π.
Proof. Note that for this limit, c > 0. Then The equivalent computation shows lim It is also clear that lim c→±∞ Ξ(c) = 0.
We conclude that ξ ∈ (− √ π, 0) ∪ (0, √ π). We have reached a point in the calculations where, if we can represent c as a function of ξ, we would be able to unwind the above calculations and find a representation of the transformed solution. We now seek the inverse of Ξ(c). Graphically, the function Ξ(c) appears to be a strictly decreasing function (on each connected piece of the domain). We will show that Ξ(c) is strictly decreasing, thus proving that Ξ(c) is a one-to-one function. Hence Ξ(c) is invertible.
Claim 11. On each connected component of the domain of Ξ(c) (4.11), Ξ(c) is a strictly decreasing function.
Proof. Without loss of generality, let c 1 > c 2 and c i in (−∞, 0). Then is an even function in v. It will be sufficient to understand the resultant integral on [0, ∞). Note that the integrand is negative on (0, √ c 1 c 2 ) and positive on ( √ c 1 c 2 , ∞). Splitting the integral, we have dv.

The Solution Class of Grossly Determined Solutions.
We are now ready to prove Theorem 1.
Thus a class of grossly determined solutions, each solution dependent upon its own density field, is given by x − y) dy.

Conclusions
In the terms of Truesdell and Muncaster's conjectures on grossly determined solutions, we have established the existence of a class of grossly determined solution for a Boltzmannlike equation. Specifically, given a gas' density at an initial time, we are able to state the convolution solution (for all time) for an inhomogeneous transport equation with modified linearized collisions operator. In a companion paper, we will demonstrate that the class of general solutions to (1.2) does have the property that, in time, each member decays to a solution from the subclass of grossly determined solutions.