Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law
In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the following steady problem:
where is a bounded domain in , , is the Knudsen number and is a Maxwellian with non-constant (non-isothermal) wall temperature . Based on new constructive coercivity estimates for both steady and dynamic cases, for and any fixed value of , we construct a unique solution to (0.1) and (0.2), continuous away from the grazing set and exponentially asymptotically stable. This solution is a genuine non equilibrium stationary solution differing from a local equilibrium Maxwellian. As an application of our results we establish the expansion and we prove that, if the Fourier law holds, the temperature contribution associated to must be linear, in the slab geometry. This contradicts available numerical simulations, leading to the prediction of breakdown of the Fourier law in the kinetic regime.
CB3 0WA, UK44footnotetext: Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma, Italy.
1 Introduction and notation
According to the Boltzmann equation (1.2), a rarefied gas confined in a bounded domain, in contact with a thermal reservoir modeled by (0.2) at a given constant temperature (isothermal), has an equilibrium state described by the Maxwellian
If the temperature at the boundary is not uniform in (0.2), such a statement is not true and the existence of stationary solutions and the rate of convergence require a much more delicate analysis, because rather complex phenomena are involved. For example, suppose that the domain is just a slab between two parallel plates at fixed temperatures and with . Then, one expects that a stationary solution is reached, where there is a steady flow of heat from the hotter plate to the colder one. The approach to the stationary solution may involve convective motions, oscillations and possibly more complicated phenomena. Even the description of the stationary solution is not obvious, and the relation between the heat flux and the temperature gradient, (e.g. the Fourier law (1.1)), is not a priori known. A first answer to such questions can be given confining the analysis to the small Knudsen number regime (i.e., K in (0.1)), where the particles undergo a large number of collision per unit time and a hydrodynamic regime is established. In this case, it can be formally shown, using expansion techniques [9, 11], that the lowest order in K is a Maxwellian local equilibrium and the evolution is ruled by macroscopic equations, such as the Navier-Stokes equations. In particular, the heat flux vector turns out to be proportional to the gradient of temperature, as predicted by the Fourier law
with the heat conductivity depending on the interaction potential. This was first obtained by Maxwell and Boltzmann [28, 7] which relates the macroscopic heat flow to the microscopic potential of interaction between the molecules. The rigorous proof of such a statement was given in [16, 17] in the case of the slab geometry and provided that is sufficiently small (uniformly in the Knudsen number K). This is a special case of a problem which has received recently a large attention in the Statistical Mechanics community, the derivation of the Fourier law from the microscopic deterministic evolutions ruled by the Newton or Schrödinger equation or from stochastic models [8, 30, 6, 1].
The aim of this paper is to analyze the thermal conduction phenomena in the kinetic regime. This problem was studied in the slab geometry, for small Knudsen numbers in [16, 17] and for large Knudsen number (K) in . Here we are interested in a general domain and in a regime where the Knudsen number K is neither small nor large. In this regime, the only construction of solutions to (0.1) and (0.2) we are aware of, was achieved in  in a slab for large , with techniques closer in the spirit to the DiPerna-Lions renormalized solutions [13, 14] (see also  and references quoted therein). However, the uniqueness and stability of such solutions are unknown. To develop the quantitative analysis we have in mind, the theory of the solutions close to the equilibrium ([35, 36, 10, 22]) is better suited. For this reason we confine ourselves to temperature profiles at the boundary which do not oscillate too much. More precisely, we will assume that the temperature on is given by with a small parameter and a prescribed bounded function on such that
This will allow us to use perturbation arguments in the neighborhood of the equilibrium at the uniform temperature .
We shall consider the Boltzmann equation
with the probability density that a particle of the gas at time is in a small cell of the phase space centered at . Here is a bounded domain in , with a smooth boundary . The function is required to be a positive function on such that is fixed for any . The right hand side of (1.2), , is the Boltzmann collision operator (non-symmetric)
where with (hard potential), (angular cutoff) is the collision cross section and are the incoming velocities in a binary elastic collision with outgoing velocities and impact parameter :
The contact of the gas with thermal reservoirs is described by suitable boundary conditions. We confine ourselves to the simplest interesting case of the diffuse reflection (0.2), although more general boundary data could be studied . On (supposed to be a -smooth surface with external normal well defined in each point ) we assume the condition:
for and , where is the Maxwellian at temperature ,
normalized so that
Throughout this paper, is a connected and bounded domain in , for and the velocity and such that
We denote the phase boundary in the phase space as , and split it into the outgoing boundary , the incoming boundary , and the grazing boundary :
The backward exit time is defined for
and . Furthermore, we define the singular grazing boundary , a subset of , as:
and the discontinuity set in :
We will use the short notation for the Maxwellian
Moreover, to denote the global Maxwellian at temperature , , we will simply use the symbol :
Since (1.7) is valid for all , we have
We denote by the standard linearized Boltzmann operator
with the collision frequency for . Moreover, we set
Finally, we define
Thanks to (1.13), , viewed as function on for any fixed , is a -projection with respect to the measure for any boundary function defined on .
We denote either the norm or the norm in the bulk, while is either the norm or the norm at the boundary. Also we adopt the Vinogradov notation: is equivalent to where is a constant not depending on and . We subscript this to denote dependence on parameters, thus means . Denote . Our main results are as follows.
If with is another solution such that, for
then . Furthermore, if is continuous on then is continuous away from . In particular, if is convex then . On the other hand, if is not convex, we can construct a continuous function on in (1.25) with such that the corresponding solution in not continuous.
We stress that the solution is a genuine non equilibrium steady solution. Indeed, it is not a local Maxwellian because it would not satisfy equation (0.1), nor a global Maxwellian because it would not satisfy the boundary condition (0.2).
We also remark that in addition to the wellposedness property, the continuity property in this theorem is the first step to understand higher regularity of in a convex domain. The robust estimates used in the proof enable us to establish the following expansion of , which is crucial for deriving the necessary condition of the Fourier law (1.1). In the rest of this paper we assume the normalization .
with for all from (1.7), and set
Then there exist with , for , such that the following -expansion is valid
with for . In particular, satisfies
We have the following dynamical stability result:
For every fixed , there exist and , depending on , such that if
If the domain is convex, is continuous on and moreover is continuous away from and satisfies the compatibility condition
then is continuous away from .
The asymptotic stability of further justifies the physical importance of such a steady state solution. We remember that, when , then fails to be a global Maxwellian or even a local Maxwellian, and its stability analysis marks a drastic departure from relative entropy approach (e.g. ). Moreover, such an asymptotic stability plays a crucial role in our proof of non-negativity of .
An important consequence of Theorem 1.2 is the Corollary below which specializes the result to the case of a slab between two parallel plates kept at temperatures and .
Let and be the solution to
where is the component in the direction of the velocity .
Then , the first order in correction to according to the expansion of Theorem 1.2, is the unique solution to
In order to establish a criterion of validity of the Fourier law, let us remember that the temperature associated to the stationary solution is given by
where and . The heat flux associated to the distribution is the vector field defined as
We state the Fourier law in the following formulation: there is a positive function , the heat conductivity, such that (1.1) is valid for .
If the Fourier law holds for , then is a linear function over .
Available numerical simulations, Figure 1,  indicate the linearity is clearly violated for all finite Knudsen number K ( in Figure 1).
We therefore predict that the general Fourier law (1.1) is invalid and inadequate in the kinetic regime with finite Knudsen number K. It is necessary to at least solve the linear Boltzmann equation (1.19) to capture the heat transfer in the Boltzmann theory.
Without loss of generality, we may assume and K throughout the rest of the paper. Therefore in the rest of this paper we assume
The key difficulty in the study of the steady Boltzmann equation lies in the fact that the usual entropic estimate for , coming from , is absent. The only a-priori estimate is given by the entropy production , which is very hard to use . In the context of small perturbation of , only the linearized dissipation rate is controlled 111We denote by the projection of on the null space of ., and the key is to estimate the missing hydrodynamic part , in term of . This is a well-known basic question in the Boltzmann theory. Motivated by the studies of collisions in a plasma, a new nonlinear energy method in high Sobolev norms was initiated in the Boltzmann study, to estimate () in terms of . For , remembering the definitions (1.14) of the linearized Boltzmann operator and (1.15) of the quadratic nonlinear collision operator, the Boltzmann equation (1.2) can be rewritten for the perturbation as
If the operator were positive definite, then global solutions (for small ) could be constructed “easily” for (1.26). However, is only semi-positive
where is the -weighted norm. The kernel (the hydrodynamic part) is given by
Note that we use slight different definitions of and from  to capture the crucial total mass constraint. The so-called ‘hydrodynamic part’ of , , is the -projection on the kernel of , for every given . The novelty of such energy method is to show that is indeed positive definite for small solutions to the nonlinear Boltzmann equation (1.26). The key macroscopic equations for connect and via the Boltzmann equation as in page 621 of :
where denotes some second order differential operator in . Such hidden ellipticity in () implies that the hydrodynamic part , missing from the lower bound of , can be controlled via the microscopic part , so that can control the full from (1.27). Such nonlinear energy framework has led to resolutions to several open problems in the Boltzmann theory [19, 20, 25].
It should be noted that all of these results deal with idealized periodic domains, in which the solutions can remain smooth in for . Of course, a gas is usually confined within a container, and its interaction with the boundary plays a crucial role both from physical and mathematical view points. Mathematically speaking, the phase boundary is always characteristic but not uniformly characteristic at the grazing set and . In particular, many of the natural physical boundary conditions create singularities in general domains ([22, 26]), for which the high Sobolev estimates break down in the crucial elliptic estimates (1.29). Discontinuities are expected to be created at the boundary, and then propagate inside a non-convex domain. Therefore completely new tools need to be developed.
A new –framework is developed in  (see  for a short summary of the method) to resolve such a difficulty in the Boltzmann theory, which leads to resolution of asymptotic stability of Maxwellians for specular reflection in an analytic and convex domain, and for diffuse reflection (with uniform temperature!) in general domains (no convexity is needed). We remark that the non-convex domains occur naturally for non-isothermal boundary (e.g. two non flat separated boundaries). Furthermore, the solutions to the boundary problems are shown to be continuous if the domain is strictly convex. Different methods have been used in [2, 3, 5, 16, 17] in particular geometries.
The new framework introduced in  has two parts:
Positivity: Assume the wall temperature is constant in (1.5). It suffices to establish the following finite-time estimate
The natural attempt is to establish estimate for from the macroscopic equation (1.29). However, this is very challenging due to the fact that only has trace in the sense of Green’s identity (Lemma 2.2), neither nor even make sense on the boundary of a general domain. Instead, the proof of (1.30) given in  is based on a delicate contradiction argument because it is difficult to estimate via directly in a setting in the elliptic equation (1.29), in the presence of boundary conditions. The heart of this argument, lies in an exact computation of which leads to the contradiction. As a result, such an indirect method fails to provide a constructive estimate of (1.30) with explicit constants.
Bound: The method of characteristics can bootstrap the bound into a point-wise bound to close the nonlinear estimate. Let be the semigroup generated by and the semigroup generated by , with the prescribed boundary conditions. By two iterations, one can establish:
From the compactness property of , the main contribution in (1.31) is roughly
where denotes the generalized characteristics associated with specific boundary condition. A change of variable from to would transform the and -integration in (1.32) into and integral of , which decays from the theory. The key is to check if
is valid. Without boundary, is simply , and most of the time. For specular or diffuse reflections, each type of characteristic trajectories repeatedly interact with the boundary. To justify (1.33), various delicate arguments were invented to overcome different difficulties, and analytic boundary and convexity are needed for the specular case.
Since its inception, this new approach has already led to new results in the study of relativistic Boltzmann equation , in hydrodynamic limits of the Boltzmann theory(  ), in stability in the presence of a large external field ( ).
Our current study of non-isothermal boundary ( is non-uniform) is naturally based on such a framework. The main new difficulty in contrast to , however, is that the presence of a non-constant temperature creates non-homogeneous terms in both the linear Boltzmann (steady and unsteady) as well as in the boundary condition, so that the exact computation, crucial to the estimate (1.30), breaks down, and the scheme  collapses.
The main technical advance in this paper is the development of a direct (constructive) and robust approach to establish (1.30) in the presence of a non-uniform temperature diffuse boundary condition (0.2). Instead of using these macrosopic equations (e.g. (1.29)), whose own meaning is doubtful in a bounded domain, we resort to the basic Green’s identity for the transport equation and choose proper test functions to recover ellipticity estimates for , , and hence directly. In light of the energy identity, is controlled at the boundary , but not in (1.16). The essence of the method is to choose a test function which can eliminate the contribution at the boundary, and to control the , , component of respectively in the bulk at the same time. The choice of the test function for is rather direct: we set
for some constant to be determined, with and on . On the other hand, the test function for is rather delicate. In fact, two different sets of test functions have to be constructed:
with and on . In particular, a unique constant can be chosen to deduce the estimates for , thanks to the special structure of the transport equation and the diffuse boundary condition. The choice of the test function for requires special attention. It turns out that in order to eliminate the contribution , we need to choose the test function
with and Neumann boundary condition
This is only possible if the total mass of is zero, i.e.,
This illustrates the importance of the mass constraint, which unfortunately is not valid for the steady problem (0.1) and (0.2). So we are forced to use a penalization procedure (see below) to deal with it. The key lemma which delivers the basic estimates is Lemma 3.4. Furthermore, in the dynamical case, also the time derivatives , and need to be controlled in negative Sobolev spaces. This is possible due to the special structure of the Boltzmann equation as well as the diffuse boundary condition. Once again, the total mass zero condition (1.34) is essential. This is the key to prove the crucial Lemma 6.2. Even though this new unified procedure can be viewed as a ‘weak version’ of the macroscopic equations (1.29), the estimates we obtain via this approach are more general. For instance, the (1.29) was only valid for dimension , but the new estimates are valid for any dimension. There seems to be a very rich structure in the linear Boltzmann equation.
To bootstrap such a estimate into a estimate, we define the stochastic cycles for the generalized characteristic lines interacting with the boundary:
Definition 1.6 (Stochastic Cycles)
Fixed any point with , let . For such that , define the -component of the back-time cycle as
Define , and let the iterated integral for be defined as
where is a probability measure.
We note that the ’s () are all independent variables, and depend on for . However, the phase space implicitly depends on . Our method is to use the Vidav’s two iterations argument() and estimate the -norm along these stochastic cycles with corresponding phase spaces . The key is to estimate measures of various sets in in Lemma 4.2. We designed an abstract and unified iteration (4), which is suitable for both steady and unsteady cases. New precise estimate (4.16) of non-homogeneous terms resulting from the non-constant temperature are obtained in Prop. 4.1. Based on a delicate change of variables in Lemma 2.3, such an estimate is crucial in the proof of formation of singularity for a non-convex domain in Theorem 1.1.
In order to keep the mass zero condition and to start iterating scheme, it is essential to introduce a penalization to solve the problem
The presence of ensures the critical zero mass condition (1.34). It is important to note that our estimate are intertwined with at every step of the approximations, which ensures the preservation of the continuity for a convex domain. The continuity properties of our final solutions follows from the limit at every step. Moreover, the proof of continuity away from the singular set in (1.11) in a general domain is consequence of a delicate result for in .
To illustrate the subtle nature of our construction, we remark that for the natural positivity-preserving scheme:
we are unable to prove the convergence, due to breakdown of , whence the mass zero constraint (1.34) fails to be satisfied. Consequently, we are unable to prove in our construction. Such a positivity is only proven via the dynamical asymptotic stability of , in which the initial positivity plus the choice of a small time interval are crucial to guarantee the convergence of the analog of (1.37) in the dynamical setting.
Our estimates are robust and allow us to expand our steady state in terms of , the magnitude of the perturbation. This leads to the first order precise characterization of by , which satisfies (1.19).
It should be pointed out the our rather complete study of the non-isothermal boundary for the Boltzmann theory for forms a mathematically solid foundation for influential work in applied physics and engineering such as in , where the existence of such steady solutions to the Boltzmann equation is a starting point, but without mathematical justification. We expect our solutions as well as our new estimates would lead to many new developments along this direction.
The plan of the paper is the following. In next section we present some background material and in particular a version of the Ukai Trace Theorem and the Green identity as well as a new estimate at the boundary in Lemma 2.3. Section 3 is devoted to the construction of solutions to the stationary linearized problem. In particular, we prove Lemma 3.4 which provides the basic estimate of the norms of in terms of the norm of . In Section 4, after introducing an abstract iteration scheme suitable for proving bounds, we prove, in Proposition 4.1, the existence of the solution to the linearized problem in and related bounds. In Section 5 we combine the results of the previous sections to construct the stationary solution to the Boltzmann equation and discuss its regularity properties. In particular, we give the proof of the -expansion and use it to establish a necessary condition for the validity of the Fourier law. In Section 6 we extend the estimates to the time dependent problem. Section 7 contains the extension of the estimates to the time dependent problem and the proof of the exponential asymptotic stability of the stationary solution. From this we then obtain its positivity.
In this section we state basic preliminaries. First we shall clarify the notations of functional spaces and norms: we use for both of the norm and the norm, and for the standard inner product. Moreover we denote and . For the phase boundary integration, we define where is the surface measure and define and the corresponding space as