Diffusive limit for a Boltzmann-like equation with non-conserved momentum

TL;DR

This paper proves that a kinetic model with non-conserved momentum converges to a diffusive limit described by coupled diffusion equations for density and temperature, with explicit error estimates.

Contribution

It establishes the diffusive limit for a Boltzmann-like equation with non-conserved momentum, including rigorous error bounds.

Findings

The rescaled distribution converges to a Maxwellian with density and temperature solving coupled diffusion equations.

Explicit error estimates in $L^2_{x,v}$ norm are provided.

The model accounts for collisions with fixed scatterers and hard-sphere interactions.

Abstract

We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution $f(x,v,t)$. There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled $f^\varepsilon(x,v,t)=f(\varepsilon^{-1}x,v,\varepsilon^{-2}t)$, as $\varepsilon\to 0$ tends to a Maxwellian $M_{\rho, 0, T}=\frac{\rho}{(2\pi T)^{3/2}}\exp[{-\frac{|v|^2}{2T}}]$, where $\rho$ and $T$ are solutions of coupled diffusion equations and estimate the error in $L^2_{x,v}$.

Equations

$${\partial }_{t}F+v\cdot \nabla F=Q(F),$$

$$\begin{gathered} Q_B(F,H)(v):= \\ \frac{1}{2}{\int }_{{\mathbb{R}}^3}\mathrm{d}w{\int }_{{\mathbb{S}}^2}\mathrm{d}\omega B(v-w,\omega )[F(v^{\prime })H(w^{\prime })+F(w^{\prime })H(v^{\prime })-F(v)H(w)-F(w)H(v)] \end{gathered}$$

$${\int }_{{\mathbb{R}}_v^3}dv{Q}_d(F)=0,{\int }_{{\mathbb{R}}_v^3}dv|v{|}^2{Q}_d(F)=0,$$

$${\int }_{{\mathbb{R}}_v^3}dvv{Q}_d(F)\ne 0,$$

$$D_d(F)=\int dx\int dv{Q}_d(F)\log F\le 0.$$

$$M_{\rho ,u,T}=\frac{\rho }{(2\pi T{)}^{3/2}}{e}^{-\frac{(v-u{)}^2}{2T}}.$$

$$\eta (\rho ,u,T):=-D_d(M_{\rho ,u,T})\ge 0,\text{ with}\eta (\rho ,u,T)=0\text{iff }u=0.$$

$$Q_d(F)={\int }_{{\mathbb{S}}^2}d\omega [F(v-2(v\cdot \omega )\omega )-F(v)]|v\cdot \omega |.$$

$$Q_B(F,F)+\alpha Q_d(F)=0\text{iff }F=M_{\rho ,0,T}.$$

$$D_B(F)+\alpha D_d(F)=0.$$

$${\partial }_t{F}^{\epsilon }+{\epsilon }^{-1}v\cdot \nabla F^{\epsilon }={\epsilon }^{-2}Q(F^{\epsilon }),$$

$$F^{\epsilon }=\mu +\sum _{i=1}^3{\epsilon }^i{F}_i+{\epsilon }^{5/2}\sqrt{\mu }f,$$

$${\int }_{\Omega \times {\mathbb{R}}_v^3}dxdv{F}^{\epsilon }={\int }_{\Omega \times {\mathbb{R}}_v^3}dxdv\mu ,{\int }_{\Omega \times {\mathbb{R}}_v^3}dxdv|v{|}^2{F}^{\epsilon }={\int }_{\Omega \times {\mathbb{R}}_v^3}dxdv|v{|}^2\mu .$$

$$\begin{cases}\partial_{t}\rho=\displaystyle{\nabla\cdot\Big{[}H\frac{\nabla\rho}{\rho}\Big{]}+\nabla\cdot\Big{[}H^{\prime}\frac{\nabla T}{T}\Big{]}}=0\\ \displaystyle{\frac{3}{2}}\rho\partial_{t}T=\displaystyle{\nabla\cdot\Big{[}TH^{\prime}\frac{\nabla\rho}{\rho}\Big{]}+\nabla\cdot\Big{[}TH^{\prime}_{1}\frac{\nabla T}{T}\Big{]}}=0\end{cases}$$

$$H=\int dvv_{i}\mathcal{L}^{-1}(\mu v_{i});\quad H^{\prime}=\int dvv_{i}\Big{(}\frac{|v|^{2}}{2T}-\frac{3}{2}\Big{)}\mathcal{L}^{-1}(\mu v_{i});$$

$$H^{\prime}_{1}=\int dvv_{i}\Big{(}\frac{|v|^{2}}{2T}-\frac{3}{2}\Big{)}\mathcal{L}^{-1}\Big{(}v_{i}\Big{(}\frac{|v|^{2}}{2T}-\frac{3}{2}\Big{)}\mu\Big{)}$$

$$\mathcal{L}F={\mathcal{L}}_{B}F-\alpha Q_d(F),$$

$${\mathcal{L}}_{B}F=-2Q_B(\mu ,F),$$

$$F_1=-{\mathcal{L}}^{-1}[v\cdot \nabla \mu ]+\mu \left[\frac{{\rho }_1}{\rho }+\frac{|v{|}^2-3T}{2\rho T^2}{T}_1\right].$$

$${\int }_{\Omega }dx{\rho }_1=0,{\int }_{\Omega }dx{T}_1=0.$$

$$\underset{0\le s\le t}{\sup }\sum _{k,\ell }(\Vert {\nabla }^k{\partial }_t^{\ell }\rho (s){\Vert }_2+\Vert {\nabla }^k{\partial }_t^{\ell }T(s){\Vert }_2)<C(\rho ,T).$$

$$\Vert {\mu }^{-1/2}(F-\mu ){\Vert }_2\le C\epsilon .$$

$$\|f\|_{p}=\Big{[}\int dxdv|f(x,v,t)|^{p}\Big{]}^{1/p};\quad\|f\|_{\infty}={\rm ess}\sup_{x,v}|f(x,v,t)|.$$

$$-$$

$$v\cdot \nabla \mu +\mathcal{L}{F}_1=0.$$

$$\int dvv\cdot \nabla \mu =0,\int dv|v{|}^{2}v\cdot \nabla \mu =0.$$

$$F_1=-{\mathcal{L}}^{-1}[v\cdot \nabla \mu ]+\mu \left[\frac{{\rho }_1}{\rho }+\frac{|v{|}^2-3T}{2\rho T^2}{T}_1\right],$$

$${\int }_{\Omega }dx{\rho }_1=0,{\int }_{\Omega }dx{T}_1=0.$$

$${\partial }_t\mu +v\cdot \nabla F_1-Q_B(F_1,F_1)+\mathcal{L}{F}_2=0,$$

$$\int dv[{\partial }_t\mu +v\cdot \nabla F_1]=0,\int dv|v{|}^2[{\partial }_t\mu +v\cdot \nabla F_1]=0.$$

Full text

(R. E.) International Research Center M&MOCS, Università dell’Aquila, Italy

(P.G.)Instituto Carlos I de Fisica Teorica y Computacional. Universidad de Granada. E-18071 Granada, Spain

(J. L.)Departments. of Mathematics and Physics, Rutgers University, USA

(R. M.) Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma, Italy

1. Introduction and results

We study a kinetic model investigated by Garrido and Lebowitz in [11] in which only the mass and the energy are conserved by the evolution but not the momentum. This models the flow of a gas (or fluid) in a porous medium. It can also be seen as the Grad-Boltzmann limit of a hard sphere system elastically scattered by randomly distributed obstacles. It thus serves as a simplified example for the derivation of macroscopic equations from mesoscopic kinetic ones: the number of conserved quantities is reduced from five to two. There are at present no rigorous derivations of hydrodynamic equations in the diffusive limit when there are five conserved quantities and density and temperature are space-time dependent to the lowest order. Here we extend the heuristic analysis of this system in two dimensions described in [11] and give a fully rigorous derivation of the appropriate coupled diffusion equations.

The model is defined in the following way:

Let $\Omega$ be the three dimensional unit torus. The kinetic equation on $\Omega\times\mathbb{R}^{3}_{v}$ is

[TABLE]

with $Q(F)=Q_{B}(F,F)+\alpha Q_{d}(F)$, $\alpha\geq 0$, where $Q_{B}(F,H)$ is the symmetrized Boltzmann collision operator for hard spheres [4] defined as

[TABLE]

with $v^{\prime}=v-[(v-w)\cdot\omega]\omega,\ w^{\prime}={v}+[(v-w)\cdot\omega]\omega{,}$ and $B(V,\omega)=|V\cdot\omega|$ is the hard spheres cross section and $Q_{d}(F)$ models elastic collisions with randomly distributed infinite mass scatterers at rest. $Q_{d}(F)$ is a linear operator conserving only mass and energy, not momentum, so that

[TABLE]

for any $F$, but

[TABLE]

for some $F$. We also require that the corresponding entropy dissipation is negative

[TABLE]

Letting $M_{\rho,u,T}$ be the local Maxwellian with $\rho$, $u$ and $T$ possibly depending on space and time

[TABLE]

We set

[TABLE]

We can model the non momentum conserving collisions with the background by various choices of $Q_{d}(F)$ [11].

We prefer here, for simplicity of presentation, to consider the operator

[TABLE]

The results in this paper apply to all choices in [11]. Note that, since $|v-2\omega(\omega\cdot v)|=|v|$, $Q_{d}(F)=0$ if $F$ depends on $v$ only through $|v|$. By the Boltzmann $H$ theorem, $D_{B}(F)=\int_{\Omega}dx\int_{\mathbb{R}^{3}_{v}}dv\log FQ_{B}(F,F)\leq 0$ and vanishes if and only if $F=M_{\rho,u,T}$. Moreover, if $\alpha>0$, we have also

[TABLE]

In fact, if $Q_{B}(F,F)+\alpha Q_{d}(F)=0$, multiplying by $\log F$ and integrating, we obtain

[TABLE]

But both are non positive, so we must have $D_{B}(F)=0$ and $D_{d}(F)=0$. The first implies $F=M_{\rho,u,T}$. By the second of (1.7) then $u=0$ and we get the conclusion.

From now on we assume $\alpha>0$.

To look at the behavior of the solution on the diffusive space-time scale [5] we consider the equation for $F^{\varepsilon}(x,v,t)=F(\varepsilon^{-1}x,v,\varepsilon^{-2}t)$. $F^{\varepsilon}$ so defined satisfies the equation

[TABLE]

and we seek for its solution in the form

[TABLE]

where $\mu=M_{\rho,0,T}$.

Note that, by total mass and total energy conservation, there is no loss of generality in assuming

[TABLE]

We prove that, as $\varepsilon\to 0$, $F^{\varepsilon}(x,v,t)$ tends to the Maxwellian $M_{\rho,0,T}$ where $\rho$ and $T$ are solutions of the following set of two coupled diffusion equations for the density $\rho$ and the temperature :

[TABLE]

where $H,H^{\prime},H_{1}$ are transport coefficients whose expressions are (independent of the index $i$)

[TABLE]

[TABLE]

Here

[TABLE]

where

[TABLE]

We remark that the transport coefficients $H$ and $H^{\prime}$ diverge as $\alpha\to 0$ because $v_{i}\mu$ are in the null space of $\mathcal{L}_{B}$, and $\mathcal{L}_{B}^{-1}$ is not well defined on the function $\mu v_{i}$.

Moreover, we determine also $F_{1}$ as

[TABLE]

where $\rho_{1}$, $T_{1}$ are solutions of linear diffusion equations such that

[TABLE]

$F_{i}$, for $i>1$, will be specified later.

The main result of this paper is the following

Theorem 1.1**.**

Let $t>0$ be fixed and assume that the solution $(\rho,T)$ to (3.20) and (3.21) have positive lower bounds and there is $C(\rho,T)\ll 1$ such that, for $0\leq k,\ell\leq 4$ with $1\leq k+\ell\leq 4$,

[TABLE]

Assume also that the initial value of $F$ is positive and satisfies (4.27) below. Then, if $\varepsilon\ll 1$, (1.10) has a positive solution $F$ such that

[TABLE]

Here, for $p\geq 1$, the $\|\,\cdot\,\|_{p}$-norm is defined as

[TABLE]

2. Strategy of the proof

The proof of theorem (1.1) will be given in Section 4 and here we only present the main ideas of the proof.

Once the Maxwellian $\mu$ and the terms of the expansion $F_{i}$, $i=1,\dots,3$ in (1.11) are computed, the main technical problem is to obtain bounds uniform in $\varepsilon$ for the remainder $f$, which solves a non linear problem. To deal with the non linearity we use an iterative procedure based on two steps. The first step is to study the linear problem obtained by pretending that the non linear term is computed using the solution of the previous step of the iteration. The aim is to bound the $L^{2}$-norm of the solution to the linear problem (see Proposition 4.4). The novelty with respect to previous work using this ideas, e.g. [7], is the fact that the Maxwellian $\mu$ depends on $x,t$ through $\rho$ and $T$. This produces a term singular in $\varepsilon$ in the inequality for $\|f\|_{2}^{2}$ which has to be dealt with. The most dangerous part of this term, depending on $(\mathbf{P}f)^{2}$ (here $\mathbf{P}$ is the projector on the space spanned by the conserved quantities) vanishes thanks to the fact that the Maxwellian $\mu$ has mean velocity $u=0$.

The other terms which have to be dealt with contain a polynomial of degree three in $v$ which gives troubles for large velocities. To this end, following [10], we introduce a global Maxwellian $\mu_{T}$ with temperature given by the $\min T(x,t)$ assumed strictly positive and bound the high velocity tail of $f$ in terms of the $L^{\infty}$ norm of $h\sim f\sqrt{\mu}/\sqrt{\mu}_{T}$. Then $h$ is bounded in $L^{\infty}$ by Proposition 4.3. The presence of $\|h\|_{\infty}$ in the energy inequality is a serious obstacle to obtaining a global in time statement. Theorem 1.1 is in fact established for arbitrary $t>0$, but with constants depending on $t$.

Once the linear problem is solved, we need to get bounds on the non linear term. Here we have another novelty with respect to [7]. In that paper the non linear term is bounded in terms of $L^{p}([0,t],\Omega,\mathbb{R}^{3}_{v})$ norms of $f$ and its time derivative $f_{t}$, with $p=3$ and $6$. Here we cannot use this method because the equation for $f_{t}$ involves a term which is too singular in $\varepsilon$. Therefore we can only use $L^{2}$ and $L^{\infty}$ norms. But the singularity $\varepsilon^{-3/2}$ of the $L^{\infty}$ norm of $h$ (see Proposition 4.3) has to be controlled by a sufficiently high power of $\varepsilon$ in front of the non linear term and hence we need to look for a remainder in (1.11) of order $\varepsilon^{5/2}$ while in [7] $\varepsilon$ was sufficient. We remark that, as a consequence, in the present case we need to assume some regularity properties of the limiting solution, while in [7] the convergence is proved without such assumption.

Remark 2.1**.**

We conclude this section by observing that if the random collisions operator $Q_{d}$ is absent ($\alpha=0$), the problem of deriving in a rigorous way hydrodynamic equations in the diffusive limit, with non homogeneous density and temperature at time zero, is completely open. A formal expansion shows (see e.g. [5, 2, 13]) that the limiting equations are different from the Navier-Stokes equations. In stationary non homogeneous situations there are few results, see for example [1].

3. The expansion

We start presenting the expansion. Following a strategy similar to [7], [8] (where only the first two terms of the expansion are considered) we look for a solution of the form (1.11),(1.12). By substituting (1.11) into equation (1.10) we get

[TABLE]

We now examine all the terms in the equation above. The most diverging term $\varepsilon^{-2}[Q_{B}(\mu,\mu)+Q_{d}(\mu)]$ vanishes because $\mu$ is a local Maxwellian with vanishing mean velocity.

To cancel the diverging term of order $\varepsilon^{-1}$ we impose

[TABLE]

Since $\int dv\mathcal{L}F=0$ and $\int dv|v|^{2}\mathcal{L}F=0$, the previous equation has a solution under the solvability conditons

[TABLE]

But $v\cdot\nabla\mu$ is odd in $v$, so conditions (3.2) are satisfied and one can write the most general solution to (3.1) in the form

[TABLE]

where the second term is the component of $F_{1}$ in the null space of $\mathcal{L}$. One can choose $\rho_{1}$, $T_{1}$ such that

[TABLE]

The term of order $\varepsilon$ satisfies the equation

[TABLE]

This equation has a solution if the following solvability conditions are satisfied

[TABLE]

Hence

[TABLE]

By using the expression of $F_{1}$ in (3.3) the solvability conditions (3.6) provide two diffusion equations for the density and temperature. In fact, writing (3.6) explicitly we get

[TABLE]

[TABLE]

The last term in both equations vanishes because it is odd in $v$. Thus, the equations reduce to

[TABLE]

[TABLE]

The term of order $\varepsilon$ is canceled by requiring

[TABLE]

We use this to find $F_{3}$:

[TABLE]

provided that

[TABLE]

Finally, we are left with an equation for $f$ which will be discussed later.

Now we proceed by finding the explicit expression of the hydrodynamic equations (3.10) and (3.11).

We have:

[TABLE]

[TABLE]

We notice that $\mathcal{L}^{-1}(v_{j}\mu(\frac{|v|^{2}}{2T}-\frac{3}{2}))$ and $\mathcal{L}^{-1}(v_{j}\mu)$ are well defined since the functions $v_{j}\mu(\frac{|v|^{2}}{2T}-\frac{3}{2}$ and $(v_{j}\mu)$ are in the space orthogonal to the null space of $\mathcal{L}$. We can use that $\mathcal{L}^{-1}$ is self-adjoint with respect to the $L^{2}$ scalar product with weight ${\mu}^{-1}$ to write

[TABLE]

Define the transport coefficients:

[TABLE]

[TABLE]

For isotropy reasons $H_{ij}=\delta_{ij}H;H^{\prime}_{1ij}=\delta_{ij}H^{\prime};H^{\prime}_{{ij}}=\delta_{ij}H^{\prime}_{1}$.

Then, the previous equations take the form:

[TABLE]

[TABLE]

where $e=\frac{3}{2}\rho T$ is the internal energy density.

To relate the above transport coefficients to the Onsager coefficients, we introduce the fugacity $z$ (related to the chemical potential $\mu$ through $\frac{\mu}{T}=\log z$) given for the perfect gas in $3d$ by $\log z=\displaystyle{\log\frac{\rho}{T^{3/2}}}$ and write the equations in the form

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

so that

[TABLE]

[TABLE]

By comparison,

[TABLE]

In [11] the form of the transport coefficients for a different choice of the linear Boltzmann operator and in dimension $2$ is discussed.

To get the regularity properties of the transport coefficients we can for example use the method in [3], where the eigenvalues of the linearized and the linear Boltzmann operators are studied by an expansion in spherical functions.

Plugging the expression for $F_{1}$ (3.3) and the one for $F_{2}$ (3.7) in the first equation of (3.14) we get

[TABLE]

The first and the third term in the first integral and the first term in the second integral do not give contribution. We are left with

[TABLE]

By using the relation $Q_{D}(h,h)=-\mathcal{L}(\mu^{-1}h^{2})$ and noting that $\int dvv_{i}\mathcal{L}_{B}^{-1}[\mathcal{L}_{B}\displaystyle{\frac{F_{1}^{2}}{\mu}}]=0$ by oddness, we get

[TABLE]

which is a linear diffusive non homogenous equation for $\rho_{1}$

By proceeding in the same way starting from the second compatibility condition, we get a linear diffusive non homogeneous equation for $T_{1}$.

The compatibility conditions (3.6), (3.14) thus turn out to be diffusion equations for $\rho$, $T$, $\rho_{1}$ and $T_{1}$. On the other hand $\rho_{2}$ and $T_{2}$ are determined by the orthogonality condition (4.19) discussed in Section 4, which are also diffusion equations; the procedure to get the equations for them is the same and we omit it. $\rho_{3}$ and $T_{3}$ are determined by a different condition (4.20), also discussed in Section 4.

4. Proofs

In this technical section we construct a solution for the equation for the remainder and prove Theorem 1.1.

4.1. General setup

We need some notation. We denote

[TABLE]

Since $\mu(v-2\omega(\omega\cdot v))=\mu(v)$, we have

[TABLE]

where

[TABLE]

Below, depending on the context, $(f,g)$ denotes the standard $L^{2}$ inner product in $L^{2}(\mathbb{R}^{3}_{v})$ or in $L^{2}(\Omega\times\mathbb{R}^{3}_{v})$. As is well known (see e.g. [4]), the quadratic form $(f,{L}_{B}f)$ is non negative and strictly positive if $f$ belongs to the orthogonal complement of the null space of $\mathcal{L}_{B}$, which is spanned by the orthonormal functions

[TABLE]

On the other hand, a direct check shows that there is $\lambda_{d}>0$ such that

[TABLE]

Let $\mathcal{N}$ be the null space of ${L}$. By the previous observations we immediately conclude that it is the linear subspace spanned by the normalized vectors

[TABLE]

In fact, let us consider the quadratic form

[TABLE]

If $f\in\mathcal{N}$ then $(f,Lf)=0$. Since the two terms in (4.7) are both non negative, then $(f,L_{B}f)=0$ and $(f,L_{d}f)=0$. Thus $f\in\mathcal{N}_{B}$ and $(f,\psi_{i})=0$ for $i=1,\dots,3$. Hence $\mathcal{N}$ is spanned by (4.6).

Denote by $\mathbf{P}$ the projector on such a subspace and by $(\mathbf{I}-\mathbf{P})$ the projector on the orthogonal subspace. Then

[TABLE]

For the Boltzmann collisions Grad proved (see [12]),

[TABLE]

where $K_{B}$ is a compact operator and $\nu_{B}$ satisfies the following bounds: there are positive $\tilde{\nu}_{0}$ and $\tilde{\nu}_{1}$ such that

[TABLE]

with $\langle v\rangle=(1+|v|^{2})^{\frac{1}{2}}$. The statement can be easily extended to the present operator $L$ which can thus be decomposed as

[TABLE]

where $K$ is a compact operator and $\nu$ satisfies the following bounds which follow immediately from (1.8): there are positive $\nu_{0}$ and $\nu_{1}$ such that

[TABLE]

The following spectral inequality holds:

[TABLE]

for a positive $\lambda$ and

[TABLE]

To prove (4.13), we note, as is well known (see e.g. [4]), that for the linear Boltzmann operator $L_{B}$ the following spectral inequality holds:

[TABLE]

where $\mathbf{P}_{B}f=\mathbf{P}f+\sum_{j=1}^{3}(f,\psi_{j})\psi_{j}$ and $\psi_{j}=\sqrt{\frac{\mu}{\rho T}}v_{j}$. By (4.5) we obtain (4.13) with $\lambda=\min(\lambda_{B},\lambda_{d})$. Note that $\lambda_{d}\sim\alpha$. Thus $\lambda_{d}\to 0$ as $\alpha\to 0$.

The main core of the proof of Theorem 1.1 is the control of the remainder $\sqrt{\mu}f$ which satisfies the following equation

[TABLE]

where

[TABLE]

and

[TABLE]

For the estimate of $f$ it will be essential that the part of $\tilde{A}$ in the null space of $\mathcal{L}$ vanishes. For this reason we also impose that

[TABLE]

[TABLE]

As before, (4.19) becomes a couple on linear non homogeneous parabolic equations.

Finally, we use the freedom of choice of $\rho_{3}$ and $T_{3}$ to ensure (4.20).

We have the following

Proposition 4.1**.**

Assume $0<\rho_{0}(x)$, $0<T_{0}(x)$ with finite $L^{\infty}$ norms. Then, for any $\bar{t}>0$ the equations (3.20) and (3.21) have smooth solutions. Moreover, if (1.19) is verified at time $t=0$, then it stays true for $t\in[0,\bar{t}]$. The functions $F_{i}$ satisfy the inequalities

[TABLE]

for $i=1,\dots,3$.

Proposition 4.1 is a simple consequence of the parabolic regularity and we omit the proof (see e.g. [9]).

It is convenient to write

[TABLE]

We note that

[TABLE]

The equation for $f$ then becomes

[TABLE]

where

[TABLE]

and

[TABLE]

where we have set $f_{i}=\mu^{-1/2}F_{i}$. Equation (4.24) has to be solved with initial datum $f(0)$ such that

[TABLE]

It is standard to check (see e.g. [7]) that it is possible to construct $f(0)$ so that (4.27) is satisfied.

It will be essential to have $\mathbf{P}A=0$. To this end we have required (4.19) and (4.20) which imply

[TABLE]

and hence

[TABLE]

We note the presence in (4.24) of the divergent term $\frac{1}{2}f[\varepsilon^{-1}v\cdot\nabla\log\mu]$, which is a source of extra difficulties. It is due to the use of the decomposition $F=\mu+\varepsilon F_{1}+\varepsilon^{2}F_{2}+\varepsilon^{3}F_{3}+\varepsilon^{5/2}\sqrt{\mu}f$ instead of $F+\varepsilon F_{1}+\varepsilon^{2}F_{2}+\varepsilon^{3}F_{3}+\varepsilon^{5/2}R$. However the first decomposition is useful to take advantage of the spectral properties of the operator $L$ in $L^{2}(\mathbb{R}^{3}_{v})$. Alternatively, we could use the second decomposition, but then the spectral properties we need should be sought for in $L^{2}(\mathbb{R}^{3}_{v},\mu^{-1})$ and, when integrating by parts, the weight $\mu^{-1}$ would produce a similar term in the resulting equation for $\|\mu^{-1/2}R\|_{2}$. As we shall see, the spectral properties are crucial in our proof, so we need to deal with this extra term.

We will follow the approach in [10] and [7]. Instead of repeating all the proofs in these papers, we only outline and give explicit proofs when the previous approach has to be modified to adapt to the case we study. The difference with respect to [10] is that the scaling in this case is diffusive instead of hyperbolic. The main difference with respect to [7] is in getting a $L^{2}$ bound for the linear equation due to the presence of the third term in (4.35) below. This term appears because $\mu$ depends on $x$ and $t$.

First of all we remark that in this term there will be a contribution including a higher power of velocity, $|v|^{3}f$,which is not present in the case studied in [7]. Following [10], we introduce $L^{2}$ and $L^{\infty}$ polynomial norms to control it.

By the assumption that $C(\rho,T)$ is sufficiently small, we see that there is $T_{M}>0$ such that for any $0\leq s\leq t$ and $x\in\Omega$,

[TABLE]

We define the global Maxwellian

[TABLE]

The inequalities (4.30) imply that there exist constants $c_{1}$ and $c_{2}$ such that for some $1/2<\alpha<1$ and for each $(t,x,v)$

[TABLE]

We stress that the previous bounds are true under the assumption of a slowly varying $T(x,t)$. Furthermore, we introduce the polynomial $w_{\sigma}(v)=(1+|v|^{2})^{\sigma}$ for the control of the cubic power of velocity. We define $h$ by the position

[TABLE]

and choose $\sigma>0$ later. Note that in consequence of (4.32) we have

[TABLE]

for some $C>0$.

4.2. The linear Problem

We start with the linear equation

[TABLE]

with some $g\in L^{2}(\Omega\times\mathbb{R}_{v})$, such that,

[TABLE]

In applying the result, we shall use $g$ of the form

[TABLE]

so that (4.36) is satisfied.

As we shall see, the use of the spectral inequality (4.13) in the energy inequality provides the control of the $L^{2}$-norm of $(\mathbf{I}-\mathbf{P})f$. What is missing is the control of the $L^{p}$-norm of $\mathbf{P}f$. This is achieved as in [7], which can be extended to the present setup. We have the following

Proposition 4.2**.**

Suppose $\mathbf{P}g=0$. If $\varepsilon>0$ is sufficiently small, there exists a function $G(t)$ such that, for all $0\leq s\leq t$, $|G(s)|\leq\|f(s)\|_{2}^{2}$ and a constant $C$ so that

[TABLE]

The proof of this proposition is postponed to the next section. Note that in [7] it is only requested that $\int dxdvg=0$, clearly implied by the stronger condition $\mathbf{P}g=0$.

As we shall see, the control of the cubic in $v$ term appearing in the energy inequality requires an $L^{\infty}$ estimate of the function $h$ defined in (4.33). This will be achieved by using the proposition below. As a consequence of (4.35), $h$ satisfies the equation

[TABLE]

with

[TABLE]

where $w=(1+|v|^{2})^{\sigma}$.

Proposition 4.3**.**

If $f$ solves (4.35) with $g$ given by (4.37) and $\rho$, $T$ are smooth solutions to (3.22) and (3.23) with stricly positive lower bounds uniform in $t$, such that inequalities (4.32) are satisfied, then there is a constant $C$ such that, for $\varepsilon$ sufficiently small,

[TABLE]

The proof is given in [7] and [10].

This proposition will be used within an iterative procedure where, at some step we know $\bar{h}$ and want to compute $h$ using (4.39) with

[TABLE]

where

[TABLE]

We shall use the bounds [10]:

[TABLE]

[TABLE]

To get an $L^{2}$ bound on the linear equation we multiply it by $f$ and integrate in $x$ and $v$ to obtain:

[TABLE]

Proposition 4.4**.**

Suppose that $C(\rho,T)$ is sufficiently small. If $f$ solves (4.35) with $g$ satisfying (4.36) and $\rho$, $T$ are smooth solutions to (3.22) and (3.23) with stricly positive lower bounds uniform in $t$, then, fixed $t>0$, there is a constant $C$ such that, for $\varepsilon$ sufficiently small,

[TABLE]

Proof.

In equation (4.46) the essential difference w.r.t. [7] are the second and third terms. To bound the third term we split $f^{2}=[\mathbf{P}f]^{2}+[(\mathbf{I}-\mathbf{P})f]^{2}+2\mathbf{P}f(\mathbf{I}-\mathbf{P})f$ and compute the contributions separately. The most singular (in $\varepsilon$) one is due to $|\mathbf{P}f|^{2}$, whose size in $L^{2}$ is $\varepsilon^{-2}$-times larger than $\|(\mathbf{I}-\mathbf{P})f\|_{2}^{2}$ by Proposition 4.2. Fortunately, we have

[TABLE]

because $v\cdot\nabla\log\mu$ is odd in $v$, while $a+c(|v|^{2}-3T)$ is even in $v$. Thus the largest term vanishes. As for the term $\varepsilon^{-1}\int dxdv\mathbf{P}f(\mathbf{I}-\mathbf{P})fv\cdot\nabla\log\mu$, we note that

[TABLE]

because the factor $\sqrt{\mu}$ in $\mathbf{P}f$ controls the polynomial $\nu^{-1/2}v\cdot\nabla\log\mu$. Thus

[TABLE]

and the first term is controlled using Proposition 4.2.

Now, as in [10] we introduce a cut-off on the velocity ${\kappa}{\varepsilon^{-a}}$ for some $a>0$ to be chosen, and estimate separately the term with low and high velocity.

We bound $\frac{1}{2}\varepsilon^{-1}\int dxdv((\mathbf{I}-\mathbf{P})f)^{2}v\cdot\nabla\log\mu$. We have

[TABLE]

The first term is bounded as

[TABLE]

The high velocity part is bounded as

[TABLE]

where $C_{k}={\frac{1}{(\varepsilon^{2a}+\kappa^{2})^{2}}}$. Moreover we have used that $\mu_{M}\leq C\mu$ and $|(1+v^{2})^{3/2}\nu^{-/2}f|\leq|(1+v^{2})^{-2}h|$ for $\sigma>\frac{5}{4}+2$ and $a=\frac{1}{2}$. Now,

[TABLE]

by choosing $\delta=\varepsilon^{3}$.

The term

[TABLE]

also contains a contribution involving $|\mathbf{P}f|^{2}$ which in this case is not zero. We have

[TABLE]

so that Proposition 4.2 can be used. The other terms can be controlled as before. Note that there is no $\varepsilon^{-1}$ factor.

Next, since $\mathbf{P}g=0$, we have the bound

[TABLE]

Summarizing, by using (4.13) we have

[TABLE]

Integrating on time between [math] and $t$ we obtain:

[TABLE]

By using Proposition 4.2 we can replace $\int_{0}^{t}ds\|\mathbf{P}f(s)\|_{2}^{2}$ by the right hand side of (4.38) so that (4.55) becomes

[TABLE]

Now we choose the parameters $\kappa$, $\gamma$ and $C(\rho,T)$ is such a way that

[TABLE]

so that (4.57) becomes

[TABLE]

∎

Proof of Proposition 4.2:

Proof.

It can be shown along the lines of [6]. We consider the following weak version of (4.35), obtained by multiplying (4.35) by a smooth function $\psi$, integrating for $(x,v,s)\in\Omega\times\mathbb{R}^{3}_{v}\times[0,t]$ and integrating by parts to move all the derivatives on $\psi$:

[TABLE]

for smooth test function $\psi$. We apply this for $\psi=\sqrt{\mu}\zeta$. Then from (4.58) we get

[TABLE]

We remark that the bad term disappears due to a cancellation. We can write

[TABLE]

for some funtions $a(x,t)$, $c(x,t)$ such that

[TABLE]

The conditions (4.61) are satisfied in consequence of the assumption (1.12).

To get bounds on $a$ and $c$ we use some particular functions $\zeta_{a}$ and $\zeta_{c}$. In fact we choose

[TABLE]

[TABLE]

where $\phi_{a}$ solves

[TABLE]

and $\phi_{c}$ solves

[TABLE]

and $\beta_{a}$ and $\beta_{c}$ are constants to be chosen as in [6, 7]. The zero average conditions for $a$ and $c$ (4.61) are essential to ensure the solvability of (4.62) and (4.63) and they are compatible with the equation as a consequence of the (1.12). The estimates (4.38) of $\|a\|_{2}$ and $\|c\|_{2}$ are obtained as in [6, 7]. ∎

4.3. The non linear problem

Now we remind that $g$ is given by (4.37). The strategy to construct the solution to (4.24) is to define a sequence $\{f^{(n)}\}_{n=0}^{n=\infty}$ of solutions to the following linear problems: $f^{(0)}=0$ and, for $n>0$

[TABLE]

with given $g^{(n)}\in L^{2}(\Omega\times\mathbb{R}_{v})$. In view of (4.24) the choice of $g^{(n)}$ will be

[TABLE]

for $n\geq 1$. In consequence, we also define $h^{(n)}$ so that

[TABLE]

By (4.67), we have

[TABLE]

We have the following

Lemma 4.5**.**

[TABLE]

The proof of the lemma follows as in [7], using also Proposition 4.1 for (4.71) and (4.72) and we do not repeat it.

As a consequence, reminding (4.67), and using $\|f^{n-1}\|_{\infty}\leq C\|h^{n-1}\|_{\infty}$, we have

[TABLE]

[TABLE]

By using (4.73) and (4.47) we have

[TABLE]

By using (4.74) and (4.41) we have

[TABLE]

We assume that

[TABLE]

Inductive hypothesis: Fixed $\bar{t}>0$, assume

[TABLE]

By using this assumption we have

[TABLE]

We choose $\varepsilon$, $c_{0}$, $C(\rho,T)$, $C_{\kappa}$ so that

[TABLE]

so that

[TABLE]

Then

[TABLE]

We choose $\varepsilon$, $c_{1}$, and $\eta_{0}$ so that

[TABLE]

so that

[TABLE]

Therefore the inductive hypothesis is verified up to $n$. This shows that the sequence $\{f^{(n)}\}_{n=0}^{n=\infty}$ is uniformly bounded in $L^{2}$ by $\eta_{0}$ and in $L^{\infty}$ by $\eta_{1}$. By similar arguments one can show that $\|f^{(n)}-f^{(n-1)}\|_{2}\leq\theta\|f^{(n-1)}-f^{(n-2)}\|_{2}$ for some $\theta<1$ and hence the sequence is convergent and the limit solves uniquely (4.24). The proof of the positivity of $F$ is standard [7]. This concludes the proof of Theorem 1.1.

Acknowledgements. This work was supported in part by AFOSR [grant FA-9550-16-1- 0037]. PLG was supported also by the Spanish governement project FIS2013-43201P.