Diffusion limit for a kinetic equation with a thermostatted interface

TL;DR

This paper proves that under diffusive scaling, the energy density in a harmonic chain with an interface converges to a heat equation with a fixed boundary temperature, linking microscopic stochastic dynamics to macroscopic heat flow.

Contribution

It establishes the diffusion limit for a kinetic phonon equation with an interface, connecting microscopic stochastic models to macroscopic heat equations with boundary conditions.

Findings

Solutions converge to a heat equation with boundary temperature T

Validates the diffusive limit for the phonon Boltzmann equation

Provides a rigorous link between microscopic and macroscopic models

Abstract

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature $T$ in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution $\rho(t,y)$ of a heat equation with the boundary condition $\rho(t,0)\equiv T$.

Equations

$$\begin{array}[]{ll}\partial_{t}W(t,y,k)+\bar{\omega}^{\prime}(k)\partial_{y}W(t,y,k)=\gamma_{0}LW(t,y,k),&\quad(t,y,k)\in{\mathbb{R}}_{+}\times{\mathbb{R}}_{*}\times{\mathbb{T}}_{*},\end{array}$$

$${\mathbb{R}}_{+}:=(0,+\infty ),{\mathbb{R}}_{\ast }:=\mathbb{R}\setminus \{0\},{\mathbb{T}}_{\ast }:=\mathbb{T}\setminus \{0\},$$

$$LF(k):={\int }_{\mathbb{T}}R(k,k^{\prime })\left[F\left(k^{\prime }\right)-F\left(k\right)\right]d{k}^{\prime },k\in \mathbb{T}$$

$${\bar{\omega }}^{\prime }(k)=\frac{{\omega }^{\prime }(k)}{2\pi },k\in \mathbb{T},$$

$$\begin{array}{rl}W(t,0^{+},k)& =p_{-}(k)W(t,0^{+},-k)+p_{+}(k)W(t,0^{-},k)+T\mathfrak{g}(k),for\; 0<k\le 1/2,\\ W(t,0^{-},k)& =p_{-}(k)W(t,0^{-},-k)+p_{+}(k)W(t,0^{+},k)+T\mathfrak{g}(k),for-1/2<k<0.\end{array}$$

$$p_{+}(k)+p_{-}(k)+\mathfrak{g}(k)=1.$$

$$\begin{array}{rl}& {\partial }_t{W}^{\epsilon }(t,y,k)+\frac{1}{\epsilon }{\bar{\omega }}^{\prime }(k){\partial }_y{W}^{\epsilon }(t,y,k)=\frac{\gamma }{{\epsilon }^2}{\int }_{\mathbb{T}}R(k,k^{\prime })\left[W^{\epsilon }\left(t,y,k^{\prime }\right)-W^{\epsilon }\left(t,y,k\right)\right]d{k}^{\prime },y\ne 0,\\ & W^{\epsilon }(0,y,k)=W_0(y,k),\end{array}$$

$${\int }_{\mathbb{T}}\frac{{\omega }^{\prime }(k{)}^2}{R(k)}dk<+\infty ,$$

$$\underset{ϵ\to 0}{\lim }{\int }_{\mathbb{R}\times \mathbb{T}}{W}^{\epsilon }(t,y,k)G(y,k)dydk={\int }_{\mathbb{R}\times \mathbb{T}}\rho (t,y)G(y,k)dydk,$$

$${\partial }_t\rho (t,y)=D{\partial }_y^2\rho (t,y),(t,y)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{\ast },$$

$$\rho (t,0^{\pm })\equiv T$$

$$\rho (0,y)={\rho }_0(y):={\int }_{\mathbb{T}}{W}_0(y,k)dk.$$

$$D=\frac{1}{\gamma }\int {\omega }^{\prime }(k)(-L{)}^{-1}{\omega }^{\prime }(k)dk$$

$$\tilde{W}(t,0^{+},k)=p_{-}(k)\tilde{W}(t,0^{+},-k)+p_{+}(k)\tilde{W}(t,0^{-},k)),for\; 0<k\le 1/2,$$

$$\tilde{W}(t,0^{-},k)=p_{-}(k)\tilde{W}(t,0^{-},-k)+p_{+}(k)\tilde{W}(t,0^{+},k),for-1/2<k<0.$$

$$0={\int }_0^{+\infty }{\int }_{\mathbb{R}\times \mathbb{T}}\tilde{W}(t,y,k)\left[{\partial }_t\phi (t,y,k)+{\bar{\omega }}^{\prime }(k){\partial }_y\phi (t,y,k)+\gamma L\phi (t,y,k)\right]dtdydk$$

$$+{\int }_{\mathbb{R}\times \mathbb{T}}{W}_0(y,k)\phi (0,y,k)dydk$$

$$R(k,k^{\prime })=R(k^{\prime },k),$$

$$R(k,k^{\prime })>0,(k,k^{\prime })\in {\mathbb{T}}_{\ast }^2$$

$$R(k):={\int }_{\mathbb{T}}R(k,k^{\prime })d{k}^{\prime }$$

$$p(k,k^{\prime }):=\frac{R(k,k^{\prime })}{R(k)}\in C^{\infty }({\mathbb{T}}^2).$$

$$R(k)\sim R_0|\sin (\pi k){|}^{\beta },|k|\ll 1$$

$${\omega }^{\prime }(k)\sim 2{\omega }_0^{\prime }\mathrm{sign}(k)|\sin (\pi k){|}^{\kappa },|k|\ll 1$$

$$0\le \beta <1+2\kappa .$$

$$\tilde{g}(\lambda ):={\left(1+\gamma {\int }_{\mathbb{T}}\frac{\lambda dk}{{\lambda }^2+{\omega }^2(k)}\right)}^{-1},\mathrm{Re}\lambda >0.$$

$$|\tilde{g}(\lambda )|\le 1,\lambda \in {\mathbb{C}}_{+}:=[\lambda \in \mathbb{C}:\mathrm{Re}\lambda >0].$$

$$\nu (k):=\underset{ϵ\to 0+}{\lim }\tilde{g}(ϵ-i\omega (k))$$

$$\mathfrak{g}(k):=\frac{\gamma |\nu (k){|}^2}{|{\bar{\omega }}^{\prime }(k){|}^2},\mathfrak{P}(k):=\frac{\gamma \nu (k)}{2|{\bar{\omega }}^{\prime }(k)|}$$

$$p_{+}(k):={\left|1-\mathfrak{P}(k)\right|}^2$$

$$p_{-}(k):={\left|\mathfrak{P}(k)\right|}^2.$$

$$\mathrm{Re}\nu (k)=\left(1+\frac{\gamma }{2|{\bar{\omega }}^{\prime }(k)|}\right)|\nu (k){|}^2.$$

$$\underset{ϵ\to 0}{\lim }{\int }_0^{+\infty }dt{\int }_{\mathbb{R}\times \mathbb{T}}{W}^{\epsilon }(t,y,k)\phi (t,y,k)dydk={\int }_0^{+\infty }dt{\int }_{\mathbb{R}\times \mathbb{T}}\rho (t,y)\phi (t,y,k)dydk,$$

$${\partial }_t\rho (t,y)=D{\partial }_y^2\rho (t,y),(t,y)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{\ast },$$

Full text

T.K. acknowledges the support of the National Science Centre: NCN grant 2016/23/B/ST1/00492. SO’s research is supported by ANR-15-CE40-0020-01 grant LSD. Both T.K. and S.O. were partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

1. Introduction

We consider a linear phonon Boltzmann equation in contact with a heat bath at the origin. This equation describes the evolution, after a proper kinetic limit, of the phonon energy in a chain of harmonic oscillators with random scattering of velocities, where one oscillator is in contact with a heat bath at temperature $T$.

We denoting by $W(t,y,k)$ the energy density of phonons at time $t\geq 0$, with respect to their position $y\in{\mathbb{R}}$ and frequency variable $k\in{\mathbb{T}}$ - the one dimensional circle, understood as the interval $[-1/2,1/2]$ with identified endpoints. The heat bath creates an interface localized at $y=0$. Outside the interface the density satisfies

[TABLE]

Here

[TABLE]

The parameter $\gamma_{0}>0$ represents the phonon scattering rate. The scattering operator $L$, acting only on the $k$-variable, is given by

[TABLE]

for $F$ belongs to $B_{b}({\mathbb{T}})$ - the set of bounded measurable, real valued functions. Here

[TABLE]

where $\omega:{\mathbb{T}}\to[0,+\infty)$ is the dispersion relation of the harmonic chain.

The interface conditions that describe the interaction of a phonon with a thermostat (placed at $y=0$), at temperature $T>0$, are given as follows:

• the outgoing densities are given in terms of the incoming ones as

[TABLE]

where $p_{-},p_{+},{\mathfrak{g}}:{\mathbb{T}}\to(0,1]$ are continuous and

[TABLE]

In other words, $p_{-}(k)$ and $p_{+}(k)$ are the reflection and transmission coefficients across the interface, respectively. They correspond to the probabilities of the phonon being reflected, or transmitted, by the interface. The quantity $T\mathfrak{g}(k)$ is the phonon production rate by the thermostat as well as the absorption rate of the frequency $k$ phonon by the interface. The parameter $T>0$ is the heat bath temperature. This equation has been obtained in [7], without the heat bath, as the Boltzmann-Grad limit of the energy density function for a microscopic model of a heat conductor consisting of a one dimensional chain of harmonic oscillators, with inter-particle scattering conserving the energy and volume. In the presence of the thermostat, but with no scattering (the case $\gamma_{0}=0$), the limit has been proved [13]. It is believed that the limit also holds in case of the presence of scattering, i.e. when $\gamma_{0}>0$.

We are interested in the asymptotics of the solutions to (1.1) under the diffusive scaling, i.e. the limit, as $\epsilon\to 0$, for $W^{\varepsilon}(t,y,k)=W(t/\varepsilon^{2},y/\varepsilon,k)$, i.e. we consider the equation

[TABLE]

with the interface conditions (1.3). Let $R(k)=\int R(k,k^{\prime})dk^{\prime}$. In our main result, see Theorem 2.2 below, we prove that under the assumption

[TABLE]

and some other technical hypotheses, formulated in Sections 2.2 and 2.3 below, for any $G\in C_{0}^{\infty}({\mathbb{R}}\times{\mathbb{T}})$ -compactly supported, smooth function - we have

[TABLE]

where

[TABLE]

The diffusion coefficient is given by

[TABLE]

that is finite by the assumption (1.6) and the properties of $R(k,k^{\prime})$ made in section 2.2.

The result implies that only the absorption and the creation of phonons at the interface matter in this diffusive scale. Phonons that are reflected or transmitted will come back to the interface, due to scattering, and eventually get absorbed in a shorter time scale.

The diffusive limit has been considered, without the presence of interface, in [5, 11, 15]. It has been shown there that, if (1.6) is in force, then the solutions of the initial problem (1.5) converge, as in (1.7), to $\rho(t,y)$ - the solution of the Cauchy problem for the heat equation (1.8). When the condition (1.6) is violated a superdiffusive scaling may be required and the limit could be a fractional diffusion. This case has been also considered in [6, 8, 11, 15].

The case of the diffusive limit of the solution of a kinetic equation with an absorbing boundary has been considered in e.g. [14, 9, 10, 4, 3]. The diffusive limit with some other boundary conditions has also been discussed in the review paper [2], see the references contained therein. A related result for the radiative transport equation with some reflection/transmission condition has been obtained in [1] for the steady state, giving rise to different boundary conditions. We are not aware of a similar result in the dynamical case, as considered in the present paper.

The result for a fractional diffusive limit with the interface condition is a subject of the paper [12].

2. Preliminaries and the statement of the main result

2.1. Weak solution of the kinetic equation with an interface

In what follows we denote ${\mathbb{R}}_{-}:=(-\infty,0)$. Consider $\widetilde{W}(t,y,k):=W(t,y,k)-T$. It satisfies the equation (1.1) with the interface given by

[TABLE]

and

[TABLE]

Definition 2.1**.**

A function $\widetilde{W}:\bar{\mathbb{R}}_{+}\times{\mathbb{R}}\times{\mathbb{T}}\to{\mathbb{R}}$ is called a (weak) solution to equation (1.1) with the interface conditions (2.1) and (2.2), provided that it belongs to $L^{2}_{\rm loc}({\mathbb{R}}_{+},L^{2}({\mathbb{R}}\times{\mathbb{T}}))$, its restrictions $\widetilde{W}_{\iota}$ to ${\mathbb{R}}_{+}\times{\mathbb{R}}_{\iota}\times{\mathbb{T}}$, $\iota\in\{-,+\}$ extend to continuous functions on ${\mathbb{R}}_{+}\times\bar{\mathbb{R}}_{\iota}\times{\mathbb{T}}$ that satisfy (2.1) and (2.2), and

[TABLE]

for any test function $\varphi\in C_{0}^{\infty}(\bar{\mathbb{R}}_{+}\times{\mathbb{R}}_{*}\times{\mathbb{T}})$,

2.2. Assumption about the dispersion relation and the

scattering kernel.

We assume that $\omega(\cdot)$ is even, belongs to $C^{\infty}({\mathbb{T}}\setminus\{0\})$, i.e. it smooth outside $k=0$. Furthermore we assume that $\omega(\cdot)$ is unimodal, that implies that $k\omega^{\prime}(k)\geq 0$ for $k\in(-1/2,1/2)$.

We assume that the scattering kernel is symmetric

[TABLE]

positive, except for [math] frequency, i.e.

[TABLE]

and the total scattering kernel

[TABLE]

is such that the stochastic kernel

[TABLE]

In addition we assume that (1.6) is in force.

Example. Suppose that

[TABLE]

for some $\beta\geq 0$ and $R_{0}>0$ and

[TABLE]

for some $\kappa\geq 0$. Then (1.6) holds, provided that

[TABLE]

2.3. About the reflection, transmission and

absorption coefficients

In [13] the coefficient $p_{\pm}(k)$ and $\mathfrak{g}(k)$ are obtained from the microscopic dynamics and depends on the dispersion relation as follows.

Let $\gamma>0$ (the thermostat strength) and

[TABLE]

It turns out, see [13], that

[TABLE]

The function $\tilde{g}(\cdot)$ is analytic on $\mathbb{C}_{+}$. By Fatou’s theorem we know that

[TABLE]

exists a.e. in ${\mathbb{T}}$ and in any $L^{p}({\mathbb{T}})$ sense for $p\in[1,\infty)$. Denote

[TABLE]

It has been shown in Section 10 of [13] that

[TABLE]

This identity implies in particular that (1.4) is in force.

2.4. Scaled kinetic equations and the formulation of the main result

Consider $W^{\varepsilon}$ the solution of a rescaled kinetic equations (1.5). Our main result can be stated as follows.

Theorem 2.2**.**

Suppose that $W_{0}(y,k)=T+\widetilde{W}_{0}(y,k)$, where $\widetilde{W}_{0}\in L^{2}({\mathbb{R}}\times{\mathbb{T}})$. Under the assumptions made about the scattering kernel $R(\cdot,\cdot)$ and dispersion relation $\omega(\cdot)$, for any test function $\varphi\in C_{0}^{\infty}({\mathbb{R}}_{+}\times{\mathbb{R}}\times{\mathbb{T}})$ we have

[TABLE]

where $\rho(\cdot,\cdot)$ is the solution of the heat equation

[TABLE]

Here, the coefficient $D>0$ is given by (4.6) below.

Defining $\widetilde{W}^{\varepsilon}=W^{\varepsilon}-T$, one can easily see that it also satisfies (1.5) with

[TABLE]

and

[TABLE]

The initial condition $\widetilde{W}^{\varepsilon}(0,y,k)=\widetilde{W}_{0}(y,k):=W_{0}(y,k)-T$ belongs to $L^{2}({\mathbb{R}}\times{\mathbb{T}})$. This means that it is enough to prove Theorem 2.2 for $T=0$. This proof is presented in Section 4.

3. Some auxiliaries

3.1. Some functional spaces

Let $H_{+}^{1}$ be the Hilbert obtained as the completion of the Schwartz class ${\cal S}({\mathbb{R}}\times{\mathbb{T}})$ in the norm

[TABLE]

Similarly, we introduce $H^{1}_{-}$.

Let ${\cal H}$ be the Hilbert space obtained as the completion of ${\cal S}({\mathbb{R}}\times{\mathbb{T}})$ in the norm

[TABLE]

Let also

[TABLE]

3.2. Apriori bounds

Computing the time derivative we have

[TABLE]

with

[TABLE]

Taking into account (2.17) and (2.18) we obtain

[TABLE]

After straightforward calculations (recall that coefficients $p_{\pm}(k)$ are even, while $\bar{\omega}^{\prime}(k)$ is odd) we conclude that the right hand side equals

[TABLE]

Since $p_{+}(k)+p_{-}(k)\leq 1$ we have $1-p_{+}^{2}(k)-p_{-}^{2}(k)\geq 0$. In addition,

[TABLE]

Using (1.4) we conclude that the quadratic form

[TABLE]

is positive definite as long as $p_{+}(k)+p_{-}(k)<1$. The eigenvalues of the form can be determined from the equation

[TABLE]

which yields

[TABLE]

and $\lambda_{+}>\lambda_{-}$. Note that

[TABLE]

Equality (3.2) allows us to obtain the following apriori bounds

[TABLE]

By (2.17) and (2.18) we obtain that

[TABLE]

Then using the unimodality of $\omega(k)$ it follows that

[TABLE]

3.3. Uniform continuity at $y=0$

Suppose that $y>0$. Let

[TABLE]

Since $\widetilde{W}^{\varepsilon}(s,y,k)$ satisfies (1.5) we can write

[TABLE]

with

[TABLE]

Hence, using Cauchy-Schwarz inequality, we get

[TABLE]

Using the second estimate of (3.4) we conclude that for each $t_{0}>0$

[TABLE]

From (3.8) we conclude that

[TABLE]

where

[TABLE]

From (3.9) and the first estimate of (3.4) we conclude

[TABLE]

We have

[TABLE]

Since $\dot{V}_{\epsilon}(t)=\widetilde{W}_{\epsilon}(t),$ from the first estimate of (3.4) we conclude that for any $t_{0}>0$

[TABLE]

From (3.11) we get also (cf (3.1))

[TABLE]

Summarizing, we have shown the following.

Proposition 3.1**.**

For any $t_{0}>0$

[TABLE]

and

[TABLE]

Denote by $W^{1,\infty}_{0}([0,t_{0}];L^{2}({\mathbb{R}}\times{\mathbb{T}}))$ the completion of the space of smooth functions $f:[0,t_{0}]\to L^{2}({\mathbb{R}}\times{\mathbb{T}})$ satisfying $f(0)=0$, with respect to the norm

[TABLE]

As a consequence of the above proposition we immediately conclude the following.

Corollary 3.2**.**

The family $\left(V_{\varepsilon}(\cdot)\right)_{\epsilon\in(0,1]}$ is bounded in $W^{1,\infty}_{0}([0,t_{0}];L^{2}({\mathbb{R}}\times{\mathbb{T}}))\cap L^{\infty}([0,t_{0}];{\cal H})$ for aby $t_{0}>0$. Any $\star$-weak limit point $V(\cdot)$ of $V_{\varepsilon}(\cdot)$, as $\varepsilon\to 0+$, satisfies the following:

$V(t,y,k)\equiv\bar{V}(t,y):={\int}_{{\mathbb{T}}}V(t,y,k)dk$* for $(t,y,k)\in{\mathbb{R}}_{+}\times{\mathbb{R}}\times{\mathbb{T}})$*

• 2)

the mapping ${\mathbb{R}}_{+}\times{\mathbb{R}}_{\iota}\ni(t,y)\mapsto\bar{V}(t,y)$ extends to a mapping from $C(\bar{\mathbb{R}}_{+}\times\bar{\mathbb{R}}_{\iota})$, $\iota\in\{-,+\}$,

• 3)

$V(t,0^{\pm})=0$* for each $t>0$,*

• 4)

$V(0,y)\equiv 0$, $y\in{\mathbb{R}}$.

4. Proof of Theorem 2.2

Thanks to the above estimates, we conclude that the solutions $\widetilde{W}^{\varepsilon}(\cdot)$ are $\star$-weakly sequentially compact in $L^{\infty}\left([0,+\infty);L^{2}_{w}({\mathbb{R}}\times{\mathbb{T}})\right)$, where $L^{2}_{w}({\mathbb{R}}\times{\mathbb{T}})$ denotes $L^{2}({\mathbb{R}}\times{\mathbb{T}})$ with the weak topology. Suppose that $\bar{W}(t,y,k)$ is a limiting point for some subsequence $(\widetilde{W}^{\varepsilon_{n}}(s,y,k))$, where $\varepsilon_{n}\to 0$. For convenience sake we shall denote the subsequence by $(\widetilde{W}^{\varepsilon}(s,y,k))$. Thanks to (3.4) for each $t>0$ we have (cf (3.3))

[TABLE]

thus $\bar{W}(t,y,k)\equiv\rho(t,y)$, for a.e. $(t,y,k)\in{\mathbb{R}}_{+}\times{\mathbb{R}}\times{\mathbb{T}}$.

Lemma 4.1**.**

Equation

[TABLE]

has a unique solution such that

[TABLE]

and

[TABLE]

Proof.

Let $\mu$ be a Borel probability measure on ${\mathbb{T}}$ given by

[TABLE]

where

[TABLE]

We can reformulate (4.2) as

[TABLE]

where, by virtue of (1.6), the right hand side belongs to $L^{2}(\mu)$ and $P:L^{2}(\mu)\to L^{2}(\mu)$ is a symmetric operator on $L^{2}(\mu)$ given by

[TABLE]

The operator is a compact contraction and, since

[TABLE]

we conclude that $1$ is a simple eigenvalue, with the respective eigenspace spanned on the eigenvector $F_{0}\equiv 1$. Thus the conclusion of the lemma follows, as $\bar{\omega}^{\prime}/R\perp F_{0}$. ∎

Proposition 4.2**.**

For any function $\varphi\in C_{0}^{\infty}\left({\mathbb{R}}_{+}\times{\mathbb{R}}_{*}\right)$ we have

[TABLE]

with

[TABLE]

Proof.

The claim made in (4.6) follows immediately from the fact that $X_{1}\not=0$ and $D={\cal D}(\chi)$.

To prove (4.5) we apply a version of the perturbed test function technique. Let $\varphi_{\varepsilon}(t,y,k)$ be determined by

[TABLE]

where $\varphi\in C_{0}^{\infty}({\mathbb{R}}_{+}\times{\mathbb{R}}_{*})$ and $\chi_{j}(t,y,k)$, $j=1,2$ are yet to be determined.

We can write

[TABLE]

Since the support in the $y$ variable of the test function $\varphi_{\epsilon}(t,y,k)$ will turn out to be isolated from [math], the integration by parts yield the equation

[TABLE]

Substituting from (4.7) we obtain that the term in the brackets has the form

[TABLE]

with

[TABLE]

We stipulate that

[TABLE]

The first condition yields

[TABLE]

while the second implies that $\chi_{2}(t,y,k)$ is the solution of

[TABLE]

Substituting from (4.13) we get

[TABLE]

Since $(D+\bar{\omega}^{\prime}X_{1}/\gamma)/R$ belongs to $L^{2}(\mu)$ and its orthogonal to constants, we can solve the equation

[TABLE]

using the same argument as in Lemma 4.1. Then,

[TABLE]

Clearly $I\!I\!I_{\epsilon}=O(1)$. Taking the limit in (4.9) we obtain (4.5), that ends the proof of the proposition.∎

Suppose that $\bar{V}(t,y)$ is a limiting point for some subsequence

[TABLE]

where $\varepsilon_{n}\to 0$, in the sense described by Corollary 3.2. We can also assume that the respective sequence $(\widetilde{W}_{\varepsilon}(\cdot))$ $\star$-weakly converge to $\rho(t,y)$ in $L^{\infty}\left([0,+\infty);L^{2}_{w}({\mathbb{R}}\times{\mathbb{T}})\right)$.

For convenience sake we shall denote the subsequences by $(V_{\varepsilon}(\cdot))$, $(\widetilde{W}_{\varepsilon}(\cdot))$, respectively. We have

[TABLE]

Proposition 4.3**.**

For any function $\varphi\in C_{0}^{\infty}\left(\bar{\mathbb{R}}_{+}\times{\mathbb{R}}_{*}\right)$ we have

[TABLE]

with $D$ given by (4.6) and $\rho_{0}$ defined in (2.2).

Before showing the proof of the proposition we show how to finish, with its help, the proof of Theorem 2.2. According to Proposition 4.3 $\bar{V}(t,y)$ is a weak solution of the heat equation

[TABLE]

satisfying $\bar{V}(0,y)\equiv 0$. According to part 3) of Corollary 3.2 we also have $\bar{V}(t,0^{\pm})\equiv 0$, $t\geq 0$. Hence,

[TABLE]

This, combined with (4.16), implies that

[TABLE]

which satisfies the conclusion of Theorem 2.2 for $T=0$. The only thing yet to be shown is the proof of Proposition 4.3.

Proof of Proposition 4.3

According to (3.8) we have

[TABLE]

and obviously $V_{\epsilon}(0,y,k)=0$ a.e. Let $\varphi_{\varepsilon}(t,y,k)$ be given by (4.7). From (4.19) we can write

[TABLE]

Substituting from (4.7) we obtain that the term in the square brackets has the form (4.10), with $I$, $I\!I$ and $I\!I\!I_{\epsilon}$ as given in (4.16). Taking the limit in (4) we obtain (4.7).∎