Nova S´erie

MEASURE-VALUED SOLUTIONS AND WELL-POSEDNESS OF MULTI-DIMENSIONAL CONSERVATION LAWS

IN A BOUNDED DOMAIN *

C.I. Kondo and P.G. LeFloch Presented by J.P. Dias

Abstract: We propose a general framework to establish the strong convergence
of approximate solutions to multi-dimensional conservation laws in a bounded domain,
provided uniform bounds on theirL^{p} norm and their entropy dissipation measures are
available. To this end, existence, uniqueness, and compactness results are proven in a
class of entropy measure-valued solutions, following DiPerna and Szepessy. The new
features lie in the treatment of the boundary condition, which we are able to formulate
by relying only on an L^{p} uniform bound. This framework is applied here to prove the
strong convergence of diffusive approximations of hyperbolic conservation laws.

Introduction

In this paper, we are interested in the boundary and initial value problem for a hyperbolic conservation law in several space dimensions:

(1.1) ∂tu+ divf(u) = 0, u(x, t)∈R, x∈Ω, t >0 ,

(1.2) u(x,0) =u_{0}(x), x∈Ω,

where Ω is an open and bounded subset of R^{d} with a smooth boundary ∂Ω.

Here we have setx= (x_{1}, x_{2}, ..., x_{d}), divf(u) : =^{P}^{d}_{j=1}∂_{j}f_{j}(u), and∂_{j}: = ∂_{x}_{j} for

Received: April 13, 2000.

1980Mathematics Subject Classification: Primary35L65; Secondary76N10.

Keywords and Phrases: Conservation law; Entropy inequality; Measure-valued solution;

Boundary condition; Well-posedness.

* This work was performed while the first author was visiting the second author at Ecole Polytechnique for the academic year 1999–00. Thanks to the supports of PICD-Capes in Brazil and the European research project Alpha.

j= 1, ..., d. The flux-functionf = (f_{1}, f_{2}, ..., f_{d}) : R→R^{d} is a given continuous
mapping and the initial datumu0belongs to the spaceL^{p}(Ω) for somep∈(1,∞].

Furthermore along the boundary we impose a boundary datumu_{B}: ∂Ω×R+→
R^{d},

(1.3) u(x, t) =uB(x, t), x∈∂Ω, t >0,

however expressed in the weak sense of Bardos, Leroux, and Nedelec [1]. Con- cerning the boundary conditions for hyperbolic conservation laws, we refer the reader to LeFloch [10], Dubois and LeFloch [6], Szepessy [16], Cockburn, Coquel, and LeFloch [3], Joseph and LeFloch [8], Otto (see [12]), Chen and Frid [2], and the references therein.

In the present paper, in Sections 2 and 3, we develop a general framework
aimed at proving the convergence of a sequence of approximate solutionsu^{ε} to-
ward a solution of (1.1)–(1.3). The notion of entropy measure-valued solution
introduced by DiPerna [5] plays here a central role. These are Young measures
(Tartar [18]) satisfying the equation and the entropy inequalities in a weak sense.

Under some natural assumption, a sequence of approximate solutions always gen- erates an entropy measure-valued solution.

The key is given by a suitable generalization of Kruzkov’sL^{1}contraction prop-
erty [9] discovered by DiPerna [5] and extended by Szepessy [15, 16], to include
on one hand the boundary conditions and, on the other hand, measure-valued
solutions inL^{p}. Our approach in the present paper covers both approximate so-
lutions inL^{p} and a bounded domain. Recall thatL^{p} Young measures associated
with hyperbolic conservation laws were first studied by Schonbek [14]. Recall
also that DiPerna’s strategy was applied to proving the convergence of numerical
schemes by Szepessy [17], Coquel and LeFloch [4], and Cockburn, Coquel, and
LeFloch [3].

Section 4 contains the application of the above compactness framework to the vanishing diffusion problem with boundary condition.

2 – Measure-valued solutions

We aim at developing a general framework to study multi-dimensional con-
servation laws in a bounded domain, encompassing all of the fundamental issues
of existence, uniqueness, regularity and compactness of entropy solutions. First
we observe that, for the problem (1.1)–(1.3), it is easy to construct sequences
of approximate solutions u^{h} : Ω×R+ → R satisfying certain natural uniform

bounds; see (2.3)–(2.5) below. In particular,u^{h} satisfies “approximate” entropy
inequalities of the form

∂_{t}U(u^{h}) + divF(u^{h}) ≤ R^{h}_{U} −→ 0

in the sense of distributions. Such approximate solutions indeed will be con- structed explicitly in Section 4.

The main difficulty is proving that these approximations converge in a strong
topology and that the limit is an entropy solution of (1.1)–(1.2), satisfying a
relaxed version of (1.3). To this end, following DiPerna [5], we introduce the
notion ofentropy measure-valued solution, designed to handle weak limits of the
sequenceu^{h}. The key result to be proven concerns theregularity anduniqueness
of the measure-valued solution which, in fact, will coincide with a weak solution in
the standard sense. This approach relies heavily on the entropy inequalities and
on theL^{1} contraction property of the solution-operator. The classical approach
uses a compactness embedding (Helly’s theorem) instead. Another characteristic
of the present strategy is that it provides at once the strong convergence of the
sequenceu^{h} and a characterization of its limit.

Consider the problem (1.1)–(1.3) where the initial data u_{0} belongs to L^{p}(Ω)
with 1< p≤ ∞ and the boundary datum uB: ∂Ω×R^{+} 7→ R is a smooth and
bounded function. We make the following assumptions on the flux-functionf:

(1) f iscontinuous onR.

(2) Whenp <∞,f satisfies thegrowth condition at infinity

(2.1) f(u) =O^{³}1 +|u|^{r}^{´}

for somer ∈[1, p).

We also define q: = p/r. The following terminology will be used.

Definition 2.1. A continuous function satisfying (2.1) (when p < ∞) will be calledadmissible. Whenp=∞, no growth condition is imposed.

More generally, a function g =g(u, x, t) is called admissible if it continuous inu∈Rand Lebesgue measurable inx∈Ω andt∈R+ with

|g(u, x, t)| ≤ g_{1}(u) +g_{2}(x, t) ,

where the functiong_{1} ≥0 satisfies (2.1) andg_{2} ≥0 belongs to L^{∞}(R^{+}, L^{q}(Ω)).

A pair of smooth functions (U, F) : R→R×R^{d}is called atame entropy pair
if∇F =U^{0}∇f and the function U is affine outside a compact set. It is said to
be convex ifU is convex.

Note that, whenp=∞, the behavior at infinity is irrelevant. A truly remark- able property of the scalar conservation laws is the existence of a special family of one-parameter, symmetric and convex entropies, the so-called Kruzkov entropies [9],

(2.2) U˜(u, v) : =|u−v|, F˜(u, v) : = sgn(u−v)^{³}f(u)−f(v)^{´},
which play an important role in the theory of conservation laws.

For h∈(0,1), let u^{h}: Ω×R^{+}→R be a sequence of piecewise smooth func-
tions with the following properties:

(i) The uniform bound

(2.3) ku^{h}(t)k_{L}^{p}_{(Ω)}≤C(T), t∈(0, T) ,

holds for a constantC(T)>0 independent ofhand for each timeT >0.

(ii) The entropy inequalities (H_{d−1} being the (d−1)-dimensional Haussdorf
measure)

(2.4a) Z Z

Ω×R+

³U(u^{h})∂_{t}θ+F(u^{h})·gradθ^{´}dx dt +
Z

Ω

U(u^{h}(0))θ(x,0)dx

− Z Z

∂Ω×R+

B_{U}^{h}(x, t)θ(x, t) dHd−1(x)dt ≥
Z Z

Ω×R+

R^{h}_{U}(θ)dx dt
hold for every convex and tame entropy-pair (U, F) and every test-
functionθ =θ(x, t) ≥0 inC_{c}^{1}(Ω×(0,∞)). We are assuming here that
there exists a (smooth) approximate boundary flux b^{h} and an element
b∈W^{−1,∞}([0, T), W^{−1/q,q}(∂Ω)) (for allT >0 with 1/q+ 1/q^{0}= 1) such
that

(2.4b) b^{h}→b in the sense of distributions.
We have set also

B_{U}^{h} = F(u_{B})·N + U^{0}(u_{B})^{³}b^{h}−f(u_{B})·N^{´},

where N is the outside unit normal along ∂Ω. Moreover, in (2.4a),
R^{h}_{U} : Ω×R+→R are piecewise smooth functions, possibly depending
onU and converging to zero

(2.4c)

Z Z

Ω×R+

R^{h}_{U}(θ) dx dt −→ 0, h→0.

(iii) The initial traces u^{h}(0) approach the initial datum u_{0} when h → 0, in
the following weak sense

(2.5) lim sup

h→0

Z

ΩU(u^{h}(0))θ dx ≤
Z

ΩU(u0)θ dx

for all arbitraryθ=θ(x)≥0 inC(Ω) and for all convex tame entropyU. In particular (choosingU(u) =±u), (2.5) implies the weak convergence property

h→0lim Z

Ωu^{h}(0)θ dx =
Z

Ωu0θ dx .

For instance, (2.5) holds wheneveru^{h}(0) tends tou_{0} inL^{1} strongly.

It is well-known that, from a sequence u^{h} satisfying the uniform bound (2.3),
one can extract a subsequence converging in the weak topology, but not necessar-
ily converging in the strong topology. More generally, for anyadmissiblefunction
g=g(u, x, t) we have

kg(u^{h},·,·)k_{L}^{∞}_{(}_{R}_{+}_{,L}^{q}_{(Ω))} ≤ C(T)

for some uniform constant C(T) > 0 depending on g. By a weak compactness
theorem, there exists a limit ¯g ∈ L^{∞}((0, T), L^{q}) (for each T > 0) and a subse-
quence still labelledu^{h} and possibly depending ongsuch thatg(u^{h})*g¯weakly,
i.e. for everyθ=θ(x, t) in Cc(Ω×[0,∞))

(2.6) Z Z

Ω×R+

g^{³}u^{h}(x, t), x, t^{´}θ(x, t) dx dt −→

Z Z

Ω×R+

¯

g(x, t)θ(x, t) dx dt .
A fundamental difficulty must be overcome. Given a nonlinear function g=
g(u), the weak limit of the composite function g(u^{h}) of u^{h} need not coincide
with the composite function of the weak limit (say ¯u) of u^{h}. In other words

¯

g6=g(¯u). In this juncture, the Young measure provides us with a powerful tool
to express the weak limit ¯gin (2.6) from the functiong. To each (x, t)∈Ω×R^{+},
it associates a probability measureν_{x,t} on R, i.e., an element of the space of all
positive measures with unit mass, such that ¯g(x, t) be the expected value of g
with respect to the measureν_{x,t}. This means that

¯

g(x, t) =hνx,t, gi: = Z

Rg(¯u)dν_{x,t}(¯u) for a.e. (x, t) .

Recall that a measure is a linear mapping µ from the linear space C(R) of all continuous functionsg (or a subset of them) such that for someC >0

|hµ, gi|: =

¯

¯

¯

¯ Z

Rg(¯u)dµ(¯u)

¯

¯

¯

¯ ≤ Ckgk_{C(R)} ,

the mass ofµ being then

|µ|: = sup

g∈C(R) kgkC(R)=1

hµ, gi .

Recall also thatµis said to be positive iff hµ, gi ≥0 for all functions g≥0.

Theorem 2.2 (Tartar [18], Schonbek [14]). Given a sequence u^{h} satisfying
(2.3) for some p∈(1,∞], there exists a subsequence of u^{h} and a family of
probability measures {ν_{x,t}}_{(x,t)∈Ω×}_{R}_{+} with the following property. For all ad-
missible functions g=g(u, x, t), the function (x, t)7→ hνx,t, g(·, x, t)i belongs to
L^{∞}(R+, L^{q}(Ω)) and we have

(2.7) g(u^{h}, x, t)*hνx,t, g(·, x, t)i in the weak sense .
Let u¯ be the weak limit ofu^{h}. Then

u^{h} →u¯ strongly
iff

ν_{x,t} =δ_{¯}_{u(x,t)} for a.e. (x, t)∈Ω×R^{+} ,
whereδ_{u(x,t)}_{¯} denotes the Dirac measure at the pointu(x, t).¯

The mappingν constructed in Theorem 2.2 is called a Young measure associ-
ated with the sequence{u^{h}}_{h>0}. The concept of a Young measure is now applied
to the conservation law (1.1).

Definition 2.3. A Young measureν_{x,t} is anentropy measure-valued solution
of the problem (1.1)–(1.3) iff there exists an elementb∈W^{−1,∞}([0,T),W^{−1/q,q}(∂Ω))
(for allT >0 with 1/q+ 1/q^{0}= 1) such that for every convex and tame entropy
pair (U, F), every smooth functionθ=θ(x, t)≥0 and every T >0 we have
(2.8)

Z _{T}

0

Z

Ω

³hν, Ui∂_{t}θ+hν, Fi ·gradθ^{´}dx dt +
Z

Ω

U(u_{0})θ(x,0)dx

− Z T

0

Z

∂Ω

µ

F(u_{B})·N+U^{0}(u_{B})^{³}b−f(u_{B})·N^{´}

¶

θ dH_{d−1}dt ≥ 0 .
A function u ∈ L^{∞}((0, T), L^{p}(Ω)) for all T > 0 is called an entropy weak
solution of the problem (1.1)–(1.3) iff the Young measure (x, t) 7→ δ_{u(x,t)} is an
entropy measure-valued solution.

In particular, (2.8) implies that the inequality
(2.9) ∂_{t}hν, Ui+ divhν, Fi ≤ 0
holds in the sense of distributions.

Definition 2.3 is directly motivated by the following observation, which is easily deduced from the property (2.7) of the Young measure and the assumptions (2.4) and (2.5): A Young measure associated with a sequence satisfying (2.3)–

(2.5) is an entropy measure-valued solution of (1.1)–(1.2).

We discuss now the regularity of the measure-valued solutions. We show that
the initial datau_{0} is assumed in a strong sense, and we investigate in what sense
ν satisfies the boundary condition (1.3) along∂Ω.

Theorem 2.4. Letν =ν_{x,t} be an entropy measure-valued solution of(1.1)–

(1.3).

(a) For every convex and tame entropy U = U(u) and for every smooth functionθ=θ(x) with compact support inΩ, the function

(2.10) t 7−→

Z

Ωhνx,t, Uiθ dx

has locally bounded total variation and admits a trace as t→0+.

(b) For every functionU =U(u, x), that is convex inu and measurable inx
and such that|U(u, x)| ≤c|u|+|U˜(x)|where U˜ ∈L^{1}(Ω)and c≥0, we
have

(2.11) lim sup

t→0+

Z

Ωhν_{x,t}, U(·, x)idx ≤
Z

ΩU(u_{0}(x), x)dx .

(c) In particular, the Young measure assumes its initial datum u_{0} in the
following strong sense:

(2.12) lim sup

t→0+

Z

Ω

Dν_{x,t},|id−u_{0}(x)|^{E}dx = 0 .

Proof: Using in the weak formulation (2.8) a functionθ(x, t) = θ_{1}(x)θ_{2}(t),
compactly supported in Ω×[0,∞) and havingθ1, θ2 ≥0, we obtain

Z ∞ 0

dθ2

dt Z

Ωhν, Uiθ1dx dt + θ2(0) Z

ΩU(u0)θ1dx ≥ − Z ∞

0 θ2

Z

Ωgradθ1·hν, Fi dx dt

≥ −C1

Z ∞ 0

θ_{2}dt ,
for some constantC_{1} >0 depending on θ_{1}. Thus the function

V1(t) := −C1t + Z

Ωhνx,t, Uiθ1dx

satisfies the inequality

(2.13) −

Z ∞ 0

V_{1}(t)dθ_{2}

dt dt ≤ θ_{2}(0)
Z

Ω

U(u_{0})θ_{1}dx .

Using in (2.13) a test-functionθ2 ≥0 compactly supported in (0,∞), we find

−
Z _{∞}

0

V_{1}(t)dθ_{2}

dt dt ≤ 0

that is (in the sense of distributions) the functionV_{1}(t) is decreasing and, there-
fore, has locally bounded total variation. Since it is uniformly bounded, V_{1}(t)
has a limit ast→0+. This proves (a).

To establish the item (b), we fix a time t_{0} >0 and consider the sequence of
continuous functions

θ_{2}^{ε}(t) =

1 for t∈[0, t_{0}],
(t0+ε−t)/ε for t∈[t0, t0+ε],

0 for t≥t_{0}+ε .

Relying on the regularity property (a) above, we see easily that

− Z ∞

0 V_{1}(t)dθ_{2}^{ε}

dt dt −→ V_{1}(t_{0}+) .
Sinceθ^{ε}_{2}(0) = 1 andt_{0} is arbitrary, (2.13) yields

V_{1}(t_{0}) = −C1t_{0} +
Z

Ωhνx,t0, Uiθ_{1}dx ≤
Z

Ω

U(u_{0})θ_{1}dx
for allt_{0} >0 and, in particular,

(2.14) lim

t→0+

Z

Ωhνx,t, Uiθ_{1}dx ≤
Z

Ω

U(u_{0})θ_{1}dx for all θ_{1} =θ_{1}(x)≥0.
Note that the left-hand limit exists, in view of (a).

Consider the set of all linear, convex and finite combinations of the form P

jαjθ1,j(x)Uj(u), where αj ≥0, ^{P}_{j}αj = 1, the functions Uj are smooth and
convex in u and the functions θ_{1,j}(x) ≥ 0 are smooth and compactly support,
with moreover

|Uj(u)θ_{1,j}(x)| ≤ c|u|+|U˜_{j}(x)|

withc≥0 and ˜U_{j} ∈L^{1}(Ω). This set is dense (for the uniform topology inu and
theL^{1} topology in x) in the set of all functions U =U(u, x) that are convex in
uand measurable in x and satisfy

|U(u, x)| ≤ c|u|+|U˜(x)|

for somec >0 and ˜U ∈L^{1}(Ω). Therefore by density we can deduce the statement
(b) from (2.14).

The statement (c) follows from (b) by choosing U(u, x) =|u−u_{0}(x)|.

To identify some properties of the Young measure along the boundary, we will need the following:

Lemma 2.5. LetV: Ω→R^{d} be a function inL^{q}(Ω)(q >1), satisfying
divV ≤0 in the sense of distributions.

Then the function V admits a normal trace along the boundary of Ω, in the following sense. Consider the change of coordinate x = χ(¯x, y) = ¯x+y N(¯x), where (¯x, y) ∈ ∂Ω×(0, ε) for some ε > 0 sufficiently small. Call J = J(¯x, y) =

|_{∂(¯}^{∂χ}_{x,y)}| is the Jacobian of this transformation. Then for each test-function θ of
the single variabley∈(0, ε)given by

A(y) = Z

∂Ω

V(¯x, y)·N(¯x)θ(¯x)J(¯x, y) dHd−1(¯x)

is a monotone increasing function and so admits a limit, sayA_{0}, in[−∞,∞) as
y→0^{+} with

(2.15) A_{0} ≤ CkVk_{L}^{q}_{(Ω)}kθk_{W}1,q0

(∂Ω) ,

Proof: The change of coordinates x 7→ (¯x, y) is well defined in a neighbor-
hood of ∂Ω, for (¯x, y) ∈ ∂Ω×(0, ε). Given θ ∈ C(∂Ω), consider the associated
function A. For any test function ψ(x) = θ(¯x)ϕ(y) with θ ∈ C^{∞}(∂Ω), and
ϕ∈ C_{c}^{∞}((0, ε)), we have

∇x(θ ϕ) = ϕ(y)∇xθ+θ ϕ^{0}(y)∂_{x}y ,

with ∂_{x}y=−N(¯x). On the other hand from the inequality satisfied by V we

deduce that _{Z}

Ω∇x(θ ϕ)·V dx ≥ 0, i.e. using the change of variables x7→(¯x, y)

Z _{ε}

0

ϕ(y) Z

∂Ω

V(¯x, y)· ∇xθ(¯x)J(¯x, y) dHd−1(¯x)dy

− Z ε

0ϕ^{0}(y)
Z

∂ΩV(¯x, y)·N(¯x)θ(¯x)J(¯x, y)dH_{d−1}(¯x)dy ≥ 0 .

Setting

B(y) :=

Z

∂ΩV(¯x, y)· ∇xθ(¯x)J(¯x, y) dH_{d−1}(¯x) ,
we obtain

(2.16) A^{0}(y) +B(y) ≥ 0

in the sense of distributions on (0, ε). Observe that both A and B are Lebesgue
measurable functions. Now (2.16) implies that A^{0}+B is a non-negative locally
bounded Borel measure on (0, ε). Hence A(y) + ^{R}_{0}^{y}B(y^{0})dy^{0} has a pointwise
limit as y → 0+, which belongs to [−∞,∞). By assumption b ∈ L^{q}(∂Ω) so
R_{y}

0 B(y^{0})dy^{0} →0 as y→0+. This establishes that A(y) admits a limit A_{0} when
y→0+.

On the other hand, from (2.16) we deduce that for
A(y)−A_{0}+

Z y 0

B(y^{0})dy^{0} ≥ 0,
so

A_{0} ≤ 1
ε

Z ε

0|A(y)|dy + 1 ε

Z ε 0

Z y

0 |B(y^{0})|dy^{0}dy

≤ C

ε kVk_{L}^{q}_{(Ω)}kθk_{L}q0

(∂Ω) + C

ε kVk_{L}^{q}_{(Ω)}k∇θk_{L}q0

(∂Ω) , which gives (2.15). This completes the proof of Lemma 2.5.

Theorem 2.6. Letν =ν_{x,t}an entropy measure-valued solution. There exists
a Young measureν_{x,t}^{B} defined along the boundary, for(x, t)∈∂Ω×R^{+} such that
for each continuous function F: R → R^{d} satisfying the growth condition (2.1),
hν_{x,t}^{B}, F·N(x)i belongs to the distribution space W^{−1,∞}([0, T), W^{−1/q,q}(∂Ω))for
all T >0 (with 1/q + 1/q^{0} = 1). This Young measure represents the trace of
hνx,t, F·N(x)i along the boundary, in the following sense (using the notation of
Lemma 2.5), asy→0,

A(y) :=

Z

R+

Z

∂Ωhν¯x,y,t, F·N(x)iθ_{1}(¯x)θ_{2}(t)J(¯x, y) dHd−1(¯x)dt

−→

Z

R+

Z

∂Ωhν_{x,t}^{B}_{¯} , F·N(¯x)iθ_{1}(¯x)θ_{2}(t) dHd−1(¯x)dt
for every test-functionsθ_{1} and θ_{2}.

Moreover, the following boundary entropy inequality (2.17)

¿

ν^{B}, N·^{³}F(·)−F(u_{B})−U^{0}(u_{B})^{³}f(·)−f(u_{B})^{´´}

À

≥ 0

on the boundary ∂Ω×R+

in the sense of distributions, and

(2.18) b=hν^{B}, f·Ni on the boundary ∂Ω×R^{+} .

Proof: Use the weak formulation (2.8) with a function θ(x, t) =θ1(x)θ2(t),
compactly supported in Ω×[0,∞) and havingθ_{1}, θ_{2} ≥0:

Z

Ω

gradθ_{1} ·
Z _{∞}

0 hν, Fiθ_{2} dx dt + θ_{2}(0)
Z

Ω

U(u_{0})θ_{1}dx

− Z

∂Ω

Z ∞ 0

³F(u_{B})·N +U^{0}(u_{B})^{³}b−f(u_{B})·N^{´´}θ_{1}θ_{2} dt dH_{d−1}

≥ − Z

Ωθ_{1}
Z ∞

0 hν, Uidθ_{2}
dt dt dx

≥ −C_{2}(θ_{2})
Z

Ω

θ_{1}
Z _{T}

0 |hν, Ui|dt dx

= C_{2}(θ_{2})
Z

Ω∇θ1·X dx , where

C_{2}(θ_{2}) =

°

°

°

°
dθ_{2}

dt

°

°

°

°L^{∞}(R+)

andX is a solution of (see [7]) divX =

Z T

0 |hν, Ui|dt , X∈W_{0}^{1,p}(Ω).
Thus the vector-valued function

V2(x) = Z ∞

0 hνx,t, Fiθ2(t)dt − C2(θ2)X satisfies the inequality

(2.19)

− Z

Ω

V_{2}·gradθ_{1} dx

≤ − Z

∂Ω

θ_{1}
Z _{∞}

0

³F(u_{B})·N+U^{0}(u_{B})^{³}b−f(u_{B})·N^{´´}θ_{2} dt dHd−1

+ θ_{2}(0)
Z

Ω

U(u_{0})θ_{1}dx .

Using in (2.19) a test-function θ_{1} ≥ 0 compactly supported in Ω, and θ_{2} such
thatθ_{2}(0) = 0 we find

− Z

ΩV2·gradθ1dx ≤ 0 ,

that is divV_{2} ≤ 0 in the sense of distributions, with V_{2} ∈ L^{q}(Ω). Applying
Lemma 2.5 we see that the normal traceV2 exists along the boundary. We thus
define

G(F·N;θ1, θ2) := lim

y→0+

Z

∂ΩV2(¯x, y)·N(¯x)θ1(¯x)J(¯x, y) dH_{d−1}(¯x) .
We now return to the general inequality (2.19), and we let the test-function
θ_{1} tend to a zero in the interior of Ω. We find for allθ_{2} ≥0 andθ_{1}: ∂Ω→R+

−G(F·N;θ_{1}, θ_{2})
(2.19)

≤ − Z

R+

Z

∂Ω

³F(u_{B})·N +U^{0}(u_{B})^{³}b−f(u_{B})·N^{´´}θ_{1}(¯x)θ_{2}(t) dt dHd−1(¯x) .
In view of (2.15) and with the regularity available onb and u_{B}, we see that the
mappingGsatisfies the following estimate:

|G(F·N;θ1, θ2)| ≤ Ckθ1θ2k_{W}_{1,1}([0,T),W^{1,q}^{0}(∂Ω)) .
This gives us the desired regularity ofG.

Finally, rewritting Gvia a Young measureν^{B} we obtain
G(F·N;θ1, θ2) =

Z

R+

Z

∂Ωhν_{x,t}^{B}_{¯} , F·N(¯x)iθ1(¯x)θ2(t) dH_{d−1}(¯x)dt

Using similar estimates as above but with U(u) =±u we actually have the equality

Z

R+

Z

∂Ωhν_{x,t}^{B}_{¯} , fiθ_{1}(¯x)θ_{2}(t) dHd−1(¯x)dt

= Z

R+

Z

∂Ω

³f(uB)·N+b−f(uB)·N^{´}θ1(¯x)θ2(t) dH_{d−1}(¯x)dt ,
so that (2.18) holds. This completes the proof of Theorem 2.6.

3 – Existence, uniqueness, and compactness

Continuing the investigation of the properties of the measure-valued solu- tions, we now arrive at a general theory of existence and uniqueness for the problem (1.1)–(1.3), based on extending to measure-valued solutions the stan- dard Kruzkov’s contraction property [9].

Theorem 3.1. Let ν_{1} and ν_{2} be two entropy measure-valued solutions.

Then, in the sense of distributions we have

(3.1) ∂_{t}hν1⊗ν_{2},U˜i+ divhν1⊗ν_{2},F˜i ≤ 0 ,

where( ˜U ,F˜)is the Kruzkov parametrized entropy pair (see(2.2)), and the tensor product of measures is defined by

hν1⊗ν_{2},U˜i :=

Z Z U˜(u_{1}, u_{2}) dν_{1}(u_{1})dν_{2}(u_{2}) .

Proof: Formally we have

∂_{t}hν_{1}⊗ν_{2},Ui˜ + divhν_{1}⊗ν_{2},F˜i

=^{D}ν_{1}^{³}∂_{t}hν_{2},U˜i+ divhν_{2},F˜i^{´E} + ^{D}ν_{2}^{³}∂_{t}hν_{1},U˜i+ divhν_{1},F˜i^{´E}

≤0 ,

where we used (2.9) and the positivity of the measuresν1 andν2. The proof can be made rigorous by regularization in the (x, t)-variable.

Theorem 3.2. Let ν_{1} and ν_{2} be two entropy measure-valued solutions
satisfying the same initial datum u_{0} ∈L^{p}(Ω). Then there exists a function u ∈
L^{∞}(R+, L^{p}(Ω))such that

(3.4) ν_{1,(x,t)}=ν_{2,(x,t)}=δ_{u(x,t)} for almost every (x, t) .

In particular, the problem (1.1)–(1.3) has exactly one entropy solution in
L^{∞}(R+, L^{p}(Ω)), which moreover satisfies its initial data in the sense

(3.5) lim sup

t→0+

Z

Ω|u(x, t)−u_{0}(x)|dx = 0 .

Furthermore, given two such solutionsu_{1}andu_{2}associated with the boundary
datauB, we have for all t≥s≥0

(3.6)

Z

Ω|u1(x, t)−u_{2}(x, t)|dx ≤
Z

Ω|u1(x, s)−u_{2}(x, s)|dx .

The existence part in Theorem 3.2 is based on the assumption that a family of approximate solutions satisfying (2.3)–(2.5) does exist, in order to generate at least one entropy measure-valued solution. Recall that Section 4 below will indeed provide such approximate solutions. In the applications, in order to establish the

strong convergence of a sequence of approximate solutions, we will appeal to the following immediate consequence of Theorem 3.2 and Theorem 2.2.

Corollary 3.3. Let u^{h} be a sequence of approximate solutions satisfying
the conditions(2.3)–(2.5). Then there exists a functionu∈L^{∞}(R^{+}, L^{p}(Ω))such
that

u^{h} →u strongly,

and u is the unique entropy solution to the problem (1.1)–(1.3). The entropy
solutionu∈L^{∞}(R+, L^{p}(Ω))of (1.1)–(1.3)satisfies the additional regularity:

(3.7) t 7→

Z

ΩU(u)θ dx has locally bounded variation

for all tame entropyU and all smooth θ≥0having compact support inΩ×R^{+}.
To prove Theorem 3.2, we shall need:

Lemma 3.4. LetuB be given inR, together with some unit vector N ∈R^{d}.
Let ν_{1} and ν_{2} be probability measures on R (acting on all admissible functions
and) satisfying the boundary entropy inequalities

(3.8a)

¿

ν_{1}, N·
µ

F(·)−F(u_{B})−U^{0}(u_{B})^{³}f(·)−f(u_{B})^{´}

¶À

≥ 0 and

(3.8b)

¿

ν2, N· µ

F(·)−F(uB)−U^{0}(uB)^{³}f(·)−f(uB)^{´}

¶À

≥ 0 for all convex and tame entropy pairs(U, F). Then we have

(3.9) hν1⊗ν_{2}, N·F˜i ≤ 0.

Proof: Using Kruzkov’s entropies, the conditions (3.8) and (3.9) are found to be equivalent to: for eachv2, v1 ∈R

(3.10a)

¿

ν_{1}, ^{³}sgn(u_{1}−v_{2})−sgn(u_{B}−v_{2})^{´ ³}f(u_{1})−f(v_{2})^{´}·N
À

≥ 0, (3.10b)

¿

ν_{2}, ^{³}sgn(u_{2}−v_{1})−sgn(u_{B}−v_{1})^{´ ³}f(u_{2})−f(v_{1})^{´}·N
À

≥ 0 . Taking succesively v2 < uB, thenv2=uB, and finallyv2 > uB, we obtain (3.11i)

Z

u1<uB

N·F(u˜ 1, u2) dν1(u1) ≥ 0, v2 < uB ,

(3.11ii)

Z

u1∈RN·F˜(u_{1}, u_{B})dν_{1}(u_{1}) ≥ 0 ,
and

(3.11iii)

Z

u1>uB

N·F(u˜ _{1}, u_{2}) dν_{1}(u_{1}) ≥ 0, v_{2} > u_{B} .
Similarly we get

(3.12i)

Z

u2<uB

N ·F˜(u_{1}, u_{2})dν_{2}(u_{2}) ≥ 0, v_{1} < u_{B} ,

(3.12ii)

Z

u2∈RN·F˜(u_{2}, u_{B})dν_{2}(u_{2}) ≥ 0 ,
and

(3.12iii)

Z

u2>u_{B}N ·F˜(u_{1}, u_{2})dν_{2}(u_{2}) ≥ 0, v_{1} > u_{B} .
These conditions (3.11)–(3.12) imply immediately that
(3.13)

Z Z

Q1∪Q3

N ·F˜(u2, u_{B}) dν1(u1)dν2(u2) ≥ 0 ,
where Q_{1} :={u1 ≥u_{B}, u_{2}≥u_{B}} and Q_{3}:={u1≤u_{B}, u_{2} ≤u_{B}}.

To estimate the sign in the regionQ4 :={u1 > u_{B}, u2 < u_{B}}, we use (3.11iii)
which gives us

Z

u1>v2

N·f(u_{1}) dν_{1}(u_{1}) ≥
Z

u1>v2

N ·f(v_{2}) dν_{1}(u_{1}), v_{2} > u_{B} .
We use also (3.12i) which gives

Z

u2<v1

N·f(u_{2}) dν_{2}(u_{2}) ≤
Z

u2<v1

N ·f(v_{1}) dν_{2}(u_{2}), v_{1} < u_{B} .
Combining these two inequalities we arrive at

Z

u2<uB

Z

u1>v2

N·f(u_{1}) dν_{1}(u_{1})dν_{2}(u_{2}) ≥
Z

u2<uB

Z

u1>v2

N·f(v2) dν_{1}(u_{1})dν_{2}(u_{2})

−→

Z

u2<uB

Z

u1>uB

N·f(u_{B}) dν_{1}(u_{1})dν_{2}(u_{2}),
asv_{2}→u_{B}. We also have

Z

u1>u_{B}

Z

u2<v1

N·f(u2) dν1(u1)dν2(u2) ≥ Z

u1>u_{B}

Z

u2<v1

N·f(v1) dν1(u1)dν2(u2)

−→

Z

u1>uB

Z

u2<uB

N·f(uB)dν1(u1)dν2(u2),

asv_{1}→u_{B}. This implies exactly
(3.14)

Z Z

Q4

N ·F˜(u2, uB) dν1(u1)dν2(u2) ≥ 0 .
A similar argument applies on Q_{2} :={u1 < u_{B}, u_{2}> u_{B}}
(3.15)

Z Z

Q2

N ·F˜(u2, u_{B}) dν1(u1)dν2(u2) ≥ 0 .
This completes the proof of Lemma 3.4.

Proof of Theorem 3.2: Consider two solutionsν1 and ν2 associated with
a pair of data u_{01}, u_{B} and u_{02}, u_{B}, respectively. With the Green formula and
Theorem 3.1, together with the existence of the normal trace (Theorem 2.6), we
obtain immediately for test-functionsθ_{1}, θ_{2}≥0

− Z

R+

Z

Ωhν1⊗ν_{2},U˜iθ_{1}(x)θ^{0}_{2}(t) dx dt

− Z

R+

Z

Ωhν1⊗ν_{2}, N ·F˜i · ∇θ1(x)θ^{0}_{2}(t) dx dt ≤ 0,
and so

(3.17)

− Z

R+

Z

Ωhν1⊗ν_{2},U˜iθ_{1}(x)θ_{2}^{0}(t) dx dt
+

Z

R+

Z

∂Ωhν_{1}^{B}⊗ν_{2}^{B}, N·Fi˜ θ_{1}(x)θ^{0}_{2}(t) dHd−1(x)dt ≤ 0.
In view of Lemma 3.4, we have

−B(t) ≤ 0 . therefore we arrive at

(3.18) dA

dt (t) +B(t) ≤ 0 , where

A(t) :=

Z

Ωhν1⊗ν2,U˜iθ1(x) dx .

We now turn to evaluate of A. Since ˜U(λ_{1}, λ_{2}) = |λ2−λ_{1}|, the term A(t) is
regarded as the L^{1} norm between the two solutions. On the other hand, from
(3.17)–(3.18) we deduce

(3.19) A(t)−A(s) ≤ 0, 0< s≤t .

First of all, suppose that ν_{1} and ν_{2} assume the same boundary and initial
data u_{B} and u0. Since the Young measures satisfy the initial condition in the
strong sense (2.12), we obtain for allt >0

A(t) ≤ Z

Ω

Dν_{1}⊗ν_{2}, |¯u_{1}−u_{0}|+|¯u_{2}−u_{0}|^{E} dx

≤ Z

Ωhν1,|¯u_{1}−u_{0}|idx +
Z

Ωhν2,|¯u_{2}−u_{0}|idx ,
thus

lim sup

t→0+

A(t) = 0 . Therefore lettings→0 in (3.19),

A(t)≡0, t≥0

and thus _{Z}

Ωhν1⊗ν_{2},U˜idx = 0.

Thus, for almost every (x, t), the measuresν_{1} =ν_{1,(x,t)}andν_{2}=ν_{2,(x,t)}satisfy
(3.20)

Z Z

|¯u_{2}−u¯_{1}|dν_{1}(¯u_{1})dν_{2}(¯u_{2}) = hν1⊗ν_{2},U˜i = 0.

Fix (x, t) such that (3.20) holds. We claim that there exists some w ∈ R such that

ν_{1} =ν_{2} =δ_{w} .

Otherwise there would exist w1 ∈ suppν1 and w2 ∈suppν2 with w1 6=w2. By
definition of the support of a measure, there exist continuous functions ϕ_{j} ≥0
such that suppϕj ⊂ B(wj, ε) ⊂ Ω (the ball with center wj and radius ε) and
hνj, ϕ_{j}i 6= 0. One can always assume thatεis so small thatB(w_{1}, ε)∩B(w2, ε) =∅.

To conclude, we observe that 0 <

Z Z

ϕ_{1}ϕ_{2} dν_{1}⊗dν_{2} ≤

°

°

°

°
ϕ_{1}ϕ_{2}

¯
u_{2}−u¯_{1}

°

°

°

°_{∞}
Z Z

|¯u_{2}−u¯_{1}|dν_{1}(¯u_{1})dν_{2}(¯u_{2}) = 0 ,
which is a contradiction. The proof of Theorem 3.2 is completed.

4 – Zero diffusion limit

The theory developed in Sections 2 and 3 is now applied to analyze a singular limit problem. We treat a class of multi-dimensional conservation laws containing

vanishing diffusion. Precisely, we consider the problem (1.1)–(1.3), where the
flux-function satisfies a growth condition like (2.1) withr= 1. Given a diffusion
parameter ε > 0 and a (uniformly in x) positive-definite matrix (b_{ij}(x))_{1≤i,j≤d}
depending smoothly onx, we study the equation

(4.1) u^{ε}_{t} + divf(u^{ε}) = ε

d

X

i,j=1

∂_{i}(b_{ij}∂_{j}u^{ε}),
together with

u(x,0) =u^{ε}_{0}(x), x∈Ω,
(4.2)

u(x, t) =u^{ε}_{B}(x, t), x∈∂Ω, t >0.
(4.3)

Here u^{ε}_{0} and u^{ε}_{B} are sufficiently smooth initial and boundary data. Standard
existence results show that for allε >0, the problem (4.1)–(4.3) admits a unique
smooth solutionu^{ε} defined globally in time. The aim of this section is to prove
the convergence ofu^{ε} toward the entropy solution of (1.1)–(1.3).

Theorem 4.1. Suppose that the fluxf satisfies the growth condition

(4.4) f^{0}(u) =O(1)

and consider an initial datum u_{0} in L^{2}(Ω) and a sequence of smooth data u^{ε}_{0}
satisfying the uniform bound

(4.5) ku^{ε}_{0}k_{L}^{2}_{(Ω)}≤C

and a boundary datau_{B} such that

(4.6) u_{B} ∈ L^{∞}_{loc}^{³}[0,∞), H^{1/2}(∂Ω)^{´} and ∂_{t}u_{B} ∈ L^{2}_{loc}^{³}[0,∞), L^{2}(∂Ω)^{´} .
Then for each T >0the solutions u^{ε} of (4.1)–(4.3)satisfy for

(4.7) ku^{ε}(t)k_{L}^{2}_{(Ω)}+ ε^{1/2}k∇u^{ε}k_{L}2((0,T),L^{2}(Ω)) ≤ C(T), t∈(0, T) ,
for some constantC(T)>0, and converge strongly to the unique entropy solution
u∈L^{∞}_{loc}([0,∞), L^{2}(Ω)) of the hyperbolic problem(1.1)–(1.3).

Proof: Let ˜u_{B}: Ω×[0,∞) 7→R be an extension of the function u_{B} to the
whole domain. In view of (4.6) we can assume that ˜u_{B} ∈L^{∞}_{loc}([0,∞), H^{1}(Ω)) and

∂_{t}u˜_{B}∈L^{2}_{loc}([0,∞), H^{1/2}(Ω)). Define
F^{³}u^{ε}(x, t),u˜B(x, t)^{´} =

Z u^{ε}(x,t)

˜ uB(x,t)

³v−u˜B(x, t)^{´}f^{0}(v)dv .

We have

div^{³}F^{³}u^{ε}(x, t),u˜_{B}(x, t)^{´´} = (u^{ε}−u˜_{B})f^{0}(u^{ε})∇u^{ε} −
Z u^{ε}

˜

u_{B}∇˜u_{B}f^{0}(v)dv

= (u^{ε}−u˜_{B})f^{0}(u^{ε})∇u^{ε} + ∇˜u_{B}^{³}f(˜u_{B})−f(u^{ε})^{´} .
Using this and (4.1) we obtain

d dt

Z

Ω

1

2(u^{ε}−u˜_{B})^{2}dx

= Z

Ω

(u^{ε}−u˜_{B})
µ

−divf(u^{ε}) + ε

d

X

i,j=1

∂_{i}(b_{ij}∂_{j}u^{ε})

¶ dx −

Z

Ω

(u^{ε}−u˜_{B})∂_{t}u˜_{B} dx

= Z

Ω

µ

∇˜u_{B}·^{³}f(˜u_{B})−f(u^{ε})^{´}− divF(u^{ε},u˜_{B})

¶ dx + ε

d

X

i,j=1

Z

Ω(u^{ε}−u˜B)∂i

³bij∂j(u^{ε}−u˜B)^{´}dx

+ ε

d

X

i,j=1

Z

Ω(u^{ε}−u˜B)∂i

³bij∂j(˜uB)^{´}dx −
Z

Ω(u^{ε}−u˜B)∂tu˜B dx .
By integration by parts using that F(˜uB,u˜B) = 0 we find

d dt

Z

Ω

1

2(u^{ε}−u˜_{B})^{2}dx

= Z

Ω∇˜u_{B}·^{³}f(˜u_{B})−f(u^{ε})^{´}dx − ε

d

X

i,j=1

Z

Ω∂_{i}(u^{ε}−u˜_{B})b_{ij}∂_{j}(u^{ε}−u˜_{B}) dx
+

Z

Ω

(u^{ε}−u˜_{B})
µ

−∂tu˜_{B} + ε

d

X

i,j=1

∂_{i}(b_{ij}∂_{j}u˜_{B})

¶ dx .

Since ^{P}^{d}_{i,j=1}α_{i}b_{ij}α_{j} ≥c ^{P}^{d}_{i,j=1}α_{i}^{2} for some c >0, and since f is Lipschitz
continuous by (4.4), we find with Cauchy–Schwarz inequality

d dt

1

2ku^{ε}(t)−u˜_{B}(t)k^{2}_{L}2(Ω)

≤ k∇u˜_{B}(t)k_{L}^{2}_{(Ω)} Lip(f)ku^{ε}(t)−u˜_{B}(t)k_{L}^{2}_{(Ω)}

− ε ck∇u^{ε}(t)−u˜_{B}(t)k^{2}_{L}2(Ω) + C(˜u_{B})ku^{ε}(t)−u˜_{B}(t)k_{L}^{2}_{(Ω)} ,
whereC(˜u_{B}) is bounded by the conditions (4.6).

We find that for some constant C >0 depending on u_{B} and f
d

dtku^{ε}(t)−u˜_{B}(t)k^{2}_{L}2(Ω)+ε ck∇u^{ε}(t)− ∇˜u_{B}(t)k^{2}_{L}2(Ω) ≤ C+ku^{ε}(t)−u˜_{B}(t)k^{2}_{L}2(Ω)

Then, by Gronwall inequality, for anyT >0, there existsC(T)>0 such that for allt∈[0, T],

ku^{ε}(t)−u˜_{B}(t)k^{2}_{L}2(Ω) + ε
Z _{T}

0 k∇u(t)− ∇˜u_{B}(t)k^{2}_{L}2(Ω) dt

≤ C(T)^{³}1 +ku0−u˜_{B}(0)k^{2}_{L}2(Ω)

´

≤ C^{0}(T) .
In view of (4.6) we thus have proved (4.7).

To apply the framework of Sections 3 and 4, we need to check several assump- tions. First of all (for a subsequence at least) we claim that

X

i,j=1,...,d

ε b_{ij}∂_{j}u^{ε}_{|∂Ω}N_{i} converges in the sense of distributions

to some q∈H^{−1}^{³}(0, T), H^{−1/2}(∂Ω)^{´}.
Namely, multiplying the equation (4.1) by a test function ϕ = ϕ(x, t) and
integrating on Ω×(0, T), we obtain

(4.8)

d

X

i,j=1

Z T 0

Z

∂Ω

ε b_{ij}ϕ ∂_{j}u^{ε}N_{i} dHd−1dt

= Z

Ωu^{ε}(x, T)ϕ(x, T)dx −
Z

Ωu^{ε}(x,0)ϕ(x,0)dx

−
Z _{T}

0

Z

Ω

³u^{ε}∂_{t}ϕ+f(u^{ε})·∇ϕ^{´}dx dt +
Z _{T}

0

Z

∂Ω

ϕ f(u_{B})·N dHd−1dt
+ ε

d

X

i,j=1

Z _{T}

0

Z

∂Ω

b_{ij}∂_{i}ϕ ∂_{j}u^{ε}dx dt .

Using the bounds (4.7) we estimate the boundary flux in the following way:

(4.9)

¯

¯

¯

¯

¯

d

X

i,j=1

Z _{T}

0

Z

∂Ω

ε b_{ij}ϕ ∂_{j}u^{ε}N_{i} dHd−1dt

¯

¯

¯

¯

¯

≤ Ckϕ(T)k_{L}^{2}_{(Ω)}+ Ckϕ(0)k_{L}^{2}_{(Ω)}

+ Ck∂tϕk_{L}1((0,T),L^{2}(Ω)) + Ck∇ϕk_{L}1((0,T),L^{2}(Ω))
+ Ckϕk_{L}1((0,T),L^{1}(∂Ω)) + ε^{1/2}Ck∇ϕk_{L}2((0,T),L^{2}(Ω)) .
It follows that ^{P}^{d}_{i,j=1}ε b_{ij}∂_{j}u^{ε}_{∂Ω}N_{j} is uniformly bounded in some distribution
space and, for a subsequence at least, admits a limit, say q, in the sense of

distributions. Furthermore this limit satisfies the inequality (4.10)

¯

¯

¯

¯

¯
Z _{T}

0

Z

∂Ωq ϕ dHd−1dt

¯

¯

¯

¯

¯

≤ Ckϕ(T)k_{L}^{2}_{(Ω)}+ Ckϕ(0)k_{L}^{2}_{(Ω)}+ Ck∂tϕk_{L}1((0,T),L^{2}(Ω))
+ Ck∇ϕk_{L}_{1}((0,T),L^{2}(Ω)) + Ckϕk_{L}_{1}((0,T),L^{1}(∂Ω)) .
Restricting attention to test-functions compactly supported in time in [0, T) we
haveϕ(T)≡0 and

kϕ(0)k_{L}^{2}_{(Ω)} ≤ k∂tϕk_{L}1((0,T),L^{2}(Ω)) .
Therefore we arrive at

(4.11)

¯

¯

¯

¯

¯
Z _{T}

0

Z

∂Ωq ϕ dHd−1dt

¯

¯

¯

¯

¯

≤ Ck∂tϕk_{L}1((0,T),L^{2}(Ω)) + Ck∇ϕk_{L}1((0,T),L^{2}(Ω)) + Ckϕk_{L}1((0,T),L^{1}(∂Ω)) .
On the other hand, as we are interested in the trace along the boundary only, we
can always pick up anyϕon ∂Ω and extend it to the whole of Ω so that

k∂tϕk_{L}1((0,T),L^{2}(Ω)) + k∇ϕk_{L}1((0,T),L^{2}(Ω))

≤ Ck∂tϕk_{L}_{1}((0,T),H^{−1/2}(∂Ω)) + Ckϕk_{L}_{1}((0,T),H^{1/2}(∂Ω)) .
Finally we obtain

(4.12)

¯

¯

¯

¯

¯ Z T

0

Z

∂Ωq ϕ dH_{d−1}dt

¯

¯

¯

¯

¯

≤ Ck∂tϕk_{L}_{1}((0,T),H^{−1/2}(∂Ω)) + Ckϕk_{L}_{1}((0,T),H^{1/2}(∂Ω)) + Ckϕk_{L}_{1}((0,T),L^{1}(∂Ω))

≤ Ckϕk_{W}1,1((0,T),H^{1/2}(∂Ω)) .

This proves that the limiting traceq satisfies at least
(4.13) q ∈ W^{−1,∞}^{³}(0, T), H^{−1/2}(∂Ω)^{´}.

It remains to check the conditions (2.4) of Section 2.

Multiplying the equation (4.1) by U^{0}(u^{ε})θ with θ = θ(x, t) ≥ 0 in C_{c}^{1}(Ω×
[0,∞)) and integrating over Ω×[0,∞), we obtain

(4.14) Z Z

Ω×R+

³U(u^{ε})∂_{t}θ+F(u^{ε})·gradθ^{´} dx dt +
Z

Ω

U(u^{ε}_{0}(0))θ(x,0)dx

− Z Z

∂Ω×R+

B^{ε}_{U}(x, t)θ(x, t) dH_{d−1}(x)dt =
Z Z

Ω×R+

R˜^{ε}_{U}dx dt ,