Emmrich E., Wittbold P. (editors) Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series

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Coupling of a scalar conservation law with a parabolic problem 125

Then, from (2.14), we deduce that

Ω

|∇(z − ˆz)|

(s)dx + αku − ˆuk

)

≤

Ω

b(x)(f(u) − f ( ˆu)) · ∇(z − ˆz)dxdt.

(2.15)

As b is bounded and f is Lipschitz continuous on [m, M], we see that



Ω

b(x)(f(u) − f( ˆu)) · ∇(z − ˆz)dx



≤ kbk

∞

(Ω)

ku − ˆuk

)

kz − ˆzk

(0,s;H

(Ω))

≤

ku − ˆuk

)

+ C

k∇(z − ˆz)k

)

, (2.16)

where C

> 0 depends on α, b and f .

Finally, from (2.15) and (2.16), we deduce

Ω

|∇(z − ˆz)|

(s)dx ≤ C

k∇(z − ˆz)k

)

The conclusion follows by using Gronwall’s lemma (see Lemma 1.2).

3 The hyperbolic problem

In this section, we consider the special case Ω

= ∅ so that Ω = Ω

. The problem

we aim to solve can be formally written as follows: Find a measurable and essentially

bounded function u such that in the sense of distributions











∂

u + ∇ · (b(x)f(u)) = 0 in Q,

u(0, ·) = u

on Ω,

u = 0 (on a part of) Γ.

(3.1)

To begin with, we introduce some notations and deﬁnitions: The outward unit

normal vector of Ω deﬁned a.e. on Γ is denoted by ν. An element of Σ is denoted

by σ, while σ is an element of Γ so that σ = (t, σ). For all a, b ∈ R, we set

F (a, b) = (f(a)− f(b))sgn(a− b). The function F is Lipschitz continuous on [m, M]

and usually called the “Kruzhkov ﬂux”.

For all τ, k ∈ R, we deﬁne

(τ, k) =

(F (τ, 0) − F (k, 0) + F (τ, k)).

Deﬁnition 3.1 (regular entropy pair). Let η ∈ C

(R) be a convex function and q ∈

(R). Then (η, q) is called a regular entropy pair to problem (3.1) if, for all r ∈ R,

(r) = f

(r)η

(r).

126 Julien Jimenez

Remark 3.2. In [15, Def. 3.22], (η, q) is called an entropy-entropy ﬂux pair.

Following [17], we also give

Deﬁnition 3.3 (boundary entropy-entropy ﬂux pair). The pair (H, J) with H ∈ C

J ∈ C

) is said to be a boundary entropy-entropy ﬂux pair if (H(·, w), J(·, w)) is a

regular entropy pair and

H(w, w) = 0, J(w, w) = 0, ∂

H(w, w) = 0

for all w ∈ R, where ∂

denotes the partial derivative with respect to the ﬁrst argument.

In the sequel, we will use the particular boundary entropy-entropy ﬂux pair below.

Example 3.4. Let δ > 0. We deﬁne on R

the functions H

and J

(τ, k) = ((dist(τ, I[0, k]))

+ δ

)

1/2

− δ

and

(τ, k) =

∂

(λ, k)f

(λ)dλ.

Here I[0, k] denotes the closed interval with the endpoints 0 and k. Then, for any δ,

, J

) is a boundary entropy-entropy ﬂux pair. Moreover, the pair converges uni-

formly towards (dist(τ, I[0, k]), F

) as δ → 0

. This example of boundary entropy-

entropy ﬂux pair will be used to obtain the boundary condition given for (3.1).

3.1 The notion of entropy solution

It is well known (see for example [10]) that a weak solution to (3.1) may not be unique.

So we need an additional condition to select one solution among all the weak solu-

tions, which has to be the physically relevant solution. To this end we introduce the

notion of entropy solution. Another difﬁculty is to formulate a boundary condition for

(3.1). Indeed, it is not possible to deﬁne a trace, in a strong sense, for a L

∞

-function.

This difﬁculty has been overcome by F. Otto (see [17]) using the so-called boundary

entropy-entropy ﬂux pair deﬁned above.

Deﬁnition 3.5. A function u ∈ L

∞

(Q) is said to be an entropy solution to (3.1) if

(i) for all nonnegative ϕ ∈ C

∞

(Q) and all k ∈ R

(|u − k|∂

ϕ + sgn(u − k)(f (u) − f(k))b(x) · ∇ϕ)dxdt ≥ 0, (3.2)

(ii)

esslim

t→0

Ω

|u(t, x) − u

(x)|dx = 0, (3.3)

Coupling of a scalar conservation law with a parabolic problem 127

(iii) for all nonnegative ϕ ∈ L

(Σ) and all k ∈ R

ess lim

s→0

−

(u(σ + sν), k)b(σ) · ν ϕdσ ≥ 0. (3.4)

Remark 3.6. The entropy inequality (3.2) implies that an entropy solution u is a solu-

tion in the sense of distributions.

3.2 Uniqueness: the method of doubling variables

This part is inspired from the book of J. Málek et al. [15, Chap. 2] and from the article

of F. Otto [17]. We present the method of doubling variables, due to S. N. Kruzhkov

[13]. More precisely, we prove the following theorem.

Theorem 3.7. Let u

and u

be two entropy solutions to (3.1) for initial data u

0,1

and

0,2

, respectively. Then, for almost all t ∈ (0, T ),

Ω

(t, x) − u

(t, x)|dx ≤

Ω

0,1

(x) − u

0,2

(x)|dx.

We need ﬁrst some preliminary results. We refer to [15, Chap. 2, Lemma 7.12

and 7.34] and [17] for the quite technical proofs of these results.

Lemma 3.8. Let u ∈ L

∞

(Q) satisfying (3.2) and (3.3). Then, for all nonnegative

ϕ ∈ C

∞

(R × R

) and all k ∈ R,

ess lim

s→0

−

F (u(σ + sν), k)b(σ) · ν ϕdσ exists. (3.5)

Moreover,

−

{|u − k|∂

ϕ + b(x)F (u, k) · ∇ϕ}dxdt

≤

Ω

− k|ϕ(0, x)dx − ess lim

s→0

−

F (u(σ + sν), k)b(σ) · ν ϕdσ. (3.6)

For j ∈ N, (t, x,

t, ˜x) ∈ Q×Q, we set

(t, x,

t, ˜x) = ϕ



t +

x + ˜x



(t −

i=1

− ˜x

where (ρ

)

j∈N

is a standard sequence of molliﬁers on R and ϕ ∈ C

∞

(Q), ϕ ≥ 0.

To simplify the notation, we set p = (t, x), ˜p = (

t, ˜x),

(x − ˜x) =

i=1

− ˜x

) and ρ

(p − ˜p) = ρ

(t −

t)ρ

(x − ˜x),

128 Julien Jimenez

so that

(p, ˜p) = ϕ



p + ˜p



(p − ˜p).

Let us remark that, for almost all σ ∈ Σ,

0 ≤

(σ − ˜p)d ˜p ≤ 1 and lim

j→∞

(σ − ˜p)d ˜p =

In this framework, the next statement holds.

Lemma 3.9. Suppose u ∈ L

∞

(Q) fulﬁlls (3.6). Then there holds for all ϕ ∈ C(Q)

lim

j→∞

F (u(p), 0)b( ˜σ)ψ

(p, ˜σ)d ˜σdp

ess lim

s→0

−

F (u(σ + sν), 0)b(σ) · νϕ(σ)dσ,

as well as

lim

j→∞

ess lim

s→0

−

F (u(σ + sν), 0)b(σ) · νψ

(σ, ˜p)dσd ˜p

ess lim

s→0

−

F (u(σ + sν), 0)b(σ) · νϕ(σ)dσ.

We turn now to the proof of Theorem 3.7. First, we prove the following lemma

that gives a so-called Kruzhkov inequality for the difference of two entropy solutions

to (3.1).

Lemma 3.10. Let u

and u

be two entropy solutions to (3.1). Then there holds

(|u

− u

|∂

ϕ + F (u

, u

)b(x) · ∇ϕ)dxdt ≥ 0 (3.7)

for all nonnegative ϕ ∈ C

∞

(R × R

) with ϕ(0, ·) = 0.

Proof. First, we deduce from (3.5) that there exists θ

∈ L

∞

(Σ) (i = 1, 2) such that for

any ϕ ∈ L

(Σ)

ess lim

s→0

−

F (u

(σ + sν), 0)b(σ) · ν ϕ(σ)dσ =

(σ)ϕ(σ)dσ.

From the deﬁnition of F

, the boundary condition (3.4) and inequality (3.6), we ﬁnd

−

(|u

− k|∂

ϕ + b(x)F (u

, k) · ∇ϕ)dxdt

≤ −

F (k, 0)b(σ) · νϕdσ + ess lim

s→0

−

F (u

(σ + sν), 0)b(σ) · ν ϕdσ (3.8)

Coupling of a scalar conservation law with a parabolic problem 129

for all nonnegative ϕ ∈ C

∞

(R × R

) with ϕ(0, ·) = 0 and all k ∈ R.

Now we use the method of doubling variables that can be divided into three steps:

Step 1. For ˜p = (

t, ˜x) ﬁxed, we choose k = u

( ˜p), ϕ = ψ

(·, ˜p) in (3.8) written for

(p). We then integrate with respect to ˜p over Q. It follows (recall that p = (t, x))

−

(p) − u

( ˜p)|∂



p + ˜p



(p − ˜p)dpd ˜p

−

b(x)F (u

(p), u

( ˜p)) · ∇ϕ



p + ˜p



(p − ˜p)dpd ˜p

−

(p) − u

( ˜p)|∂

(p − ˜p)ϕ



p + ˜p



dpd ˜p

−

b(x)F (u

(p), u

( ˜p)) · ∇

(p − ˜p)ϕ



p + ˜p



dpd ˜p

≤

(σ)ψ

(σ, ˜p)dσd ˜p −

F (u

( ˜p), 0)b(σ) · νψ

(σ, ˜p)dσd ˜p.

Step 2. In the same way, for p ﬁxed, we choose k = u

(p), ϕ = ψ

(p, ·) in (3.8)

written for u

( ˜p). Now we integrate with respect to p over Q which yields (using

Fubini’s theorem and the fact that −∂

(p − ˜p) = ∂

(p − ˜p), −∇

˜x

(p − ˜p) =

∇

(p − ˜p))

−

( ˜p) − u

(p)|∂



p + ˜p



(p − ˜p)dpd ˜p

−

b( ˜x)F (u

( ˜p), u

(p)) · ∇ϕ



p + ˜p



(p − ˜p)dpd ˜p

( ˜p) − u

(p)|∂

(p − ˜p)ϕ



p + ˜p



dpd ˜p

b( ˜x)F (u

( ˜p), u

(p)) · ∇

(p − ˜p)ϕ



p + ˜p



dpd ˜p

≤

(˜σ)ψ

(p, ˜σ)d ˜σdp −

F (u

(p), 0)b( ˜σ) · νψ

(p, ˜σ)d ˜σdp.

We sum up the two previous inequalities and remark that the terms involving ∂

(p− ˜p)

vanish. Therefore, we ﬁnd

−

(p) − u

( ˜p)|∂



p + ˜p



(p − ˜p)

−

F (u

(p), u

( ˜p))(b(x) + b(˜x)) · ∇ϕ



p + ˜p



(p − ˜p)dpd ˜p

130 Julien Jimenez

−

F (u

(p), u

( ˜p))(b(x) − b( ˜x)))ϕ



p + ˜p



· ∇

(p − ˜p)dpd ˜p

≤

(σ)ψ

(σ, ˜p)dσd ˜p +

(˜σ)ψ

(p, ˜σ)d ˜σdp

−

F (u

( ˜p), 0)b(σ) · νψ

(σ, ˜p)dσd ˜p −

F (u

(p), 0)b( ˜σ)ψ

(p, ˜σ)d ˜σdp.

(3.9)

Step 3. We take the limit with respect to j. By Lemma 3.9, we have

lim

j→∞

F (u

(p), 0)b( ˜σ)ψ

(p, ˜σ)d ˜σdp

(σ)ϕ(σ)dσ = lim

j→∞

(σ)ψ

(σ, ˜p)dσd ˜p

and

lim

j→∞

F (u

( ˜p), 0)b(σ)ψ

(σ, ˜p)dσd ˜p

(σ)ϕ(σ)dσ = lim

j→∞

(˜σ)ψ

(p, ˜σ)d ˜σdp.

Consequently, the right-hand side of (3.9) goes to zero as j tends to ∞.

We study now the ﬁrst line of (3.9). To this end, we set

(p) − u

( ˜p)|ρ

(p − ˜p)∂



p + ˜p



dpd ˜p,

I :=

(p) − u

(p)|∂

ϕ(p)dp.

Let us show that lim

j→∞

= I. By deﬁnition of ρ

, we have

(p) − u

(p)|∂

ϕ(p)dp =

(p) − u

(p)|∂

ϕ(p)ρ

(p − ˜p)dpd ˜p

if j is sufﬁciently large. Therefore, we obtain

− I| =

(|u

(p) − u

( ˜p)| − |u

(p) − u

(p)|)ρ

(p − ˜p)∂



p + ˜p



dpd ˜p

(p) − u

(p)|



∂



p + ˜p



− ∂

ϕ(p)



(p − ˜p)dpd ˜p,

and thus

− I| ≤ k∂

ϕk

∞

(p) − u

( ˜p)|ρ

(p − ˜p)dpd ˜p

+ku

− u

∞



∂



p + ˜p



− ∂

ϕ(p)



(p − ˜p)dpd ˜p.

Coupling of a scalar conservation law with a parabolic problem 131

The ﬁrst term on the right-hand side of the foregoing inequality tends to zero in view of

the property of Lebesgue points. The second one goes to zero since ∂

ϕ is continuous

on Q. Therefore, we may conclude that lim

j→∞

= I.

In a similar way, we prove that

lim

j→∞

F (u

(p), u

( ˜p))(b(x) + b(˜x)) · ∇



p + ˜p



(p − ˜p)dpd ˜p

F (u

, u

)b(x) · ∇ϕ(p)dp.

We continue with the study of the term

F (u

(p), u

( ˜p))(b(x) − b( ˜x)))ϕ



p + ˜p



· ∇

(p − ˜p)dpd ˜p

that can be decomposed as L

= L

1,j

+ L

2,j

with

1,j

(F (u

(p), u

( ˜p)) − F (u

( ˜p), u

( ˜p)))(b(x) − b(˜x))·

∇

(p − ˜p)ϕ



p + ˜p



dpd ˜p,

2,j

F (u

( ˜p), u

( ˜p))(b(x) − b( ˜x))∇

(p − ˜p)ϕ



p + ˜p



dpd ˜p.

Thanks to an integration by parts and (1.2), we have

2,j

= −

F (u

( ˜p), u

( ˜p))(b(x) − b( ˜x))ρ

(p − ˜p)∇



p + ˜p



dpd ˜p

F (u

( ˜p), u

( ˜p))(b(σ) − b( ˜x))ρ

(σ − ˜p)ϕ



σ + ˜p



dσd ˜p.

As b is continuous on Ω, we also ﬁnd lim

j→∞

2,j

= 0.

Moreover, using the Lipschitz continuity of f and b, we may estimate

1,j

| ≤

(p) − u

( ˜p)||x − ˜x||∇

(x − ˜x)|ρ

(t −

t)ϕ



p + ˜p



dpd ˜p.

In view of the deﬁnition of ρ

, there exists a constant C

> 0 such that for all (t, x) ∈ Q,

|∇

(x − ˜x)|ρ

(t −

t) ≤ C

n+2

Therefore, there exists a constant C

> 0 such that

1,j

| ≤ C

n+1

{(p, ˜p)∈Q

; |p− ˜p|≤

}

(p) − u

( ˜p)|dpd ˜p.

132 Julien Jimenez

Using the property of Lebesgue points, we ﬁnd

lim

j→∞

n+1

{(p, ˜p)∈Q

; |p− ˜p|≤

}

(p) − u

( ˜p)|dpd ˜p = 0.

Thus, lim

j→∞

1,j

= 0 and lim

j→∞

= 0. Finally, gathering all the limits in (3.9)

proves the assertion.

Proof of Theorem 3.7. We choose in (3.7) the test function ϕ(t, x) = α(t)β(x), where

β ∈ C

∞

), β ≡ 1 on Ω and α ∈ C

∞

(R), α ≥ 0 with α(0) = 0. This choice yields

(t, x) − u

(t, x)|α

(t)dxdt ≥ 0.

Then, for almost all t ∈ (0, T ), consider a sequence of test functions (α

)

ε>0

approxi-

mating the characteristic function χ

[0,t]

. We may suppose that α

= 0 on [ε, t − ε] and

that |α

| ≤

. From (3.3), we now obtain

0 ≤ lim

ε→0

− u

|α

(t)dxdt

= −

Ω

(t, x) − u

(t, x)|dx +

Ω

0,1

(x) − u

0,2

(x)|dx.

This completes the proof.

3.3 Existence: the vanishing viscosity method

In this section, we prove the existence of an entropy solution to (3.1) via the vanishing

viscosity method. Let µ be a positive real number. We introduce the following viscous

problem.

Find a measurable and essentially bounded function u

such that in the sense of

distributions











∂

+ ∇ · (b(x)f(u

)) = ∇ · (µ∇u

) in Q,

(0, ·) = u

on Ω,

= 0 on Σ.

(3.10)

The aim of the vanishing viscosity method is to study the limit of the sequence

)

µ>0

as µ goes to 0

and to show that this is an entropy solution to problem (3.1).

Therefore, we need to collect some a priori estimates on the sequence (u

)

µ>0

and

apply compactness arguments in order to take the limit over µ. Generally, one looks

for W

1,1

-estimates in order to be able to use a compactness argument. We refer to

[4, 9, 10] for more details. Here, with the ﬁnal aim to solve the coupled problem

(1.1), we will present a method that only requires an L

∞

-estimate on (u

)

µ>0

. To this

purpose, we use the concept of entropy process solution.

Coupling of a scalar conservation law with a parabolic problem 133

Deﬁnition 3.11. A function π ∈ L

∞

((0, 1) × Q) is said to be an entropy process solu-

tion to (3.1) if

(i) for all nonnegative ϕ ∈ C

∞

(Q) and all k ∈ R

(|π − k|∂

ϕ + sgn(π − k)(f(π) − f (k))b(x) · ∇ϕ)dxdtdα ≥ 0, (3.11)

(ii)

esslim

t→0

Ω

|π(α, t, x) − u

(x)|dxdα = 0, (3.12)

(iii) for all nonnegative ϕ ∈ L

(Σ) and all k ∈ R

ess lim

s→0

−

(π(α, σ + sν), k)b(σ) · νϕdσdα ≥ 0. (3.13)

By using the same method as in the previous section, we can prove that the entropy

process solution to (3.1) is unique.

Theorem 3.12. Assume there exists an entropy process solution π ∈ L

∞

((0, 1) × Q) to

(3.1). Then π is unique and there exists u ∈ L

∞

(Q) such that u(·) = π(α, ·) a.e. on Q

and for almost all α ∈ (0, 1). In particular, u is an entropy solution to (3.1).

Proof. We start with the uniqueness. Let π

(α, ·, ·) and π

(β, ·, ·) be two entropy pro-

cess solutions to (3.1) for the same initial data u

. We use the same reasoning as in the

proof of Theorem 3.7 to see that for almost all t ∈ (0, T )

Ω

|π

(α, t, x) − π

(β, t, x)|dxdαdβ ≤ 0.

This estimate implies that there exists u ∈ L

∞

(Q) such that a.e. on Q

u(·) = π

(α, ·) = π

(β, ·) for almost all α, β in (0, 1).

The conclusion follows.

Our purpose is now to show that there exists an entropy process solution to (3.1).

The existence of an entropy solution to (3.1) will follow from Theorem 3.12. To do

so, we need some lemmas that we will use to obtain (3.12)–(3.13) (see [15, Chap. 2,

Lemma 7.34 and 7.41] for the proofs in the framework of entropy solutions).

Lemma 3.13. Suppose that π ∈ L

∞

((0, 1) × Q) satisﬁes, for all nonnegative ϕ ∈

∞

([0, T ) × Ω) and all k ∈ R,

−

(|π − k|∂

ϕ + b(x)F (π, k) · ∇ϕ)dxdtdα ≤

Ω

− k|ϕ(0, x)dx.

Then

esslim

t→0

Ω

|π(α, t, x) − u

(x)|dxdα = 0.

134 Julien Jimenez

Lemma 3.14. Assume that π ∈ L

∞

((0, 1) × Q) fulﬁlls, for all nonnegative ϕ ∈

∞

([0, T ) × Ω) and all k ∈ R,

−

(π, k)∂

ϕ + b(x)J

(π, k) · ∇ϕ)dxdtdα ≤ 0,

where H

and J

are deﬁned in Example 3.4. Then, for any nonnegative β ∈ L

(Σ)

ess lim

s→0

−

(π(α, σ + sν), k)βb(σ) · νdσdα ≥ 0.

We come back to the viscous problem (3.10). We use the results gathered in Sec-

tion 2 to prove

Lemma 3.15. There exists a unique weak solution u

∈ W (0, T ) to (3.10), i.e.,

m ≤ u

(t, x) ≤ M for almost all (t, x) ∈ Q, (3.14)

for almost all t ∈ (0, T ) and all v ∈ H

(Ω) there holds

h∂

, vi +

Ω

(µ∇u

− b(x)f(u

)) · ∇vdx = 0, (3.15)

and the initial condition

(0) = u

in L

(Ω)

is fulﬁlled.

Proof. Setting φ(u

) = µu

, we notice that φ satisﬁes (1.3). Therefore, the assump-

tions of Theorem 2.3 are satisﬁed and the conclusion follows.

We also prove the following lemma that will be used to take the limit as µ → 0

Lemma 3.16. There exists a positive constant C independent of µ such that

|∇u

dxdt ≤ C. (3.16)

Proof. We take v = u

in (3.15) and integrate over (0, T ). An integration by parts in

the evolution term gives

h∂

, u

idt =

(T )k

(Ω)

−

(Ω)

We set g(u

) =

f(s)ds to write the convection term as

Ω

f(u

)b(x) · ∇u

dxdt =

Ω

∇g(u

) · b(x)dxdt.