Clausius theorem for conservation laws

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Clausius Theorem for Hyperbolic Scalar Conservation Laws

Sylvain Dotti∗ September 20, 2019

Abstract

I give for hyperbolic scalar conservation laws with non-linear flux, the ideas behind the definitions of entropic solution and kinetic solution in different cases (homogeneous, deterministic source term, stochastic source term): they come from the Clausius theorem.

Keywords: Hyperbolic scalar conservation laws, entropic solution, kinetic solution, stochastic source term, Clausius theorem

MSC Number: 35-01 (35L65, 35L60, 35R60)

1 Introduction

The definition of solution of hyperbolic scalar conservation laws with non-linear flux is very important to solve the Cauchy problems. Kruzkov in [Kru70] solved the problem of uniqueness of weak solutions by adding a physical relevant condition called entropy condition. Since that, Di Perna in [DiP83], Eymard, Gallou¨et and Herbin in [EGH95], Lions, Perthame and Tadmor in [LPT94], E Mazel Khanin and Sinai in [E+00], Dotti Vovelle in [DV18] used this idea of monotonicity of the mathematical entropy of the solution of conservation laws to define their solution in order to prove existence and uniqueness. This idea is a physical idea, it is actually the Clausius theorem. With the same physical idea, the kinetic formulations of the entropic inequalities can also be established. It gives the definitions of kinetic solution and kinetic entropy defect measure of Perthame [Per02].

2 Case of homogeneous hyperbolic scalar conservation laws

2.1 Entropy solution

Let me explain what I understand of the Clausius theorem for the hyperbolic scalar conservation law

∂t(u (x, t)) + divx(A (u (x, t))) = 0, t ∈ R+, x ∈ Rd, d ∈ N∗, (2.1) with u : Rd× R+_{7→ R called the conserved quantity and A : R 7→ R}d _{the flux. Without} loss of generality, I suppose that A(0) = 0.

When the physical phenomenon corresponding to (2.1) is reversible, the mathemati-cal function u is regular. In this case, the quantity of entropy η (u (x, t)) is conserved over time, and thus is a solution of the conservation law

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η : R 7→ R is called the mathematical entropy, it is a convex function (see [Kru70], [Lax71]), φ : R 7→ Rd is called the η entropy flux. Thermodynamic entropy is concave. The link between the two entropies is well described in the case of gas dynamics by Dubois [Dub90].

If A, η and φ are regular functions, those two conservation laws can be written

∂t(u (x, t)) + A 0 (u (x, t)) .∇x(u (x, t)) = 0 and η0(u (x, t)) ∂t(u (x, t)) + φ 0 (u (x, t)) .Ox(u (x, t)) = 0.

By identification of those two conservation laws, we get the definition (up to an additive constant) of the entropy flux

φ0(ξ) = A0(ξ) η0(ξ) .

But when the physical phenomenon corresponding to (2.1) is irreversible, that is when u is not regular, a defect of mathematical entropy is created over time, that is written

∂t(η (u (x, t))) + divx(φ (u (x, t))) ≤ 0. (2.2)

The inequality (2.2) is to be taken in the weak sense. It means to multiply by a test function ϕ ∈ C_c∞(Rd× R+_{; R}+_{), to integrate against the time variable t ∈ R}d_{, against} the space variable x ∈ Rd, to integrate by parts d + 1 times (against each real variable t and xi, i ∈ {1, ..., d}) to obtain under the initial data u(x, 0) = u0(x) ∈ L∞(Rd) the entropy inequality: − Z Rd×R+ η (u(x, t)) ∂tϕ (x, t) + φ (u (x, t)) .∇ϕ (x, t) ! dxdt − Z Rd η (u0(x)) ϕ (x, 0) dx ≤ 0.

Remark 2.1. Kruzkov decided to use the mathematical entropies

{u ∈ R 7→ |u − ξ| such that ξ ∈ R}

to solve the problem of uniqueness of weak solutions of J. Leray [Ler33]. They are convex functions. If he took the opposite of those functions, which are concave, the inequality (2.2) would be in the opposite sense, and we could say that mathematical entropy increase over time, just like physical entropy. But it is not the case !

Remark 2.2. The equivalence between kruzkov entropies and convex functions (mathe-matical entropies) in the entropy inequality (2.2) is proved in detail in paragraph 6.

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2.2 Kinetic solution

2.2.1 A first method

Lions, Perthame and Tadmor replaced the entropy inequality by an equality called ’ki-netic equality’, introducing a supplementary variable ξ ∈ R thanks to Kruzkov entropies and corresponding entropy fluxes well chosen.

More precisely, let us apply the inequality (2.2) to the entropies ηξ(u) = |u − ξ| − |ξ| and to the entropy fluxes φξ(u) = sgn(u − ξ) (A(u) − A(ξ)) − sgn(ξ)A(ξ) :

∂t |u (x, t) − ξ| − |ξ| +divx sgn(u(x, t) − ξ) (A(u(x, t)) − A(ξ)) − sgn(ξ)A(ξ) ≤ 0. (2.3) The ξ was a constant for Kruzkov, it becomes a variable for Lions Perthame and Tadmor. They denote the left-hand side of (2.3) −2m(x, t, ξ) and can write the entropy inequality

m(x, t, ξ) ≥ 0.

The kinetic formulation of the entropy inequality (2.2) or simply of the hyperbolic scalar conservation law (2.1) is the equation giving the relation between ∂ξm(x, t, ξ) and the partial derivatives (in the weak sense) of χ(ξ, u(x, t)) := 1u(x,t)>ξ− 10>ξ.

From the equality

− 2m(x, t, ξ) = ∂_t |u (x, t) − ξ| − |ξ|

+ divx sgn(u(x, t) − ξ) (A(u(x, t)) − A(ξ)) − sgn(ξ)A(ξ)

and the partial derivatives

• ∂ξ |u (x, t) − ξ| − |ξ| = −2χ(ξ, u(x, t))

• ∂ξ sgn(u(x, t) − ξ) (Ai(u(x, t)) − Ai(ξ)) − sgn(ξ)Ai(ξ) = −2A

0

i(ξ)χ(ξ, u(x, t)) we obtain the kinetic formulation:

∂t(χ (ξ, u (x, t))) + A

0

(ξ) .∇x(χ (ξ, u (x, t))) = ∂ξm(x, t, ξ)

where m is a bounded non-negative measure.

Remark 2.3. when the physical process is reversible, that is when u is regular, no entropy is absorbed over time, the kinetic entropy defect measure m is the null measure. when the physical process is irreversible, that is when u is not regular,

m (or rather 2m) measurethe entropy defect created over time, in other words entropy absorption over time.

Remark 2.4. The choice of Lions, Perthame and Tadmor in [LPT94] of couples en-tropy/entropy flux

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is done to find, differentiating with respect to ξ, the integrable function χ(ξ, u(x, t)). The choice of Debussche and Vovelle in [DV10] of couples entropy/entropy flux

|u (x, t) − ξ| − ξ, sgn(u(x, t) − ξ) (A(u(x, t)) − A(ξ)) − A(ξ),

gives a kinetic formulation with the locally integrable function 1u(x,t)>ξ in place of the χ(ξ, u(x, t)). Other choices are possible.

2.2.2 A second method For the conservation law

∂t(u (x, t)) + divx(A (u (x, t))) = 0,

the second method to obtain the kinetic formulation consist, when u is regular (and A also), in doing the operation ⊗δu(x,t)(dξ): we obtain the equality (weak in ξ)

∂t(χ (ξ, u (x, t))) + A

0

(ξ) .∇x(χ (ξ, u (x, t))) = 0

with the use of

∂t(χ (ξ, u (x, t))) = ∂tu (x, t) ⊗ δu(x,t)(dξ)

A0(ξ) .∇x(χ (ξ, u (x, t))) = A

0

(u (x, t)) .∇x(u (x, t)) ⊗ δu(x,t)(dξ) .

When the quantity of entropy is conserved, that is when

∂t(η (u (x, t))) + divx(φ (u (x, t))) = 0,

by doing the operation ⊗δu(x,t)(dξ), we obtain the equality (weak in ξ)

η0(ξ) ∂t(χ (ξ, u (x, t))) + φ

0

(ξ) .∇x(χ (ξ, u (x, t))) = 0. (2.4)

Defining m up to an additive constant by

∂ξ(m(x, t, ξ)) = ∂t(χ (ξ, u (x, t))) + A

0

(ξ) .∇x(χ (ξ, u (x, t))) ,

we can write (2.4) as

η0(ξ)∂ξm(x, t, ξ) = 0.

When the physical process is irreversible, that is when u is not regular, an entropy defect is created which is written

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or, with the kinetic variable ξ Z Rξ η0(ξ) ∂t(χ (ξ, u (x, t))) + φ 0 (ξ) .∇x(χ (ξ, u (x, t))) dξ ≤ 0,

or, with the kinetic entropy defect measure m Z

Rξ

η00(ξ)m(x, t, dξ) ≥ 0.

Thus, it is the measure η00(ξ)m(x, t, ξ) which measures the entropy of the quantity u(x, t) over time. We can also say that measuring the entropy η(u(x, t)) over time, is the same as measuring η00 with m.

The kinetic formulation of the hyperbolic scalar conservation law is

∂t(χ (ξ, u (x, t))) + A

0

(ξ) .∇x(χ (ξ, u (x, t))) = ∂ξm(x, t, ξ),

where the non-negativity of the measure m is equivalent to the entropy inequality.

3 Case of the hyperbolic scalar conservation law with

de-terministic source term

For the conservation law with source term

∂t(u (x, t)) + divx(A (u (x, t))) = G (u (x, t)) , (3.1)

let me tell what I understand of the Clausius theorem.

When the physical phenomenon corresponding to (3.1) is reversible, that is when u is rgular, the quantity of entropy η (u (x, t)) is conserved thus

∂t(η (u (x, t))) + divx(φ (u (x, t))) = H (u (x, t))

where H (u (x, t)) is the source of entropy.

If A, η and φ are regular functions, these two conservation laws can be written

∂t(u (x, t)) + A 0 (u (x, t)) .Ox(u (x, t)) = G (u (x, t)) and η0(u (x, t)) ∂t(u (x, t)) + φ 0 (u (x, t)) .Ox(u (x, t)) = H (u (x, t)) .

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By identification of these two conservation laws, we get the definition of the entropy source

H (ξ) = G (ξ) η0(ξ)

But when the physical phenomenon corresponding to (3.1) is irreversible, that is when u is not regular, a defect of entropy is created over time which is written

∂t(η (u (x, t))) + divx(φ (u (x, t))) − η

0

(u (x, t)) G (u (x, t)) ≤ 0. (3.2)

To find the kinetic formulation of (3.1) or of (3.2),

we apply the inequality (3.2) to the entropies ηξ(u) = |u − ξ| − |ξ| and the entropy fluxes φξ(u) = sgn(u − ξ) (A(u) − A(ξ)) − sgn(ξ)A(ξ) :

∂t |u (x, t) − ξ| − |ξ| + divx sgn(u(x, t) − ξ) (A(u(x, t)) − A(ξ)) − sgn(ξ)A(ξ)

− sgn(u(x, t) − ξ)G (u (x, t)) ≤ 0. (3.3)

The ξ was a constant for Kruzkov, it becomes a variable for Lions Perthame and Tadmor. They denote the left-hand side of (3.3) −2m(x, t, ξ) and can write the entropy inequality

m(x, t, ξ) ≥ 0.

The kinetic formulation of the entropy inequality (3.2) or simply of the hyperbolic scalar conservation law (3.1) is the equation giving the relation between ∂ξm(x, t, ξ) and the partial derivatives (in the weak sense) of χ(ξ, u(x, t)).

Frome the equality

− 2m(x, t, ξ) = ∂t |u (x, t) − ξ| − |ξ|

+ divx sgn(u(x, t) − ξ) (A(u(x, t)) − A(ξ)) − sgn(ξ)A(ξ)

− sgn(u(x, t) − ξ)G (u (x, t))

and the weak derivatives

• ∂_ξ |u (x, t) − ξ| − |ξ| = −2χ(ξ, u(x, t))

• ∂ξ sgn(u(x, t) − ξ) (Ai(u(x, t)) − Ai(ξ)) − sgn(ξ)Ai(ξ) = −2A

0

i(ξ)χ(ξ, u(x, t)) • ∂_ξ sgn(u(x, t) − ξ)G (u (x, t)) = −2G (u(x, t)) δ_u(x,t)(dξ)

we obtain the kinetic formulation :

∂t(χ (ξ, u (x, t))) + A

0

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Remark 3.1. The inequality

∂ξ χ (ξ, u (x, t)) = δ0(dξ) − δu(x,t)(dξ) allows us to write the kinetic formulation

∂t(χ (ξ, u (x, t))) + A

0

(ξ) .∇x(χ (ξ, u (x, t)))

+ G (u(x, t)) ∂ξ χ (ξ, u (x, t)) − G (u(x, t)) δ0(dξ) = ∂ξm(x, t, ξ).

4 Case of the hyperbolic scalar conservation law with

de-terministic flux and source term depending on the space

and time variables

If now, we are interested by the conservation law with source term

∂t(u (x, t)) + divx(A (x, t, u (x, t))) = G (x, t, u (x, t)) (4.1)

with A : Rdx× R+t × Rξ→ Rd et G : Rdx× R+t × Rξ → R, This is what I understand of the Clausius theorem.

When the physical phenomenon corresponding to (4.1) is reversible, that is when u is regular, the entropy quantity η (u (x, t)) is conserved thus

∂t(η (u (x, t))) + divx(φ (x, t, u (x, t))) = H (x, t, u (x, t)) ,

H (u (x, t)) being the entropy source.

With A, η and φ being regular functions, those two conservation laws can be written

∂t(u (x, t)) + (divxA) (x, t, u(x, t))

+ ∂ξA (x, t, u (x, t)) .∇x(u (x, t)) = G (x, t, u (x, t))

and

η0(u(x, t)) ∂t(u (x, t)) + (divxφ) (x, t, u(x, t))

+ ∂ξφ (x, t, u (x, t)) .∇x(u (x, t)) = H (x, t, u (x, t)) .

By identification of these two conservation laws, we have only one possibility for the definition of the entropy flux :

∂ξφ(x, t, ξ) = η

0

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Multiplying the first conservation law by η0(u(x, t)) and subtracting the second conser-vation law, it comes

η0(u(x, t)) (divxA) (x, t, u(x, t)) − (divxφ) (x, t, u(x, t))

= η0(u(x, t))G (x, t, u (x, t)) − H (x, t, u (x, t)) . That gives the definition of the entropy source :

H (x, t, ξ) = η0(ξ)G (x, t, ξ) − η0(ξ) (divxA) (x, t, ξ) + (divxφ) (x, t, ξ)

In the case where η(ξ) = |ξ − κ| is a Kruzkov entropy, the corresponding entropy flux can be written

φ(x, t, ξ) = sgn(ξ − κ) (A(x, t, ξ) − A(x, t, κ)) , that gives (in that particular case), the definition of entropy source

H (x, t, ξ) = η0(ξ)G (x, t, ξ) − η0(ξ) (divxA) (x, t, κ) .

But when the physical phenomenon corresponding to (4.1) is irreversible, that is when u is not regular, an entropy defect is created over time which can be written

∂t(η (u (x, t))) + divx(φ (x, t, u (x, t))) − H (x, t, u (x, t)) ≤ 0 (4.2) or, with the test functions (x, t) 7→ ϕ(x, t) ∈ C_c∞(Rd× R+ _{; R}+_{) :}

− Z Rd η(u(x, 0))ϕ(x, 0)dx − Z Rd×R+ η(u(x, t))∂tϕ(x, t)dxdt − Z Rd×R+ φ (x, t, u (x, t)) .∇xϕ(x, t)dxdt − Z Rd×R+

η0(u(x, t)) G − (divxA) (x, t, u(x, t)) ϕ(x, t)dxdt −

Z

Rd×R+

(divxφ) (x, t, u(x, t)) ϕ(x, t)dxdt ≤ 0

To give the kinetic formulation of (4.1) or of (4.2),

we apply the inequality (4.2) to the entropies ηξ(u) = |u − ξ| − |ξ| and to the entropy fluxes φξ(x, t, u) = sgn(u − ξ) (A(x, t, u) − A(x, t, ξ)) − sgn(ξ)A(x, t, ξ) :

∂t |u (x, t) − ξ| − |ξ|

+ divx sgn(u(x, t) − ξ) (A(x, t, u(x, t)) − A(x, t, ξ)) − sgn(ξ)A(x, t, ξ)

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with

Hξ(x, t, u) = sgn(u − ξ) G(x, t, u) − (divxA)(x, t, ξ) − sgn(ξ) (divxA) (x, t, ξ) = sgn(u − ξ)G(x, t, u) − 2χ(ξ, u) (divxA) (x, t, ξ),

which gives

∂t |u (x, t) − ξ| − |ξ|

+ divx sgn(u(x, t) − ξ) (A(x, t, u(x, t)) − A(x, t, ξ)) − sgn(ξ)A(x, t, ξ) − sgn(u(x, t) − ξ)G(x, t, u(x, t)) + 2χ(ξ, u(x, t)) (divxA) (x, t, ξ) ≤ 0.

The ξ was a constant for Kruzkov, it becomes a variable for Lions Perthame and Tadmor. They denote the left-hand side of (4.3) −2m(x, t, ξ) and can write the entropy inequality

m(x, t, ξ) ≥ 0.

The kinetic formulation of the entropy inequality (4.2) or simply of the hyperbolic scalar conservation law (4.1) is the equation giving the link between ∂ξm(x, t, ξ) and the weak partial derivatives of χ(ξ, u(x, t)).

From the equality

− 2m(x, t, ξ) = ∂t |u (x, t) − ξ| − |ξ|

+ divx sgn(u(x, t) − ξ) (A(x, t, u(x, t)) − A(x, t, ξ)) − sgn(ξ)A(x, t, ξ) − sgn(u(x, t) − ξ)G(x, t, u(x, t)) + 2χ(ξ, u(x, t)) (div_xA) (x, t, ξ)

and the weak derivatives

• ∂_ξ |u − ξ| − |ξ| = −2χ(ξ, u)

• ∂ξ φξ,i(x, t, u) = −2∂ξ Ai(x, t, ξ)χ(ξ, u)

• ∂_ξ sgn(u − ξ)G(x, t, u) = −2G (x, t, u) δ_u(x,t)(dξ) •

∂ξ χ(ξ, u) (divxA) (x, t, ξ)

= ∂ξ χ(ξ, u) (divxA) (x, t, ξ) + χ(ξ, u)∂ξ (divxA) (x, t, ξ)

we obtain the kinetic formulation :

∂t(χ (ξ, u (x, t))) + ∂ξA (x, t, ξ) .∇x(χ (ξ, u (x, t)))

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Remark 4.1. The equality

∂ξ χ (ξ, u (x, t)) = δ0(dξ) − δu(x,t)(dξ) allows to write the kinetic formulation

∂t(χ (ξ, u (x, t))) + ∂ξA (x, t, ξ) .∇x(χ (ξ, u (x, t)))

+ G (x, t, u(x, t)) − (divxA) (x, t, ξ)∂ξ χ (ξ, u (x, t))

− G (x, t, u(x, t)) δ₀(dξ) = ∂ξm(x, t, ξ).

5 Case of the hyperbolic scalar conservation law with

stochas-tic source term

If we study the conservation law with stochastic source term

d (u (x, t, ω)) + divx(A (u (x, t, ω))) dt = Φ (x, u (x, t, ω)) dW (t, ω) , (5.1)

with W a cylindrical Wiener process defined on a separable Hilbert space H and Φ : Rd× R → L2(H, R) a continuous function (see [DV18] for details on assumptions). To simplify the calculations, one can think of W as a real brownian motion and

Φ : Rd× R → R continuous, the ideas and results are similar. This is what I understand of the Clausius theorem:

If the physical phenomenon associated with the conservation law was reversible, that is if the diffusion process u was regular in the space variable x ∈ Rd, the quantity of entropy η (u (x, t, ω)) would be conserved thus

d (η (u (x, t, ω))) + divx(φ (u (x, t, ω))) dt = d (G (u (x, t, ω)))

where G (u (x, t, ω)) is the entropy source.

If A, η and Φ were regular functions, the two conservation laws could be written

d (u (x, t, ω)) + A0(u (x, t, ω)) .∇x(u (x, t, ω)) dt = Φ (x, u (x, t, ω)) dW (t, ω)

and

d (η (u (x, t, ω))) + φ0(u (x, t, ω)) .∇x(u (x, t, ω)) dt = d (G (u (x, t, ω))) .

To identify the two conservation laws, we have to notice that the time ’chain rule’ is not true for diffusion processes. Instead, we use the Itˆo Lemma.

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In other words, we don’t have d (η (u (x, t, ω))) = η0(u (x, t, ω)) d (u (x, t, ω)) but we have d (η (u (x, t, ω))) = η0(u (x, t, ω)) d (u (x, t, ω)) + 1 2η 00 (u (x, t, ω)) kΦ (x, u (x, t, ω)) k2_L₂_(H,R)dt

for any convex function η ∈ C2(R). Hence, the conservation law of the entropy would be written η0(u (x, t, ω)) d (u (x, t, ω)) +1 2η 00 (u (x, t, ω)) kΦ (x, u (x, t, ω)) k2_L₂_(H,R)dt + φ0(u (x, t, ω)) .∇x(u (x, t, ω)) dt = d (G (u (x, t, ω)))

By identification of the two conservation laws, we get the definition of the entropy source

d (G (u (x, t, ω))) = 1 2η

00

(u (x, t, ω)) kΦ (x, u (x, t, ω)) k2_L₂_(H,R)dt

+ η0(u (x, t, ω)) Φ (x, u (x, t, ω)) dW (t, ω) .

If the physical phenomenon associated with the conservation law was irreversible, that is if the diffusion process u was not regular in the space variable x ∈ Rd_{, that would} create an entropy defect which could be written

d (η (u (x, t, ω))) −1 2η 00 (u (x, t, ω)) kΦ (x, u (x, t, ω)) k2_L 2(H,R)dt + divx(φ (u (x, t, ω))) dt − η 0 (u (x, t, ω)) Φ (x, u (x, t, ω)) dW (t, ω) ≤ 0. (5.2)

This inequality is a weak in space formulation of the conservation law with stochastic source term. It is written with the test functions ϕ ∈ C_c∞(Rd; R+), for all fixed t ∈ R+ in the following way :

Z Rd ϕ(x) η (u (x, t, ω)) − η (u (x, 0, ω))dx − Z Rd Z t 0 φ (u(x, s, ω)) ds.∇xϕ(x)dx −1 2 Z Rd ϕ(x) Z t 0 η00(u (x, s, ω)) kΦ (x, u (x, s, ω)) k2_L 2(H,R)dsdx − Z Rd ϕ(x) Z t 0 η0(u (x, s, ω)) Φ (x, u (x, s, ω)) dW (s, ω) dx ≤ 0.

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To get the kinetic formulation of

d (u (x, t, ω)) + divx(A (u (x, t, ω))) dt = Φ (x, u (x, t, ω)) dW (t) , (5.3) we perform the operation ⊗δ_u(x,t,ω)(dξ) both sides of the equality.

If u was regular in the space variable x ∈ Rd, that is if the physical phenomenon associ-ated with (5.3) was reversible, we would obtain

d (u (x, t, ω)) + A0(u (x, t, ω)) .∇x(u (x, t, ω)) dt

⊗ δ_u(x,t,ω)(dξ)

= Φ (x, u (x, t, ω)) dW (t) ⊗ δu(x,t,ω)(dξ) .

The ’chain rule’ is untrue for Itˆo processes, instead we use a weak in ξ Itˆo Lemma . In other words, we don’t have

dχ (ξ, u (x, t, ω)) = du (x, t, ω) ⊗ δu(x,t,ω)(dξ) but we have dχ (ξ, u (x, t, ω)) = du (x, t, ω) ⊗ δ_u(x,t,ω)(dξ) −1 2∂ξ kΦ (x, ξ) k2 L2(H,R)dt ⊗ δu(x,t,ω)(dξ) . ******* Remark 5.1. Digression on the weak in ξ Itˆo Lemma

If ϕ ∈ C_c∞(R), then the Itˆo Lemma applied to the process u gives dϕ (u (x, t)) = ϕ0(u (x, t)) du (x, t) +1

2ϕ 00

(u (x, t)) kΦ (x, u (x, t)) k2_L

2(H,R)dt which can be written

d Z R ϕ0(ξ) χ (ξ, u (x, t)) dξ = Z R ϕ0(ξ) du (x, t) ⊗ δ_u(x,t)(ξ) +1 2 Z R ϕ00(ξ) kΦ (x, ξ) k2_L 2(H,R)dt ⊗ δu(x,t)(dξ) that is d Z R ϕ0(ξ) χ (ξ, u (x, t)) dξ = Z R ϕ0(ξ) du (x, t) ⊗ δu(x,t)(ξ) −1 2 D ∂ξ kΦ (x, ξ) k2_L 2(H,R)dt ⊗ δu(x,t)(dξ) , ϕ0(ξ) E

which gives the following weak in ξ Itˆo formula

d (χ (ξ, u (x, t))) = du (x, t) ⊗ δ_u(x,t)(dξ) −1 2∂ξ kΦ (x, ξ) k2_L 2(H,R)dt ⊗ δu(x,t)(dξ) .

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*******

using the weak in ξ Itˆo Lemma and the following formula

A0(ξ).∇x χ (ξ, u(x, t)) = A

0

(u(x, t)) .∇x(u(x, t)) ⊗ δu(x,t)(dξ),

our equation would be written

d (χ (ξ, u (x, t, ω))) + A0(ξ) .∇x(χ (ξ, u (x, t, ω))) dt − Φ (x, ξ) ⊗ δ_u(x,t,ω)(dξ) dW (t) +1 2∂ξ kΦ (x, ξ) k2 L2(H,R)⊗ δu(x,t,ω)(dξ) dt = 0.

If the physical phenomenon associated with the conservation law was irreversible, that is if u was not regular in x ∈ Rd, the entropy defect would be written

d (χ (ξ, u (x, t, ω))) + A0(ξ) .∇x(χ (ξ, u (x, t, ω))) dt − Φ (x, ξ) ⊗ δu(x,t,ω)(dξ) dW (t) +1 2∂ξ kΦ (x, ξ) k2 L2(H,R)⊗ δu(x,t,ω)(dξ) dt = ∂ξmω(x, dt, ξ)

with mω a non-negative entropy defect measure.

Conclusion : The kinetic formulation weak in the space variable x and in ξ (but not in the time variable) of the hyperbolic scalar conservation law with stochastic source term can be written ∀g ∈ C_c∞ Rd× Rξ , ∀0 ≤ s ≤ t < +∞, Z Rd×Rξ χ (ξ, u (x, t)) g (x, ξ) dxdξ − Z Rd×Rξ χ (ξ, u (x, s)) g (x, ξ) dxdξ = Z t s Z Rd×Rξ A0(ξ) .∇x(g (x, ξ)) χ (ξ, u (x, t)) dxdξdr + Z t s Z Rd Z Rξ g (x, ξ) Φ (x, ξ) δu(x,t)(dξ) dxdW (r) +1 2 Z t s Z Rd Z Rξ ∂ξg (x, ξ) kΦ (x, ξ) k2_L₂_(H,R)δu(x,t)(dξ) dxdr − Z t s Z Rd×Rξ ∂ξg (x, ξ) m (dx, dr, dξ)

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or can be written Z Rd×Rξ 1_u(x,t)>ξ× g (x, ξ) dxdξ − Z Rd×Rξ 1_u(x,s)>ξ× g (x, ξ) dxdξ = Z t s Z Rd×Rξ 1u(x,t)>ξ× A 0 (ξ) .∇x(g (x, ξ)) dxdξdr + Z t s Z Rd Z Rξ g (x, ξ) Φ (x, ξ) δ_u(x,t)(dξ) dxdW (r) +1 2 Z t s Z Rd Z Rξ ∂ξg (x, ξ) kΦ (x, ξ) k2L2(H,R)δu(x,t)(dξ) dxdr − Z t s Z Rd×Rξ ∂ξg (x, ξ) m (dx, dr, dξ)

6 Appendix: equivalence between Kruzkov entropies and

convex entropies in the definition of entropy solutions of

deterministic hyperbolic scalar conservation laws

Definition 6.1. We call ’Kruzkov entropy/Kruzkov entropy flux’ pair, any couple ηξφξ with ξ ∈ R defined by

ηξ: u ∈ R 7→ ηξ(u) = |u − ξ| and

φξ: u ∈ R 7→ φξ(u) = sgn(u − ξ) × (A(u) − A(ξ)) .

Remark 6.1. The weak derivative of φξ (defined for almost every u ∈ R) is : φ0_ξ(u) = sgn(u − ξ)A0(u) = (u 7→ |u − ξ|)0A0(u)

Proposition 6.1. If the entropy solution of the following Cauchy problem (6.1) verifies the entropy inequality (6.2) for all couples ’Kruzkov entropy/Kruzkov entropy flux’, then it also verifies the entropy inequality (6.2) for all couples entropy/entropy flux

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Proof of Proposition 6.1 Let u ∈ C R+; L1_loc(Rd) ∩ L∞

Rd× R+; R be the en-tropy solution in the Kruzkov sense of the Cauchy problem

∂tu (x, t) + divx(A (u (x, t))) = G (x, t, u (x, t)) , ∀ (x, t) ∈ Rd× R+∗ u (x, 0) = u0(x) , ∀x ∈ Rd, u0 ∈ L1(Rd) ∩ L∞(Rd),

(6.1)

that is the unique u ∈ C R+; L1_loc(Rd) ∩ L∞ Rd× R+; R such that ∀ξ ∈ R, ∀ϕ ∈ Cc∞ Rd× R+; R+ : Z Rd×R+ ηξ(u(x, t)) ∂tϕ (x, t) + φξ(u (x, t)) .∇ϕ (x, t) ! dxdt + Z Rd ηξ(u0(x)) ϕ (x, 0) dx − Z Rd×R+ η0_ξ(u (x, t)) G (x, t, u(x, t)) ϕ (x, t) dx ≥ 0. (6.2)

where ηξ is a Krukov entropy and φξ the associated Kruzkov entropy flux.

As we have the equality

Z Rd×R+ E × ∂tϕ (x, t) + F.∇ϕ (x, t) ! dxdt + Z Rd Eϕ (x, 0) dx − Z Rd×R+ 0 × G (x, t, u(x, t)) ϕ (x, t) dx = 0 (6.3)

for all couples (E, F ) ∈ R2, the inequality (6.2) is still true for the couples entropy/entropy flux (ηn, φn) defined ∀n ∈ N by

ηn(u) = |u − n| − n et φn(u) = sgn(u − n) × (A(u) − A(n)) − A(n), ∀u ∈ R. For all fixed u ∈ R,

lim

n→+∞ηn(u) = −u, n→+∞lim η

0

n(u) = −1 and _n→+∞lim φn(u) = −A(u),

hence

lim

n→+∞ηn(u(x, t)) = −u(x, t), n→+∞lim η

0

n(u(x, t)) = −1 and lim

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As u ∈ L∞(Rd× R+_{; R), A ∈ C}2_{(R; R}d_{) et G ∈ L}∞

(Rd× R+_{× R; R), we can apply} the dominated convergence theorem to each term of the left-hand side of the inequality (6.2), to obtain ∀ϕ ∈ C_c∞ Rd× R+; R+ Z Rd×R+ −u(x, t)∂_tϕ (x, t) − A (u (x, t)) .∇ϕ (x, t) ! dxdt + Z Rd −u0(x) ϕ (x, 0) dx + Z Rd×R+ G (x, t, u(x, t)) ϕ (x, t) dx ≥ 0.

The same method applied to the couples (ηn, φn) defined ∀n ∈ N by

ηn(u) = |u − (−n)| − n et φn(u) = sgn (u − (−n)) × (A(u) − A(−n)) + A(−n), ∀u ∈ R, shows that ∀ϕ ∈ C_c∞ Rd× R+; R+ Z Rd×R+ u(x, t)∂tϕ (x, t) + A (u (x, t)) .∇ϕ (x, t) ! dxdt + Z Rd u0(x) ϕ (x, 0) dx − Z Rd×R+ G (x, t, u(x, t)) ϕ (x, t) dx ≥ 0.

The multiplication by a positive constant doesn’t change the sense of the two last in-equalities thus, using the equality (6.3), we can say that inequality (6.2) is still true for the couples entropy/entropy ηa,E/φa,F defined by

ηa,E(u) = au + E, φa,F(u) = aA(u) + F, ∀u, a, E, F ∈ R.

Remark 6.2. K ≥ 0 and −K ≥ 0 imply K = 0. It proves that for the couples en-tropy/entropy flux ηa,E/φa,F, we have an equality to 0 instead of an inequality in (6.2).

Proposition 6.2. If the entropy solution of the Cauchy problem (6.1) verifies the en-tropy inequality (6.2) for all couples ’Kruzkov entropy/Kruzkov entropy flux’, then it also verifies the entropy inequality (6.2) for all couples entropy/entropy flux

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Proof of Proposition 6.2. η : R 7→ R being convex, its first and second deriva-tive in the classical sense are defined almost everywhere (and equal to their weak derivative), its weak second derivative is non-negative almost everywhere. Let us in-tegrate the inequality (6.2) against the measure η00(ξ)dξ on [−n, n] ⊂ Rξ, to obtain ∀ξ ∈ R, ∀ϕ ∈ Cc∞ Rd× R+; R+ Z [−n,n] η00(ξ) Z Rd×R+ |u(x, t) − ξ| ∂_tϕ (x, t) + sgn (u (x, t) − ξ) (A(u(x, t)) − A(ξ)) .∇ϕ (x, t) ! dxdtdξ + Z [−n,n] η00(ξ) Z Rd |u0(x) − ξ| ϕ (x, 0) dxdξ − Z [−n,n] η00(ξ) Z Rd×R+ sgn (u (x, t) − ξ) G (x, t, u(x, t)) ϕ (x, t) dxdtdξ ≥ 0. (6.4)

Some intermediate computations : let us fix u ∈ R, then there exists n ∈ N such that u ∈ [−n, n]. Denoting η0 the right-derivative of η, we have

Z n −n η00(ξ) |u − ξ| dξ = Z u −n η00(ξ) (u − ξ) dξ + Z n u η00(ξ) (ξ − u) dξ = −η0(−n) (u + n) + Z u −n η0(ξ)dξ + η0(n) (n − u) − Z n u η0(ξ)dξ = −η0(−n) (u + n) + η(u) − η(−n) + η0(n) (n − u) − η(n) + η(u) = 2η(u) + u − η0(−n) − η0(n) − η0(−n)n − η(−n) + η0(n)n − η(n) because η0 is monotonic and ξ ∈ R 7→ |u − ξ| is continuous (see [HP96] p 63).

Z n −n η00(ξ) sgn (u − ξ) (A(u) − A(ξ)) dξ = Z u −n η00(ξ) (A(u) − A(ξ)) dξ − Z n u η00(ξ) (A(u) − A(ξ)) dξ = −η0(−n) (A(u) − A(−n)) + Z u −n η0(ξ)A0(ξ)dξ − η0(n) (A(u) − A(n)) − Z n u η0(ξ)A0(ξ)dξ

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having chosen for example φ(u) = Z u 0 η0(ξ)A0(ξ)dξ, ∀u ∈ R. Z n −n η00(ξ) sgn (u − ξ) dξ = Z u −n η00(ξ)dξ − Z n u η00(ξ)dξ = 2η0(u) − η0(−n) − η0(n).

The entropy solution u in the sense of Kruzkov being essentially bounded, let us choose n ∈ N such that n ≥ kukL∞_(Rd_×R+₎. Let us use the remark 6.2 about the couples entropy/entropy flux, ηa,E/φa,F with a = −η

0

(−n) − η0(n) and the Fubini theorem in the inequality (6.4) to obtain ∀ϕ ∈ C_c∞ Rd× R+; R+ :

Z Rd×R+ η (u(x, t)) ∂tϕ (x, t) + φ (u (x, t)) .∇ϕ (x, t) ! dxdt + Z Rd η (u0(x)) ϕ (x, 0) dx − Z Rd×R+ η0(u (x, t)) G (x, t, u(x, t)) ϕ (x, t) dxdt ≥ 0.

for all couples η_{φ with η : R → R convex and φ defined ∀u ∈ R by} φ(u) =

Z u 0

η0(ξ)A0(ξ)dξ.

7 Conclusion

Following the thought of Alain Connes, who thinks that the physical world is part of a much larger world: the mathematical world, it is natural to think, to imagine or even to assert that physical entropy is only a particular case of the mathematical entropy which, in my opinion, also follows the Clausius Theorem: the mathematical entropy is monotonic over time!

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Clausius theorem for conservation laws

Clausius Theorem for Hyperbolic Scalar Conservation Laws

Contents

1

Introduction

2

Case of homogeneous hyperbolic scalar conservation laws

3

Case of the hyperbolic scalar conservation law with

de-terministic source term

4

Case of the hyperbolic scalar conservation law with

de-terministic flux and source term depending on the space

and time variables

5

Case of the hyperbolic scalar conservation law with

stochas-tic source term

6

Appendix: equivalence between Kruzkov entropies and

convex entropies in the definition of entropy solutions of

deterministic hyperbolic scalar conservation laws

7

Conclusion

References