2006 International Conference in Honor of Jacqueline Fleckinger. Electronic Journal of Differential Equations, Conference 16, 2007, pp. 15–28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

NONLINEAR MULTIDIMENSIONAL PARABOLIC-HYPERBOLIC EQUATIONS

GLORIA AGUILAR, LAURENT LÉVI, MONIQUE MADAUNE-TORT

Dedicated to Jacqueline Fleckinger on the occasion of an international conference in her honor

Abstract. This paper deals with the coupling of a quasilinear parabolic problem with a first order hyperbolic one in a multidimensional bounded domain Ω. In a region Ω_{p} a diffusion-advection-reaction type equation is set while in the complementary Ω_{h} ≡ Ω\Ω_{p}, only advection-reaction terms are taken into account. Suitable transmission conditions at the interface ∂Ω_{p} ∩ ∂Ω_{h} are required. We find a weak solution characterized by an entropy inequality on the whole domain.

1. Introduction

We are interested in a coupling of a quasilinear parabolic equation with an hyperbolic first-order one in a bounded domain Ω of ℝ^{n}, n ≥ 1. The main motivation for considering this problem is the study of infiltration processes in an heterogeneous porous media. For instance, in a stratified subsoil made up of layers with different geological characteristics, the effects of diffusivity may be negligible in some layers. Such a coupled problem occurs also in fluid-dynamical theory for viscouscompressible flows around a rigid profile so that near this profile the viscosity effects have to be taken into account while at a distance they can be neglected. Another example arises in heat transfer studies as mentioned in [6].

We consider the case of two layers, that is sufficient. Then, the geometrical configuration is such that:

\overline{Ω} = \overline{Ω_{h}} ∪ \overline{Ω_{p}}; Ω_{h} and Ω_{p} are two disjoint bounded domains with Lipschitz boundaries denoted by Γ_{l} = ∂Ω_{l}, l ∈ {h, p} and Γ_{hp} = Γ_{h} ∩ Γ_{p}. In addition we set Q =]0, T [×Ω and for l in {h, p}, Q_{l} =]0, T [×Ω_{l}, Σ_{l} =]0, T [×Γ_{l}. Now, for q in [0, n+1], H^{q} is the q-dimensional Hausdorff measure over ℝ^{n+1} and for l in {h, p}, ν_{l} is the outward normal unit vector defined H^{n}-a.e. on Σ_{l}. So the interface, denoted by Σ_{hp} =]0, T [×Γ_{hp}, is such that H^{n}(\overline{Σ_{hp}} ∩ (\overline{Σ_{l}\Σ_{hp}})) = 0.

2000 Mathematics Subject Classification. 35F25, 35K65.

Key words and phrases. Coupling problem; degenerate parabolic-hyperbolic equation;

entropy solution.

\begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos.

This work was supported by CTP, the Pyrenean Work Community.

Published May 15, 2007.

Now, due to a combination of conservation laws and Darcy’s law, the physical model is described as follows:

For any positive and finite real T, find a measurable and bounded function u on Q such that,

∂_{t}u − ∑_{i=1}^{n} ∂_{x_{i}}(f(u)∂_{x_{i}}P) + g(t, x, u) = 0 in Q_{h},  (1.1)

∂_{t}u − ∑_{i=1}^{n} ∂_{x_{i}}(f(u)∂_{x_{i}}P) + g(t, x, u) = Δϕ(u) in Q_{p},  (1.2)
u = 0 on ]0, T [×∂Ω,  (1.3)
u(0, .) = u_{0} on Ω.  (1.4)

Then, suitable conditions on u across the interface Σ_{hp} must be added. As for the linear problem studied by F. Gastaldi and al. in [6] or for the one dimensional nonlinear problem studied by G. Aguilar and al. in [2], these transmission conditions include the continuity property of the flux through the interface formally written here as

−f(u)∇P.ν_{h} = (∇ϕ(u) + f(u)∇P).ν_{p} on Σ_{hp}.  (1.5)

Let us mention that this problem has already been studied by the authors in [1] for a nondecreasing flux function f when ∇P.ν_{h} ≤ 0 a.e. on Γ_{hp}. Here, we still consider a nondecreasing flux function f, but we give an existence and uniqueness result holding even when ∇P.ν_{h} ≥ 0 a.e. on Γ_{hp}.

1.1. Assumptions and notation. The pressure P is a known stationary function belonging to W^{2,+∞}(Ω) and such that ΔP = 0 which is not restrictive as soon as (1.1) and (1.2) include some reaction terms. In addition,

∇P.ν_{h} has a constant sign all along Γ_{hp}.  (1.6)

The reaction function g belongs to W^{1,+∞}(]0, T [×Ω × ℝ) and we set

M′_{g} = ess sup_{(t,x,u)∈]0,T [×Ω×ℝ} |∂_{u}g(t, x, u)| and M_{0} = ess sup_{]0,T [×Ω} |g_{h}(t, x, 0)|.

The initial data u_{0} belongs to L^{∞}(Ω). Thus we can define the nondecreasing timedepending function

M : t ∈ [0, T] → M(t) = ∥u_{0}∥_{L^{∞}(Ω)}e^{M′_{g} t} + M_{0} \frac{e^{M′_{g} t} − 1}{M′_{g}}.  (1.7)

To simplify we write M = M(T).

Now, we assume local hypotheses on f and ϕ.

(i) The flux function f is a nondecreasing Lipschitzian function on [−M, M] with constant M′_{f} and such that f(0) = 0. To express the boundary conditions on the frontier of the hyperbolic area, we introduce the nonnegative function F defined on [−M, M]^{3} by

F(a, b, c) = \frac{1}{2}\{|f(a) − f(b)| − |f(c) − f(b)| + |f(a) − f(c)|\}.  (1.8)

(ii) ϕ is an increasing Lipschitzian function on [−M, M] such that ϕ^{−1} is Hölder continuous and ϕ(0) = 0.

(iii) f ∘ ϕ^{−1} is Hölder continuous with exponent θ in [1/2, +∞[ that is there exists a positive constant C such that

∀(x, y) ∈ [−M, M]^{2}, |(f ∘ ϕ^{−1})(x) − (f ∘ ϕ^{−1})(y)| ≤ C|x − y|^{θ}.  (1.9)

Remark 1.1. The monotonicity of f and the condition (1.6) involve that

if a.e. on Γ_{hp}, ∇P.ν_{h} ≤ 0, then Σ_{hp} is included in the set of outward characteristics for the first-order operator in the hyperbolic domain and along the interface the information is leaving the hyperbolic domain. This property has been used in [1] to split the problem by first considering the behavior of a solution in the hyperbolic area and then in the parabolic one;

if a.e. on Γ_{hp}, ∇P.ν_{h} ≥ 0, then Σ_{hp} is included in the set of inward characteristics for the first-order operator in the hyperbolic domain and along the interface the information is now entering the hyperbolic domain. This property will also be used to first consider the behavior of a solution in the parabolic area and then in the hyperbolic one.

At last, for any positive real μ, sgn_{μ} is the Lipschitzian approximation of the function sgn defined by:

∀x ∈ [0, +∞[, sgn_{μ}(x) = min(\frac{x}{μ}, 1), sgn_{μ}(−x) = − sgn_{μ}(x).  (1.10)

For the rest of this work, σ (resp. σ̄) denotes the variable on Σ_{l} (resp. Γ_{l}), l ∈ {h, hp, p}. This way, for any t of [0, T], σ = (t, σ̄).

1.2. Functional spaces. In the sequel, W (0, T) is the Hilbert space

W (0, T) ≡ {v ∈ L^{2}(0, T; H_{0}^{1}(Ω)); ∂_{t}v ∈ L^{2}(0, T; H^{−1}(Ω))}

equipped with the norm ∥w∥_{W (0,T)} = (∥∂_{t}w∥_{L^{2}(0,T;H^{−1}(Ω))}^{2} + ∥∇w∥_{L^{2}(Q)^{n}}^{2})^{1/2} and V is the Hilbert space

V = {v ∈ H^{1}(Ω_{p}), v = 0 a.e. on Γ_{p}\Γ_{hp}}

equipped with the norm ∥v∥_{V} = ∥∇v∥_{L^{2}(Ω_{p})^{n}}.

We denote ⟨., .⟩ the pairing between V and V′.

At last BV (O) with O = Ω_{h} or O = Q_{h} is the space of summable functions v with bounded total variation on O where the total variation is given by

T V_{O}(v) = sup { ∫_{O} v(x)divΦ(x)dx, Φ ∈ (D(O))^{p}, ∥Φ∥_{(L^{∞}(O))^{p}} ≤ 1}

where p is the dimension of the open set O. Moreover, we denote by γv the trace on Γ_{hp} or Σ_{hp} of a function v belonging to BV (O).

The concept of a weak entropy solution to (1.1)-(1.5) is defined in Section 2 through an entropy inequality in the whole domain, the boundary conditions on the outer frontier of the hyperbolic area being expressed by referring to [8]. Then, we show some properties of such a solution in the hyperbolic area and in the parabolic one. The proof of the existence result is given in Section 3 and the uniqueness property is established in Section 4.

2. The Entropy Formulation

2.1. Weak entropy solution. The definition of a weak entropy solution to (1.1)- (1.5) has to include an entropy criterion in Q_{h} where the quasilinear first-order hyperbolic operator is set. Problem (1.1)-(1.5) can be viewed as an evolutional problem for a quasilinear parabolic equation that strongly degenerates in a fixed subdomain Q_{h} of Q. As in [2] or [1], we propose a weak formulation through a global entropy inequality in the whole Q, the latter giving rise to a variational equality in the parabolic domain and to an entropy inequality in the hyperbolic one so as to ensure the uniqueness.

Definition 2.1. A function u is a weak entropy solution to the coupling problem (1.1)-(1.5) if u ∈ L^{∞}(Q), ϕ(u) ∈ L^{2}(0, T; V) and for all φ ∈ D(Q), φ ≥ 0, for all k ∈ ℝ,

\begin{matrix}∫_{Q} |u − k|∂_{t}φ dx dt − ∫_{Q_{p}} ∇|ϕ(u) − ϕ(k)|.∇φ dx dt \\ − ∫_{Q} |f(u) − f(k)|∇P.∇φ dx dt − ∫_{Q} sgn(u − k)g(t, x, u)φ dx dt ≥ 0, \end{matrix} (2.1)

for all ζ ∈ L^{1}(Σ_{h}\Σ_{hp}), ζ ≥ 0, for all k ∈ ℝ,

ess lim_{τ→0^{−}} ∫_{Σ_{h}\Σ_{hp}} F(u(σ + τν_{h}), 0, k)∇P (σ̄).ν_{h}ζdH^{n} ≤ 0,  (2.2)
ess lim_{t→0^{+}} ∫_{Ω} |u(t, x) − u_{0}(x)|dx = 0.  (2.3)

2.2. An entropy inequality in the hyperbolic zone. We derive from (2.1) and (2.2) an entropy inequality in the hyperbolic domain.

Proposition 2.2. Let u be a weak entropy solution to the coupling problem (1.1)- (1.5). Then for any real k and any φ of D(]0, T [×ℝ^{n}), φ ≥ 0,

− ∫_{Q_{h}}(|u − k|∂_{t}φ − |f(u) − f(k)|∇P.∇φ − sgn(u − k)g(t, x, u)φ) dx dt

≤ ess lim_{τ→0^{−}} ∫_{Σ_{hp}} |f(u(σ + τν_{h})) − f(k)|∇P (σ̄).ν_{h}φ(σ)dH^{n}

+ ∫_{Σ_{h}\Σ_{hp}} |f(k)|∇P (σ̄).ν_{h}φ(σ)dH^{n}

− ess lim_{τ→0^{−}} ∫_{Σ_{h}\Σ_{hp}} |f(u(σ + τν_{h}))|∇P (σ̄).ν_{h}φ(σ)dH^{n}.

(2.4)

Proof. From (2.1) it comes that for φ in D(Q_{h}), φ ≥ 0,

∫_{Q_{h}}(|u − k|∂_{t}φ − |f(u) − f(k)|∇P.∇φ − sgn(u − k)g(t, x, u)φ) dx dt ≥ 0. (2.5)

First, by referring to F.Otto’s works in [8], we deduce from (2.5) that, for any real k and any β in L^{1}(Σ_{h}), the following limit exists:

ess lim_{τ→0^{−}} ∫_{Σ_{h}} |f(u(σ + τν_{h})) − f(k)|∇P (σ̄).ν_{h}β(σ)dH^{n}.  (2.6)

Then, it results from (2.5) (see [8]) that, for any real k and any φ in D(]0, T [×ℝ^{n}), φ ≥ 0,

− ∫_{Q_{h}}(|u − k|∂_{t}φ − |f(u) − f(k)|∇P.∇φ − sgn(u − k)g(t, x, u)φ) dx dt

≤ ess lim_{τ→0^{−}} ∫_{Σ_{h}} |f(u(σ + τν_{h})) − f(k)|∇P (σ̄).ν_{h}φ(σ)dH^{n}.

To conclude we share the frontier of Ω_{h} into Γ_{hp} and Γ_{h}\Γ_{hp} and we use boundary condition (2.2) on Σ_{h}\Σ_{hp}. □

2.3. A variational equality in the parabolic zone. We give now some information on the regularity for ∂_{t}u in Q_{p} and we derive from (2.1) a variational equality satisfied by any weak entropy solution u to the coupling problem (1.1)-(1.5).

Proposition 2.3. Let u be a weak entropy solution to the coupling problem (1.1)- (1.5). Then ∂_{t}u belongs to L^{2}(0, T; V′). Furthermore, for any φ in L^{2}(0, T; V),

\begin{matrix}∫_{0}^{T} ⟨∂_{t}u, φ⟩dt + ∫_{Q_{p}} ∇ϕ(u).∇φ dx dt + ∫_{Q_{p}} f(u)∇P.∇φ dx dt \\ + ∫_{Q_{p}} g(t, x, u)φ dx dt + ess lim_{τ→0^{−}} ∫_{Σ_{hp}} f(u(σ + τν_{h}))∇P (σ̄).ν_{h}φdH^{n} = 0. \end{matrix} (2.7)

Remark 2.4. This proposition is proved in [1, Proposition 3.4] independently of any condition on the hyperbolic characteristics on Σ_{hp}.

3. The Existence Result

In this section, we will prove the existence of a weak entropy solution.

Theorem 3.1. The coupling problem (1.1)–(1.5) has at least a weak entropy solution.

To construct a weak entropy solution to Problem (1.1)-(1.5), we work successively in the hyperbolic domain and in the parabolic one or vice-versa. Indeed, thanks to Remark 1.1, when a.e. on Γ_{hp} ∇P.ν_{h} ≤ 0, we can begin by working in the hyperbolic zone while, when a.e. on Γ_{hp} ∇P.ν_{h} ≥ 0, we can begin by working in the parabolic area.

3.1. Waves going from Q_{h} to Q_{p}. In this section we suppose that, a.e. on Γ_{hp}, ∇P.ν_{h} ≤ 0. The existence of a weak entropy solution to Problem (1.1)-(1.5) is already proved in [1] by the viscosity method. Here, we give a different proof of this result.

First, thanks to [8], there exists one and only one function w_{h} in L^{∞}(Q_{h}) such that for all φ ∈ D(Q_{h}), φ ≥ 0, for all k ∈ ℝ,

∫_{Q_{h}}(|w_{h} −k|∂_{t}φ−|f(w_{h})−f(k)|∇P.∇φ−sgn(w_{h} −k)g(t, x, w_{h})φ) dx dt ≥ 0, (3.1)

for all ζ ∈ L^{1}(Σ_{h}), ζ ≥ 0, for all k ∈ ℝ,

ess lim_{τ→0^{−}} ∫_{Σ_{h}} F(w_{h}(σ + τν_{h}), 0, k)∇P (σ̄).ν_{h}ζdH^{n} ≤ 0,  (3.2)

ess lim_{t→0^{+}} ∫_{Ω_{h}} |w_{h}(t, x) − u_{0}(x)|dx = 0.  (3.3)

Then, thanks to [5], there exists one and only one function w_{p} in L^{∞}(Q_{p}) such that ϕ(w_{p}) ∈ L^{2}(0, T; V), ∂_{t}w_{p} ∈ L^{2}(0, T; V′) and for all φ ∈ L^{2}(0, T; V),

\begin{matrix}∫_{0}^{T} ⟨∂_{t}w_{p}, φ⟩dt + ∫_{Q_{p}} ∇ϕ(w_{p}).∇φ dx dt + ∫_{Q_{p}} f(w_{p})∇P.∇φ dx dt \\ + ∫_{Q_{p}} g(t, x, w_{p})φ dx dt + ess lim_{τ→0^{−}} ∫_{Σ_{hp}} f(w_{h}(σ + τν_{h}))∇P (σ̄).ν_{h}φdH^{n} = 0, \end{matrix} (3.4)

ess lim_{t→0^{+}} ∫_{Ω_{p}} |w_{p}(t, x) − u_{0}(x)|dx = 0.  (3.5)

Indeed the mapping

φ ⟼ − ess lim_{τ→0^{−}} ∫_{Σ_{hp}} f(w_{h}(σ + τν_{h}))∇P (σ̄).ν_{h}φdH^{n}

belongs to L^{∞}(0, T; V′). Therefore to prove Theorem 3.1, we are going to establish the following lemma.

Lemma 3.2. Let u be defined by u = w_{h} in Q_{h} and u = w_{p} in Q_{p}. Then u is a weak entropy solution to the coupling problem (1.1)-(1.5).

Moreover if u_{0_{|Ω_{h}}} belongs to BV (Ω_{h}), then u_{|Q_{h}} belongs to BV (Q_{h}) and

ess lim_{τ→0^{−}} ∫_{Σ_{hp}} |f(u(σ + τν_{h})) − f(γu(σ))|dH^{n} = 0

where γu(σ) is the trace on Σ_{hp} in the BV-sense of the BV-function u_{|Q_{h}}.

Proof. First note that u ∈ L^{∞}(Q) and ϕ(u) ∈ L^{2}(0, T; V). Let φ be in D(Q), φ ≥ 0 and let k be in ℝ. As in the proof of Proposition 2.2, we derive from (3.1) the following inequality

∫_{Q_{h}}(|w_{h} − k|∂_{t}φ − |f(w_{h}) − f(k)|∇P.∇φ − sgn(w_{h} − k)g(t, x, w_{h})φ) dx dt

≥ − ess lim_{τ→0^{−}} ∫_{Σ_{hp}} |f(w_{h}(σ + τν_{h})) − f(k)|∇P (σ̄).ν_{h}φ(σ)dH^{n}.

(3.6)

Then, we choose in (3.4) the test-function φ sgn_{μ}(ϕ(w_{p}) − ϕ(k)). It follows:

− ∫_{0}^{T} ⟨∂_{t}w_{p}, sgn_{μ}(ϕ(w_{p}) − ϕ(k))φ⟩dt
− ∫_{Q_{p}} sgn_{μ}(ϕ(w_{p}) − ϕ(k)) ∇(ϕ(w_{p}) − ϕ(k)).∇φ dx dt
− ∫_{Q_{p}} sgn_{μ}(ϕ(w_{p}) − ϕ(k))(f(w_{p}) − f(k))∇P.∇φ dx dt
− ∫_{Q_{p}} g(t, x, w_{p}) sgn_{μ}(ϕ(w_{p}) − ϕ(k))φ dx dt
= ∫_{Q_{p}} sgn ′_{μ}(ϕ(w_{p}) − ϕ(k)) ∇(ϕ(w_{p}) − ϕ(k)).∇(ϕ(w_{p}) − ϕ(k))φ dx dt

+ ∫_{Q_{p}} sgn ′_{μ}(ϕ(w_{p}) − ϕ(k))(f(w_{p}) − f(k))∇P.∇(ϕ(w_{p}) − ϕ(k))φ dx dt

+ ess lim_{τ→0^{−}} ∫_{Σ_{hp}} f(w_{h}(σ + τν_{h}))∇P.ν_{h} sgn_{μ}(ϕ(w_{p}) − ϕ(k))φdH^{n}

+ ∫_{Σ_{hp}} sgn_{μ}(ϕ(w_{p}) − ϕ(k))f(k)∇P.ν_{p}φdH^{n}.  (3.7)

Thanks to (1.9) and to the Cauchy-Scharwz inequality there exists a positive constant C such that :

∫_{Q_{p}} sgn ′_{μ}(ϕ(w_{p}) − ϕ(k)) ∇(ϕ(w_{p}) − ϕ(k)).∇(ϕ(w_{p}) − ϕ(k))φ dx dt
+ ∫_{Q_{p}} sgn ′_{μ}(ϕ(w_{p}) − ϕ(k))(f(w_{p}) − f(k))∇P.∇(ϕ(w_{p}) − ϕ(k))φ dx dt
≥ −C ∫_{Q_{p}} |ϕ(w_{p}) − ϕ(k)|^{2θ} sgn ′_{μ}(ϕ(w_{p}) − ϕ(k))φ dx dt,

and the term in the right-hand side goes to 0 with μ as θ ≥ 1/2 thanks to the Lebesgue’s bounded convergence theorem.

In the first term of (3.7), we use an integration by parts formula based on a convexity inequality (see e.g. [5], the Mignot-Bamberger Lemma) to obtain

− ∫_{0}^{T} ⟨∂_{t}w_{p}, sgn_{μ}(ϕ(w_{p}) − ϕ(k))φ⟩dt = ∫_{Q_{p}} ( ∫_{k}^{w_{p}} sgn_{μ}(ϕ(r) − ϕ(k))dr)∂_{t}φ dx dt.

Therefore, we are able to pass to the limit in (3.7) when μ approaches 0^{+} in all the integrals over Q_{p}. For the one on Σ_{hp}, we argue from (3.1) and [8] that (2.6) is valid for w_{h}. Therefore there exists θ in L^{∞}(Σ_{hp}) such that for any β in L^{1}(Σ_{hp}),

ess lim_{τ→0^{−}} ∫_{Σ_{hp}} f(w_{h}(σ + τν_{h}))∇P (σ̄).ν_{h}β(σ)dH^{n} = ∫_{Σ_{hp}} θ(σ)β(σ)dH^{n}.  (3.8)

Therefore, we can use that

lim_{μ→0^{+}} ∫_{Σ_{hp}} θ(σ) sgn_{μ}(ϕ(w_{p}) − ϕ(k))φdH^{n} = ∫_{Σ_{hp}} θ(σ) sgn(ϕ(w_{p}) − ϕ(k))φdH^{n}.

After all, we obtain

∫_{Q_{p}}(|w_{p} − k|∂_{t}φ − |f(w_{p}) − f(k)|∇P.∇φ − sgn(w_{p} − k)g(t, x, w_{p})φ) dx dt

− ∫_{Q_{p}} ∇|ϕ(w_{p}) − ϕ(k)|.∇φ dx dt

≥ ess lim_{τ→0^{−}} ∫_{Σ_{hp}}(f(w_{h}(σ + τν_{h})) − f(k)) sgn(w_{p}(σ) − k)∇P.ν_{h}φdH^{n}

(3.9)

where in (3.9), w_{p}(σ) is defined as ϕ^{−1}(ϕ(w_{p}(σ))) and belongs to L^{∞}(Σ_{hp}). By adding the inequalities (3.6) and (3.9), we obtain

∫_{Q} |u − k|∂_{t}φ dx dt − ∫_{Q_{p}} ∇|ϕ(u) − ϕ(k)|.∇φ dx dt
− ∫_{Q} |f(u) − f(k)|∇P.∇φ dx dt − ∫_{Q} sgn(u − k)g(t, x, u)φ dx dt

≥ ess lim_{τ→0^{−}} ∫_{Σ_{hp}}(f(w_{h}(σ + τν_{h})) − f(k)) sgn(w_{p}(σ) − k)∇P (σ̄).ν_{h}φ(σ)dH^{n}

− ess lim_{τ→0^{−}} ∫_{Σ_{hp}} |f(w_{h}(σ + τν_{h})) − f(k)|∇P (σ̄).ν_{h}φ(σ)dH^{n}.

Now by using the condition ∇P.ν_{h} ≤ 0 a.e. on Γ_{hp}, we derive that u satisfies Inequality (2.1).

At last, thanks to (3.2), (3.3) and (3.5), we can conclude that u is a weak entropy solution to the coupling problem (1.1)-(1.5).

Now, if u_{0|Ω_{h}} belongs to BV (Ω_{h}), it results from [3] and [8] that w_{h|Q_{h}} belongs to BV (Q_{h}). Therefore u_{|Q_{h}} belongs to BV (Q_{h}) and thanks to the properties of the trace operator from BV (Q_{h}) into L^{1}(Σ_{h})

ess lim_{τ→0^{−}} ∫_{Σ_{h}} |f(u(σ + τν_{h})) − f(γu(σ))|dH^{n} = 0

where γu(σ) is the trace on Σ_{h} in the BV-sense of the BV-function u_{|Q_{h}}. □

3.2. Waves going from Q_{p} to Q_{h}. In this section we suppose that a.e. on Γ_{hp}, ∇P.ν_{h} ≥ 0.

First, thanks to [5], there exists one and only one function w_{p} in L^{∞}(Q_{p}) such that ϕ(w_{p}) ∈ L^{2}(0, T; V), ∂_{t}w_{p} ∈ L^{2}(0, T; V′) and for all φ ∈ L^{2}(0, T; V),

\begin{matrix}∫_{0}^{T} ⟨∂_{t}w_{p}, φ⟩dt + ∫_{Q_{p}} ∇ϕ(w_{p}).∇φ dx dt + ∫_{Q_{p}} f(w_{p})∇P.∇φ dx dt \\ + ∫_{Q_{p}} g(t, x, w_{p})φ dx dt + ∫_{Σ_{hp}} f(w_{p}(σ))∇P (σ̄).ν_{h}φdH^{n} = 0, \end{matrix}  (3.10)

ess lim_{t→0^{+}} ∫_{Ω_{p}} |w_{p}(t, x) − u_{0}(x)|dx = 0.  (3.11)

In (3.10), w_{p}(σ) is defined as ϕ^{−1}(ϕ(w_{p}(σ))) and belongs to L^{∞}(Σ_{hp}).

Then, thanks to [8], there exists one and only one function w_{h} in L^{∞}(Q_{h}) such that for all φ ∈ D(Q_{h}), φ ≥ 0, for all k ∈ ℝ,

∫_{Q_{h}}(|w_{h}−k|∂_{t}φ−|f(w_{h})−f(k)|∇P.∇φ−sgn(w_{h}−k)g(t, x, w_{h})φ) dx dt ≥ 0; (3.12)

for all ζ ∈ L^{1}(Σ_{h}), ζ ≥ 0, for all k ∈ ℝ,

ess lim_{τ→0^{−}} ∫_{Σ_{h}\Σ_{hp}} F(w_{h}(σ + τν_{h}), 0, k)∇P (σ̄).ν_{h}ζdH^{n} ≤ 0,  (3.13)
ess lim_{τ→0^{−}} ∫_{Σ_{hp}} F(w_{h}(σ + τν_{h}), w_{p}(σ), k)∇P (σ̄).ν_{h}ζdH^{n} ≤ 0,  (3.14)
ess lim_{t→0^{+}} ∫_{Ω_{h}} |w_{h}(t, x) − u_{0}(x)|dx = 0.  (3.15)

Therefore to prove Theorem 3.1, we establish the following lemma.

Lemma 3.3. Let u be defined by u = w_{h} in Q_{h} and u = w_{p} in Q_{p}. Then u is a weak entropy solution to the coupling problem (1.1)-(1.5).

Moreover

ess lim_{τ→0^{−}} ∫_{Σ_{hp}} |f(u(σ + τν_{h})) − f(u(σ))|dH^{n} = 0

where u(σ) is defined as ϕ^{−1}(ϕ(u(σ))).

Proof. First u ∈ L^{∞}(Q) and ϕ(u) ∈ L^{2}(0, T; V). Now let φ be in D(Q), φ ≥ 0 and let k be in ℝ.

Following the proof of (3.9) in Lemma 3.2, we deduce from (3.10) that

\begin{matrix}∫_{Q_{p}}(|w_{p} − k|∂_{t}φ − |f(w_{p}) − f(k)|∇P.∇φ − sgn(w_{p} − k)g(t, x, w_{p})φ) dx dt \\ − ∫_{Q_{p}} ∇|ϕ(w_{p}) − ϕ(k)|.∇φ dx dt \\ ≥ ∫_{Σ_{hp}} |f(w_{p}(σ)) − f(k)|∇P (σ̄).ν_{h}φ(σ)dH^{n}. \end{matrix} (3.16)

Moreover, Inequality (3.6) is still satisfied by w_{h}. By adding the inequalities (3.6) and (3.16) we obtain

∫_{Q} |u − k|∂_{t}φ dx dt − ∫_{Q_{p}} ∇|ϕ(u) − ϕ(k)|.∇φ dx dt

− ∫_{Q} |f(u) − f(k)|∇P.∇φ dx dt − ∫_{Q} sgn(u − k)g(t, x, u)φ dx dt

≥ ∫_{Σ_{hp}} |f(w_{p}(σ)) − f(k)|∇P (σ̄).ν_{h}φ(σ)dH^{n}

− ess lim_{τ→0^{−}} ∫_{Σ_{hp}} |f(w_{h}(σ + τν_{h})) − f(k)|∇P (σ̄).ν_{h}φ(σ)dH^{n}.

Then, thanks to (3.14) and to the condition ∇P.ν_{h} ≥ 0 a.e. on Γ_{hp}, we obtain that u satisfies Inequality (2.1).

Now, thanks to (3.13), (3.15) and (3.11), we conclude that u is a weak entropy solution to the coupling problem (1.1)-(1.5).

At last, it results from [7] that as Σ_{hp} is included in the set of inward characteristics for the first order operator, the solution w_{h} of Problem (3.12)-(3.15) satisfies

ess lim_{τ→0^{−}} ∫_{Σ_{hp}} |f(w_{h}(σ + τν_{h})) − f(w_{p}(σ))|dH^{n} = 0.

□

4. The Uniqueness Property

We have seen in Lemma 3.2 or Lemma 3.3 that Problem (1.1)-(1.5) has at least a weak entropy solution u for which there exists θ ∈ L^{1}(Σ_{hp}), |θ| ≤ M and

ess lim_{τ→0^{−}} ∫_{Σ_{hp}} |f(u(σ + τν_{h})) − f(θ(σ))|∇P.ν_{h}dH^{n} = 0.  (4.1)

Indeed, when ∇P.ν_{h} ≤ 0 a.e. on Γ_{hp}, as soon as u_{0|Ω_{h}} belongs to BV (Ω_{h}) then (4.1) is satisfied with θ = γu where γu is the trace on Σ_{hp} in the BV-sense of the BV-function u_{|Q_{h}}. When ∇P.ν_{h} ≥ 0 a.e. on Γ_{hp}, (4.1) is satisfied with θ = u where u is defined as ϕ^{−1}(ϕ(u)) and ϕ(u) is the trace on Σ_{hp} of ϕ(u)_{|Q_{p}}.

In this section, we prove the uniqueness property in the class of weak entropy solutions satisfying (4.1). Indeed, we have justified that Problem (1.1)-(1.5) admits such a solution (under the additional hypothesis u_{0|Ω_{h}} belongs to BV (Ω_{h}) when a.e. on Γ_{hp}∇P.ν_{h} ≤ 0).

4.1. Preliminaries. To use the method of doubling variables, we introduce a sequence of mollifiers (W_{δ})_{δ>0} on ℝ^{n+1} defined by

∀δ > 0, ∀r = (t, x) ∈ ℝ^{n+1}, W_{δ}(r) = ϖ_{δ}(t) ∏_{i=1}^{n} ϖ_{δ}(x_{i}),

where (ϖ_{δ})_{δ>0} is a standard sequence of mollifiers on ℝ. We will use classical results on the Lebesgue set of a summable function on Q and a similar property on Σ proved in [9]:

Lemma 4.1. Let v and w be in L^{∞}(Q_{h}) such that (2.5) and (4.1) hold. Then for any continuous function φ on \overline{Q_{h}},

lim_{δ→0^{+}} ∫_{Q_{h}} ∫_{Σ_{h}\Σ_{hp}} |f(v(r))|∇P (σ̄).ν_{h}φ(\frac{σ̃ + r}{2} )W_{δ}(σ̃ − r)dHn ; σ̃dr

= \frac{1}{2} ess lim_{τ→0^{−}} ∫_{Σ_{h}\Σ_{hp}} |f(v(σ + τν_{h}))|∇P (σ̄)ν_{h}φ(σ)dH^{n},

lim_{δ→0^{+}} ∫_{Q_{h}} ess lim_{τ→0^{−}} ∫_{Σ_{h}\Σ_{hp}} |f(v(σ + τν_{h}))|∇P (σ̄).ν_{h}φ(\frac{σ + r̃}{2})W_{δ}(σ − r̃)dH_{σ}^{n}dr̃

= \frac{1}{2} ess lim_{τ→0^{−}} ∫_{Σ_{h}\Σ_{hp}} |f(v(σ + τν_{h}))|∇P (σ̄).ν_{h}φ(σ)dH^{n},

and

lim_{δ→0^{+}} ∫_{Q_{h}} ∫_{Σ_{hp}} |f(θ_{v}(σ)) − f(w(r̃))|∇P (σ̄).ν_{h}φ(\frac{σ + r̃}{2})W_{δ}(σ − r̃)dH_{σ}^{n}dr̃

= \frac{1}{2} ∫_{Σ_{hp}} |f(θ_{v}(σ)) − f(θ_{w}(σ))|∇P (σ̄)ν_{h}φ(σ)dH^{n}

where θ_{v} (resp. θ_{w}) is defined by (4.1) for v (resp. w).

4.2. The uniqueness theorem.

Lemma 4.2. Let u_{1}, u_{2} be two weak solutions to (1.1)-(1.5) for initial data respectively u_{0,1}, u_{0,2} and such that (4.1) holds with f(θ_{i})∇P.ν_{h} = f(u_{i})∇P.ν_{h}, for i = 1, 2, when ∇P.ν_{h} ≥ 0 a.e. on Γ_{hp}. Then, for a.e. t of [0, T],

∫_{Ω} |u_{1}(t, .) − u_{2}(t, .)|dx ≤ e^{M′_{g} t} ∫_{Ω} |u_{0,1} − u_{0,2}|dx.

Theorem 4.3. Let u_{0} be in L^{∞}(Ω). The coupling problem (1.1)-(1.5) admits at most one weak entropy solution u such that (4.1) holds with f(θ)∇P.ν_{h} = f(u)∇P.ν_{h} when ∇P.ν_{h} ≥ 0 a.e. on Γ_{hp}.

Moreover, for initial data u_{0,1} and u_{0,2} in L^{∞}(Ω) the corresponding weak entropy solutions u_{1} and u_{2} to (1.1)-(1.5) are such that for a.e. t of [0, T],

∫_{Ω} |u_{1}(t, .) − u_{2}(t, .)|dx ≤ e^{M′_{g} t} ∫_{Ω} |u_{0,1} − u_{0,2}|dx.

Proof of Lemma 4.2. (i) We first compare the two solutions u_{1} and u_{2} in the parabolic zone. The lack of regularity of the time partial derivative of any weak entropy solution to (1.1)-(1.5) requires a doubling of the time variable.

Therefore, let χ be a nonnegative element of D(0, T). We consider δ a positive real small enough for α_{δ} : (t̃, t) ⟼ α_{δ}(t̃, t) = χ((t + t̃)/2)ϖ_{δ}((t − t̃)/2) to belong to D(]0, T [×]0, T [). Then, for μ > 0, in (2.7) for u_{1} written in variables (t, x) we consider φ(t, x) = sgn_{μ}(ϕ(u_{1})(t, x) − ϕ(u_{2})(t̃, x))α_{δ}(t̃, t) and in (2.7) written in variables (t̃, x) for u_{2}, we consider φ(t̃, x) = − sgn_{μ}(ϕ(u_{1})(t, x)−ϕ(u_{2})(t̃, x))α_{δ}(t̃, t). To simplify the writing, we add a ”tilde” superscript to any function in the t̃ variable. Moreover, thanks to (4.1) we observe that in (2.7), for i = 1, 2,

ess lim_{τ→0^{−}} ∫_{Σ_{hp}} f(u_{i}(σ + τν_{h}))∇P (σ̄).ν_{h}φdH^{n} = ∫_{Σ_{hp}} f(θ_{i}(σ))∇P.ν_{h}φdH^{n}.

Then, by adding up, it comes:

∫_{0}^{T} ∫_{0}^{T} ⟨∂_{t}u_{1} − ∂t̃ũ_{2}, sgn_{μ}(ϕ(u_{1}) − ϕ(ũ_{2}))⟩α_{δ}dtdt̃

+ ∫_{]0,T [×Q_{p}} ∇(ϕ(u_{1}) − ϕ(ũ_{2})).∇ sgn_{μ}(ϕ(u_{1}) − ϕ(ũ_{2})) α_{δ} dx dtdt̃

+ ∫_{]0,T [×Q_{p}}(f(u_{1}) − f(ũ_{2}))∇P.∇ sgn_{μ}(ϕ(u_{1}) − ϕ(ũ_{2})) α_{δ} dx dtdt̃

+ ∫_{]0,T [×Q_{p}}(g(t, x, u_{1}) − g(t̃, x, ũ_{2})) sgn_{μ}(ϕ(u_{1}) − ϕ(ũ_{2})) α_{δ} dx dtdt̃

= − ∫_{0}^{T} ∫_{Σ_{hp}} f(θ_{1}(σ))∇P.ν_{h} sgn_{μ}(ϕ(u_{1}) − ϕ(u_{2}(σ̃)))α_{δ}dH_{σ}^{n}dt̃

+ ∫_{0}^{T} ∫_{Σ_{hp}} f(θ_{2}(σ̃))∇P.ν_{h} sgn_{μ}(ϕ(u_{1}) − ϕ(u_{2}(σ̃)))α_{δ}dHn ; σ̃dt.

(4.2)

In the left-hand side, we use the calculus of the proof of Lemma 3.2. So, we are able to pass to the limit in (4.2) when μ approaches 0^{+}. Therefore,

− ∫_{]0,T [×Q_{p}} |u_{1} − ũ_{2}|(∂_{t}α_{δ} + ∂t̃α_{δ}) dx dtdt̃
≤ ∫_{]0,T [×Q_{p}} |g(t, x, ũ_{2}) − g(t̃, x, ũ_{2})|α_{δ} dx dtdt̃

− ∫_{0}^{T} ∫_{Σ_{hp}}(f(θ_{1}(σ)) − f(θ_{2}(σ̃)))∇P.ν_{h} sgn_{μ}(ϕ(u_{1}) − ϕ(u_{2}(σ̃)))α_{δ}dH_{σ}^{n}dt̃.

Now, we come back to the definition of α_{δ} to express the sum ∂_{t}α_{δ} + ∂t̃α_{δ}. Then we are able to take the limit with respect to δ through the notion of the Lebesgue’s set of a summable function on ]0, T [. Therefore, as g is Lipschitzian, for any χ in D(0, T), χ ≥ 0,

\begin{matrix}− ∫_{Q_{p}} |u_{1} − u_{2}|χ′(t) dx dt \\ ≤ M′_{g} ∫_{Q_{p}} |u_{1} − u_{2}|χ(t) dx dt \\ − ∫_{Σ_{hp}}(f(θ_{1}(σ)) − f(θ_{2}(σ)))∇P.ν_{h} sgn(ϕ(u_{1}) − ϕ(u_{2}))χ(t)dH^{n}. \end{matrix}  (4.3)

(ii) Now, we work in the hyperbolic domain. We use a doubling method for all the variables . Let ψ be such that ψ ≡ χζ where χ is a function in D(0, T), χ ≥ 0, as in Part (i) and ζ is in D(ℝ^{n}) such that: ζ ≥ 0, ζ ≡ 1 on Q_{h}. We consider δ a positive real small enough in order that the mapping (t̃, t) ⟼ χ((t + t̃)/2)w_{δ}((t − t̃)/2) belongs to D(]0, T [×]0, T [). Then, for any positive δ, we define the function Ψ_{δ} in ]0, T [×ℝ^{n}×]0, T [×ℝ^{n} by Ψ_{δ}(r, r̃) = χ((t + t̃)/2)ζ((x + x̃)/2)W_{δ}(r − r̃).

Due to Proposition 2.2, Inequality (2.4) holds for u_{1} and u_{2}. We choose in (2.4) written for u_{1} in variables (t, x),

k = ũ_{2} ≡ u_{2}(t̃, x̃) and φ(t, x) = Ψ_{δ}(t, x, t̃, x̃)

and in (2.4) written for u_{2} in variables (t̃, x̃),

k = u_{1}(t, x) and φ(t̃, x̃) = Ψ_{δ}(t, x, t̃, x̃).

By integrating over Q_{h} and adding up, it comes by using (4.1):

− ∫_{Q_{h}×Q_{h}}(|u_{1} − ũ_{2}|(∂_{t}Ψ_{δ} + ∂t̃Ψ_{δ}) − |f(u_{1}) − f(ũ_{2})|(∇P.∇_{x}Ψ_{δ} + ∇P̃.∇x̃Ψ_{δ}) dr dr̃

+ ∫_{Q_{h}×Q_{h}} sgn(u_{1} − ũ_{2})(g(t, x, u_{1}) − g(t̃, x, ũ_{2}))Ψ_{δ} dr dr̃

≤ ∫_{Q_{h}} ∫_{Σ_{h}\Σ_{hp}} |f(ũ_{2})|∇_{x}P.ν_{h}Ψ_{δ}(σ, r̃)dH_{σ}^{n}dr̃

+ ∫_{Q_{h}} ∫_{Σ_{h}\Σ_{hp}} |f(u_{1})|∇x̃P̃.ν_{h}Ψ_{δ}(r, σ̃)dHn ; σ̃dr

− ∫_{Q_{h}} ess lim_{τ→0^{−}} ∫_{Σ_{h}\Σ_{hp}} |f(u_{1}(σ + τν_{h}))|∇_{x}P.ν_{h}Ψ_{δ}(σ, r̃)dH_{σ}^{n}dr̃

− ∫_{Q_{h}} ess lim_{τ→0^{−}} ∫_{Σ_{h}\Σ_{hp}} |f(u_{2}(σ̃ + τν_{h}))|∇x̃P̃.ν_{h}Ψ_{δ}(r, σ̃)dHn ; σ̃dr

+ ∫_{Q_{h}} ∫_{Σ_{hp}} |f(θ_{1}(σ)) − f(ũ_{2})|∇_{x}P.ν_{h}Ψ_{δ}(σ, r̃)dH_{σ}^{n}dr̃

+ ∫_{Q_{h}} ∫_{Σ_{hp}} |f(θ_{2}(σ̃)) − f(u_{1})|∇x̃P̃.ν_{h}Ψ_{δ}(r, σ̃)dHn ; σ̃dr.

(4.4)

Then through a classical reasoning we pass to the limit with δ on the left-hand side of (4.4). On the right-hand side, we refer to Lemma 4.1. It comes:

− ∫_{Q_{h}} |u_{1} − u_{2}|χ′(t) dx dt ≤ − ∫_{Q_{h}} sgn(u_{1} − u_{2})(g(t, x, u_{1}) − g(t, x, u_{2}))χ(t) dx dt
+ ∫_{Σ_{hp}} |f(θ_{1}(σ)) − f(θ_{2}(σ))|∇_{x}P.ν_{h}χ(t)dH^{n}.

The Lipschitz condition for g provides: for any χ of D(0, T), χ ≥ 0,

\begin{aligned}− ∫_{Q_{h}} |u_{1} − u_{2}|χ′(t) dx dt & ≤ ∫_{Σ_{hp}} |f(θ_{1}(σ)) − f(θ_{2}(σ))|∇_{x}P.ν_{h}χ(t)dH^{n} \\ & + M′_{g} ∫_{Q_{h}} |u_{1} − u_{2}|χ(t) dx dt. \end{aligned} (4.5)

By adding inequalities (4.3) and (4.5), we obtain

− ∫_{Q} |u_{1} − u_{2}|χ′(t) dx dt

≤ M′_{g} ∫_{Q} |u_{1} − u_{2}|χ(t) dx dt + ∫_{Σ_{hp}} |f(θ_{1}(σ)) − f(θ_{2}(σ))|∇_{x}P.ν_{h}χ(t)dH^{n}

− ∫_{Σ_{hp}}(f(θ_{1}(σ)) − f(θ_{2}(σ)))∇P.ν_{h} sgn(ϕ(u_{1}) − ϕ(u_{2}))χ(t)dH^{n}.

Therefore, when a.e. on Γ_{hp}, ∇P.ν_{h} ≤ 0, we have

|f(θ_{1}(σ)) − f(θ_{2}(σ))|∇P.ν_{h} ≤ (f(θ_{1}(σ)) − f(θ_{2}(σ))) sgn(ϕ(u_{1}) − ϕ(u_{2}))∇P.ν_{h}.

Now, when a.e. on Γ_{hp}, ∇P.ν_{h} ≥ 0, a.e. on Σ_{hp},

f(θ_{i}(σ))∇P.ν_{h} = f(u_{i}(σ))∇P.ν_{h}, i = 1, 2.

As a consequence, a.e. on Σ_{hp},

(f(θ_{1}(σ)) − f(θ_{2}(σ)))∇P.ν_{h} sgn(ϕ(u_{1}) − ϕ(u_{2})) = |f(θ_{1}(σ)) − f(θ_{2}(σ))|∇P.ν_{h}.

At last in both cases, we have for any χ of D(0, T), χ ≥ 0,

− ∫_{Q} |u_{1} − u_{2}|χ′(t) dx dt ≤ M′_{g} ∫_{Q} |u_{1} − u_{2}|χ(t) dx dt.

When χ is the element of a sequence approximating 𝕀_{[0,t]}, t being given outside a set of measure zero, the desired inequality of Lemma 4.2 is obtained thanks to the initial condition (2.3) for u_{1} and u_{2} and to the Gronwall Lemma. □

Comments. In this paper we have looked for solutions to the coupling problem (1.1)-(1.5). We have proved an existence and uniqueness result when along the interface all the charasteristics have the same behaviour. Either there are all leaving the hyperbolic domain, either there are all entering this domain. In the first case, we refer the reader to [1] for a study without condition (4.1) by means of the vanishing viscosity method and the notion of process solutions [4].

References

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[6] F. Gastaldi, A. Quateroni, G. Landriani Sacchi; Coupling of two-dimensional hyperbolic and elliptic equations. Comput. Methods Appl. Mech. Eng. 80 1-3 (1990), 347-354.

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Gloria Aguilar Departamento de Matemática Aplicada, Universidad de Zaragoza, CPS, Maria de Luna 3, E-50018 Zaragoza, Spain

   E-mail address: gaguilar@unizar.es

Laurent Lévi Laboratoire de Mathématiques Appliquées, UMR 5142, Université de Pau, IPRA, BP 1155, F-64013 Pau Cedex, France

E-mail address: laurent.levi@univ-pau.fr

Monique Madaune-Tort Laboratoire de Mathématiques Appliquées, UMR 5142, Université de Pau, IPRA, BP 1155, F-64013 Pau Cedex, France

E-mail address: monique.madaune-tort@univ-pau.fr