Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant

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Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant

Joachim Escher, Matthieu Hillairet, Philippe Laurencot, Christoph Walker

To cite this version:

Joachim Escher, Matthieu Hillairet, Philippe Laurencot, Christoph Walker. Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant. Indiana Uni- versity Mathematics Journal, Indiana University Mathematics Journal, 2011, 60 (6), pp.1975–2019.

�10.1512/iumj.2011.60.4441�. �hal-00495919�

GLOBAL WEAK SOLUTIONS FOR A DEGENERATE PARABOLIC SYSTEM MODELING THE SPREADING OF INSOLUBLE SURFACTANT

JOACHIM ESCHER, MATTHIEU HILLAIRET, PHILIPPE LAURENC¸ OT, AND CHRISTOPH WALKER

Abstract. We prove global existence of a nonnegative weak solution to a degenerate parabolic system, which models the spreading of insoluble surfactant on a thin liquid film.

1. Introduction

It is a widely used approach in the study of the dynamical behavior of viscous thin films to approximate the full fluid mechanical system by simpler model equations, using e.g. lubrication theory and cross-sectional averaging. In most of such models surface tension effects may then become significant, or even dominant. Therefore, also the influence of surfactant, i.e. surface active agents on the free surface of thin films, is of considerable importance. A surfactant lowers the surface tension of the liquid and the resulting gradients of surface tension induce so-called Marangoni stresses which in turn cause a spreading of the surfactant on the interface. We investigate here a model in which the surfactant is assumed to be insoluble. In addition we include gravity but neglect effects of capillarity and van der Waals forces. Writing h(t, x) for the film thickness and Γ(t, x) for the concentration of surfactant at timet >0 and positionx∈(0, L), Jensen and Grotberg derived in [9, 10] the following system:

∂th = ∂x

Gh³

3 ∂xh− h²

2 ∂xσ(Γ)

in Q∞, (1.1)

∂tΓ = ∂x

Gh²

2 Γ ∂xh+ (D−hΓ σ^′(Γ)) ∂xΓ

in Q∞. (1.2)

HereQ∞ := (0,∞)×(0, L) denotes the time-space domain of the unknowns h and Γ, with L being the spatial horizontal latitude of the system. We further impose no-flux boundary condition for h and Γ, i.e.

∂xh=∂xΓ = 0 on (0,∞)× {0, L}, (1.3) as well as initial conditions for these quantities:

(h,Γ)(0) = (h0,Γ0) in (0, L), (1.4) whereh0 and Γ0 are given. Equation (1.1) for the height functionhis a consequence of the conserva- tion of momentum and the kinematic boundary condition, reflecting the model assumption that the

Date: June 29, 2010.

This work was partially supported by the french-german PROCOPE project 20190SE.

1

velocity of the free interface balances the normal component of the liquid, cf. [7, 9, 10]. Equation (1.2) is an advection-transport equation for the surfactant concentration on the interface in whichD is a non-dimensional surface diffusion coefficient, assumed to be positive and constant. The positive constantG represents a gravitational force.

Of considerable importance in the modeling is the surface tension σ(Γ), a decreasing function of the surfactant concentration. Several equations of state giving the dependence of the surface tension σ upon the surfactant concentration Γ, including

σ(Γ) =σs−β Γ or σ(Γ) =σs−β ln

1± Γ Γ∞

,

may be found in the literature, see [4, 9, 15] and the references therein. In this paper, for technical reasons we assume that

σ∈ C³([0,∞)), σ(0) >0, 0< σ0 ≤ −σ^′ ≤σ∞, (1.5) which is satisfied in particular by the first example above. A straightforward consequence of (1.5) is the fact thatσ grows at most linearly:

|σ(r)| ≤σ(0) +σ_∞ r , r ≥0. (1.6) Observe that the coupled system (1.1), (1.2) is degenerate parabolic in the sense that parabolicity is lost ifhor Γ vanish. While modeling issues related to surfactant spreading on thin liquid films have attracted considerable interest (e.g., see [6, 9, 10, 14] and the references therein), much less research has been dedicated to analytical aspects. In [17, 18, 19] local existence results are shown. In [8]

global existence of weak solutions is derived for a variant of (1.1), (1.2) without gravity but including a fourth order term in h modeling capillarity effects. Local asymptotic stability of steady states (being simply the positive constants) is investigated in [7] for the case of soluble surfactant. These results in particular show that, starting with initial values near steady states, problem (1.1)-(1.4) admits a unique global positive classical solution.

Our aim here is to prove the existence of global nonnegative weak solutions to (1.1)-(1.4) for arbitrary nonnegative initial values. The core of our analysis is the fact that system (1.1)-(1.3) possesses an energy functional entailing various a priori estimates on (h,Γ). We regularize (1.1)- (1.4) appropriately to obtain a uniformly parabolic system with coefficients (depending nonlinearly on (h,Γ)) being regular enough to apply abstract semi-group theory to prove well-posedness of the regularized system. This approach warrants that the thereby constructed nonnegative solutions exist globally provided they are a priori bounded in W₂¹. The aforementioned energy estimates provide such bounds and inherit also compactness properties in suitable function spaces to the family of regularized solutions, which allows us to extract a subsequence converging to a weak solution.

In fact, we shall establish the following result:

Theorem 1.1. Let D, G > 0 and suppose (1.5). Given nonnegative h0,Γ0 ∈ W₂²(0, L) satisfying

∂xh0(x) =∂xΓ0(x) at x= 0 and x=L, there exists a global weak solution to (1.1)-(1.4), i.e. a pair

of nonnegative functions (h,Γ) such that h(0) = h0, Γ(0) = Γ0,

h∈L∞(0, T;L2(0, L))∩L5(0, T;C^1/5([0, L])), h^5/2 ∈L2(0, T;W₂¹(0, L), Γ∈L∞(0, T;L1(0, L))∩L2(0, T;C([0, L])), ∂xσ(Γ)∈L1((0, T)×(0, L)), j_f := −2

5 rG

3 ∂_x h^5/2 +

r3h

4G ∂_xσ(Γ)

!

∈L₂((0, T)×(0, L)), js :=−G

5 ∂x h^5/2 +√

h ∂xσ(Γ)∈L2((0, T)×(0, L)), for all T >0, and

d dt

Z L 0

h ψ dx = rG

3 Z L

0

∂xψ h^3/2 jf dx , d

dt Z L

0

Γ ψ dx = Z L

0

∂_xψ

−D ∂_xΓ +√

h Γ j_s dx ,

for all ψ ∈W_∞¹(0, L). In addition, introducing the function φ defined by φ^′′(r) =−σ^′(r)/r for r >0 and φ(1) =φ^′(1) = 0, the weak solution (h,Γ) satisfies

kh(t)k1 =kh₀k1, kΓ(t)k1 =kΓ₀k1, t≥0, (1.7) L0(t) +

Z t

0 D0(s) ds≤ L0(0), t≥0, (1.8)

with

L0(t) :=

Z L 0

G

2 |h(t, x)|²+φ(Γ(t, x))

dx , and

2D0(t) := G kjf(t)k²2+kjs(t)k²2+G² 75

∂x(h^5/2)(t)

2 2

+1 4

ph(t) ∂xσ(Γ)(t)

2

2+ 8σ0D ∂x

pΓ(t)

2 2 .

Observe that the notion of weak solutions is readily obtained by testing (1.1)–(1.4) against ψ ∈ W_∞¹(0, L) and integrating with respect to the spatial variable. Also, the weak formulation ensures the time continuity of h and Γ (for some suitable weak topology with respect to space), so that the initial data (h,Γ)(0) = (h0,Γ0) are meaningful. Observe finally that, thanks to the “energy inequality” (1.8), we actually have an improved regularity on Γ, namely √

Γ∈L2(0, T;W₂¹(0, L)).

Remark 1.2. Theorem 1.1 is actually valid under the weaker assumption h0,Γ0 ∈ W₂¹(0, L), the proof being similar to the one given below using an additional (but classical) regularization of the initial data.

We close the introduction by outlining the content of our paper. Section 2 is devoted to a regular- ized version of the original system (1.1)–(1.4). Roughly speaking, the coupling terms in (1.1)-(1.2) being of the same order as the diagonal terms (so that we are dealing with somehow a full diffusion matrix), we have to mollify them in order to be able to apply the abstract theory developed in [2] for quasilinear parabolic systems. The regularization is then crucial to establish global existence.

First, we derive the local well-posedness anda prioriestimates then ensure the global well-posedness.

Various compactness properties of the family of regularized solutions are established in Section 3, allowing us to recover a weak solution in the sense of Theorem 1.1. In the Appendix we collect some tools used for the compactness arguments in Section 3.

2. A regularized problem

For ε ∈ (0,1) and f ∈ L2(0, L), let N^ε(f) be the unique solution in W₂²(0, L) to the elliptic boundary-value problem

Nε(f)−ε² ∂_x²Nε(f) =f in (0, L), ∂xNε(f)(0) =∂xNε(f)(L) = 0. (2.1) Clearly,

Nε∈ L(L2(0, L), W₂²(0, L))∩ L(C^γ([0, L]),C^γ+2([0, L])) , γ >0, is a positive operator. We define the following functions

a1(r) :=G r³

3 , a2,ε(r) := (r−√ ε)²

2 , r≥0, (2.2)

b2,ε(r) :=r−ε , r ≥0, (2.3)

and notice that

G a²_2,ε(r)≤η r a1(r), r≥√

ε , with η:= 3

4. (2.4)

We next fixη1 ∈(η,1) and define

α₀(r, s) := η₁ s+ (1−η₁) r , α₁(r) :=

Z r 0

pa₁(ρ)dρ , r≥0, s≥0, (2.5) and

β1(r) :=

Z r 0

ρ |σ^′(ρ)| dρ , r≥0. (2.6)

The regularized problem reads

∂_th_ε = ∂_x a₁(h_ε)∂_xh_ε− a2,ε(hε)p

Nε(hε)

√hε

∂_xΣ_ε(h_ε,Γ_ε)

!

in Q_∞, (2.7)

∂tΓε = ∂x G a_2,ε(h_ε)b_2,ε(Γ_ε)p

α₀(h_ε,Nε(h_ε))

phεa1(hε) ∂xNε(α1(hε))

!

(2.8) +∂x

D β₁^′(Γε)

β₁^′(Nε(Γ_ε)) −α0(hε,Nε(hε)) Γε σ^′(Γε)

∂xΓε

in Q∞,

∂xhε = ∂xΓε= 0 on (0,∞)× {0, L}, (2.9)

(hε,Γε)(0) = (h0,ε,Γ0,ε) in (0, L), (2.10)

where Σε:= Σε(hε,Γε) solves

Σε−ε² ∂x(Nε(hε) ∂xΣε) =σ(Γε) in Q∞, ∂xΣε= 0 on (0,∞)× {0, L}. (2.11) This problem admits a unique global strong solution:

Theorem 2.1. Let h0,Γ0 ∈ W₂²(0, L) be nonnegative functions satisfying ∂xh0(x) =∂xΓ0(x) = 0 at x= 0, L. For ε∈(0,1)set

h0,ε :=h0 +√

ε , Γ0,ε := Γ0+ε . (2.12)

Then there is a unique global nonnegative solution (hε,Γε) with hε,Γε∈ C¹ (0,∞), L2(0, L)

∩ C (0,∞), W₂²(0, L)

∩ C [0,∞), W₂¹(0, L) to the regularized problem (2.7)-(2.10). Moreover,

hε(t, x)≥√

ε , Γε(t, x)≥ε , (t, x)∈[0,∞)×(0, L) . The remainder of this section is dedicated to the proof of this theorem.

2.1. Local well-posedness. We first focus our attention on the local solvability of the regularized problem. Given ε∈(0,1) fixed we use the notation

V_B^γ :=W_2,B^γ (0, L)∩ C([0, L], D0)

with D0 := (ε²,∞) and γ > 1/2, where W_2,B^γ := W_2,B^γ (0, L) coincides with the fractional Sobolev space W₂^γ := W₂^γ(0, L) if γ ≤ 3/2 or is the linear subspace thereof consisting of those u ∈ W₂^γ satisfying the Neumann boundary conditions ∂xu(0) = ∂xu(L) = 0 if γ > 3/2. Observe that V_B^γ is open in W_2,B^γ and that h0,ε,Γ0,ε ∈ V_B². In the following we use the notation C¹⁻ to indicate that a function is locally Lipschitz continuous.

The proof of the next result about Nemitskii operators can be found, e.g., in [1, Sect.15]:

Lemma 2.2. Given g ∈ C²(D0), let g^#(u)(x) := g(u(x)), x ∈ (0, L), for u : (0, L) → D0. Then g^#∈ C¹⁻(V_B^γ, W₂^γ) for γ ∈(1/2,1).

We shall also use the following continuity result about pointwise multiplication of real-valued functions.

Lemma 2.3. (i) If γ ≥ 0, then pointwise multiplication C^γ([0, L])× C^γ([0, L]) → C^γ([0, L]) is con- tinuous.

(ii) If γ >1/2, then pointwise multiplication W₂^γ×W₂^γ →W₂^γ is continuous.

(iii) If s > γ ≥0, then pointwise multiplication C^s([0, L])×W₂^γ →W₂^γ is continuous.

Proof. While assertion (i) is obvious, assertion (ii) is a consequence of [3, Thm.4.1], and assertion

(iii) is proved in [21] (see also [2, Eq.(8.3)]).

The next proposition guarantees a nonnegative maximal solution to the regularized problem (2.7)- (2.10). The crucial point is that, though the local solution which we construct belongs toW_2,B² (0, L), ana prioriestimate in W₂¹ is sufficient to obtain global existence, see (2.13) below.

Proposition 2.4. The regularized problem (2.7)-(2.10) admits a unique maximal strong solution (hε,Γε) on the maximal interval of existence J :=J(ε). The solution possesses the regularity

hε, Γε∈ C¹ J \ {0}, L2(0, L)

∩ C J \ {0}, W_2,B² (0, L)

∩ C J, W₂¹(0, L) . Moreover, if for each T >0 there is some c(T, ε)>0 such that

min

h_ε(t, x),Γ_ε(t, x) ≥ε²+c(T, ε)⁻¹ , kh_ε(t)kW2¹+kΓ_ε(t)kW2¹ ≤c(T, ε) (2.13) fort ∈ J ∩[0, T] and x∈(0, L), then J = [0,∞), i.e. the solution exists globally.

Proof. To establish the result we shall use the theory for quasilinear equations from [2, Sect.13]. We simplify the notation by omitting the subscriptε everywhere in (2.7)-(2.10) for the remainder of the proof. In the following we writeu:= (h,Γ) and introduce

a(u) :=





a1(h) 0

0 D β₁^′(Γ)

β₁^′(N(Γ)) −α0(h,N(h))Γσ^′(Γ)





and

F(u) :=∂_x

b(u)∂_x

Σ(h,Γ) N(α₁(h))

with

b(u) :=











−a2(h)p N(h)

√h 0

0 Ga2(h)b2(Γ)p

α0(h,N(h)) pha1(h)









 .

Thus, setting

A(u)w:=−∂x(a(u)∂xw), Bw:=∂xw ,

we may re-write equations (2.7)-(2.10) as a quasilinear problem of the form

∂tu+A(u)u=F(u) in (0,∞)×(0, L) , (2.14)

Bu= 0 on (0,∞)× {0, L} , (2.15)

u(0,·) =u⁰:= (h0,Γ0) on (0, L) . (2.16) We next verify the assumptions of [2, Thm.13.1] which then guarantees the existence of a weak solution to this quasilinear problem. Subsequently, we shall improve the regularity of the weak solution. In the following, we letξ∈(0,1/8) denote a sufficiently small number so that in particular, forγ := 1/2−2ξ >0,

V_B^1−ξ ֒→ C^γ :=C^γ([0, L]) . (2.17) Consequently, classical elliptic regularity applied to (2.1) ensures

N ∈ C¹⁻(V_B^1−ξ,C^γ+2) with N(f)≥ε² for all f ∈V_B^1−ξ . (2.18) Lemma 2.3(i) and (2.2), (2.17) easily yield

h7→a1(h)

∈ C¹⁻(V_B^1−ξ,C^γ), (2.19)

and we obtain from (1.5), (2.5), (2.17), (2.18), and Lemma 2.3(i) that (h,Γ)7→α0(h,N(h))Γσ^′(Γ)

∈ C¹⁻(V_B^1−ξ×V_B^1−ξ,C^γ) , (2.20) while (2.6), (2.17), and (2.18) entail

Γ7→ β₁^′(Γ) β₁^′(N(Γ))

∈ C¹⁻(V_B^1−ξ,C^γ). (2.21) Thus, (2.19), (2.20), and (2.21) imply

u= (h,Γ)7→a(u)

∈ C¹⁻ V_B^1−ξ×V_B^1−ξ,(C^γ)⁴

. (2.22)

Note that if u = (h,Γ) with h(x),Γ(x) > ε² for x ∈ (0, L), then the matrix a(u(x)) has strictly positive eigenvalues due to (1.5). Therefore, letting 2 ˆα := 3/2−3ξ so that γ > 2 ˆα−1, and using the notion of [2, Sect.4 & 8] (in particular, see [2, Eq.(8.6)]), it follows from [2, Ex.4.3.e)] that

(A,B)∈ C¹⁻ V_B^1−ξ×V_B^1−ξ,E^αˆ((0, L))

. (2.23)

That is, (A(u),B) depends Lipschitz continuously on its argument u ∈ V_B^1−ξ ×V_B^1−ξ and for each such ufixed it is normally elliptic with operatorA(u) in divergence form having C^γ-coefficients with γ >2 ˆα−1. We next study the regularity properties of the function F. Clearly, the functiong, given by g(r) :=a2(r)/√

r, r > ε², belongs to C²(D0) by (2.2) so that Lemma 2.2 applies to yield

h7→ a2(h)

√h

∈ C¹⁻(V_B^1−ξ, W₂^1−ξ) . Since (2.17) and (2.18) provide

hh7→p N(h)i

∈ C¹⁻(V_B^1−ξ,C^γ+2) ,

we obtain from Lemma 2.3(iii)

"

h7→ a2(h)p N(h)

√h

#

∈ C¹⁻ V_B^1−ξ, W₂^1−ξ

. (2.24)

As above we have by Lemma 2.2, (2.2), (2.5), and (2.18)

"

h7→ a2(h) pha1(h)

#

∈ C¹⁻ V_B^1−ξ, W₂^1−ξ

, h

h7→p

α0(h,N(h))i

∈ C¹⁻ V_B^1−ξ, W₂^1−ξ , from which we deduce, using (2.3) and Lemma 2.3(ii),

"

u= (h,Γ)7→Ga₂(h)b₂(Γ)p

α₀(h,N(h)) pha1(h)

#

∈ C¹⁻ V_B^1−ξ×V_B^1−ξ, W₂^1−ξ

. (2.25)

Sinceα1 in (2.5) is smooth in D0, we get from (2.17) and (2.18) [h 7→∂xN(α1(h))]∈ C¹⁻ V_B^1−ξ,C^1+γ

. (2.26)

The operatorf 7→f−ε²∂_x(N(h)∂_xf) is invertible inL(CB^γ+2,C^γ) forh∈V_B^1−ξby (2.18) and ellipticity, and it thus follows from (2.11), (2.18), the Lipschitz continuity (in fact: analyticity) of the inversion map ℓ7→ℓ⁻¹ for linear operators, and [Γ7→σ(Γ)]∈ C¹⁻(V_B^1−ξ,C^γ) that

(h,Γ)7→∂xΣ(h,Γ)

∈ C¹⁻(V_B^1−ξ×V_B^1−ξ,C^1+γ) . Combining this with (2.24), (2.25), and (2.26) we derive from Lemma 2.3(iii)

F ∈ C¹⁻ V_B^1−ξ×V_B^1−ξ, W₂^−ξ×W₂^−ξ

. (2.27)

At this point observe that W₂^−ξ =W_2,B^−ξ in the notation of [2] (in particular, see [2, Eq.(7.5)]) since ξ < 1/2. Thus, recalling that 2 ˆα = 3/2−3ξ and choosing the numbers (τ, r, s, σ) in [2, Eq.(13.2)]

to be (−ξ,1−ξ,1 +ξ,2 ˆα) we may apply [2, Thm.13.1] due to (2.23) and (2.27). We conclude that the quasilinear problem (2.14)-(2.16) with u⁰ = (h₀,Γ₀) ∈ V_B²×V_B² admits a unique maximal weak W₂^3/2−3ξ-solution (h,Γ) in the sense of [2, Sect.13] on some interval J; that is,

u= (h,Γ)∈ C J \ {0}, W_2,B^3/2−3ξ×W_2,B^3/2−3ξ

∩ C¹ J \ {0}, W_2,B^−1/2−3ξ×W_2,B^−1/2−3ξ .

The solution u = (h,Γ) exists globally, i.e. J = [0,∞), provided that (h,Γ)|[0,T] is bounded in W₂¹×W₂¹ and bounded away from the boundary of V_B^1−ξ for each T >0. In particular, the solution exists globally provided (2.13) holds.

We now aim at improving the regularity of u = (h,Γ) as in [2, Sect.14]. Given δ > 0 and ξ > 0 still sufficiently small, set J^δ :=J ∩[δ,∞). Then

h , Γ∈ C Jδ, W_2,B^3/2−3ξ

∩ C¹ Jδ, W_2,B^−1/2−3ξ , from which we derive

h , Γ∈ C^ρ(Jδ, W_2,B^{3/2−3ξ−2ρ}) , 0≤2ρ≤2, (2.28)

by [2, Thm.7.2]. Taking ρ := ξ and setting µ := 1−6ξ, we have W_2,B^{3/2−3ξ−2ρ} ֒→ C^µ, and it thus follows from (2.28) analogously to (2.22) that

t7→a(u(t))

∈ C^ρ(J^δ,(C^µ)⁴).

Hence, if we put 2ˆµ:= 2−8ξ so that µ > 2ˆµ−1, we obtain similarly to (2.23) from [2, Ex.4.3.e), Eq.(8.6)] that

(A(u),B)∈ C^ρ J^δ,E^µˆ((0, L))

. (2.29)

Set then 2ν:= 3/2 +ξ and note that, for ξ >0 small enough,

3/2<2ν <2ˆµ <2 and 2ρ= 2ξ > ξ= 2ν−3/2 . (2.30) Also observe that (2.27) and [2, Eq.(7.5)] ensure

F(u)∈ C^ρ(Jδ, W_2,B^−ξ×W_2,B^−ξ)֒→ C^ρ(Jδ, W_2,B^2ν−2×W_2,B^2ν−2) . (2.31) Gathering (2.29)-(2.31) and invoking [2, Thm.11.3], we conclude that the linear problem

∂tv+A(u(t))v =F(u(t)) in (Jδ\ {δ})×(0, L), (2.32) Bv = 0 on (J^δ\ {δ})× {0, L} , (2.33)

v(0,·) =u(δ,·) on (0, L), (2.34)

has a unique strongW₂^2ν-solution (in the sense of [2, Sect.11])

v ∈ C(J^δ\ {δ}, W_2,B^2ν ×W_2,B^2ν)∩ C¹(J^δ\ {δ}, W_2,B^2ν−2×W_2,B^2ν−2) .

Hence, v and u are both weak W₂^3/2−3ξ-solutions to (2.32)-(2.34) and thus u = v by uniqueness of weak solutions to linear problems. Makingδ > 0 smaller we may replaceJ^δ\{δ}byJ^δ, and using the embedding W₂^3/2+ξ ֒→ C^1+ξ for ξ sufficiently small, we get h,Γ ∈ C^ξ(J^δ,C^1+ξ). But then u = (h,Γ) satisfies

∂tu−∂x a(u)∂xu

=F(u) on Jδ×(0, L)

subject to the boundary condition Bu= 0 with a(u)∈ C^ξ(Jδ,C¹) andF(u)∈ C^ξ(Jδ, L2) from which we readily conclude that

h , Γ∈ C(Jδ, W_2,B² )∩ C¹(Jδ, L2)

with δ >0 arbitrarily small by invoking [2, Thm.10.1] with (E₀, E₁) := (L₂, W_2,B² ). This proves the

proposition.

2.2. Global well-posedness. Let (hε,Γε) denote the unique strong solution to (2.7)-(2.10) on the maximal interval of existenceJ =J(ε) provided by Proposition 2.4. We now show that (2.13) holds which impliesJ = [0,∞). Introducing the abbreviations

Hε:=N^ε(hε), Aε :=N^ε(α1(hε)), Bε :=N^ε(Γε), Σε := Σε(hε,Γε), (2.35)

and subsequently omitting the subscriptεeverywhere in (2.7)-(2.10) to simplify notation, the strong solution (h,Γ) = (hε,Γε) thus satisfies

∂_th = ∂_x a₁(h) ∂_xh− a2(h)√

√ H

h ∂_xΣ

!

in J \ {0} ×(0, L), (2.36)

∂tΓ = ∂x G a₂(h)b₂(Γ)p

α₀(h, H) pha1(h) ∂xA

!

(2.37) +∂x

D β₁^′(Γ)

β₁^′(B) −α0(h, H) Γσ^′(Γ)

∂xΓ

in J \ {0} ×(0, L),

∂xh = ∂xΓ = 0 on J \ {0} × {0, L}, (2.38)

(h,Γ)(0) = (h0,Γ0) in (0, L). (2.39)

We begin with some obvious consequences of the structure of (2.36)-(2.39).

Lemma 2.5. For (t, x)∈ J ×(0, L), we have h(t, x)≥√

ε , Γ(t, x)≥ε , (2.40)

kh(t)k1 =kh₀k1, kΓ(t)k1 =kΓ₀k1. (2.41) Proof. Since a2(√

ε) = b2(ε) = 0 by (2.2) and (2.3) and since h0 ≥ √

ε and Γ0 ≥ ε by (2.12), the assertion (2.40) is a straightforward consequence of the comparison principle applied separately to (2.36) and (2.37). We next integrate (2.36) and (2.37) over (0, t)×(0, L) and use (2.38) to obtain

(2.41).

In the next lemma, we collect several properties of H, Σ, A, and B. Lemma 2.6. We have, for (t, x)∈ J ×(0, L),

kH(t)kp ≤ kh(t)kp, p∈[1,∞], √

ε≤H(t, x), (2.42)

k∂_xH(t)k²2+ 2 ε² k∂_x²H(t)k²2 ≤ k∂_xh(t)k²2, (2.43)

ε² k∂xH(t)k∞≤ kh(t)k1, (2.44)

kA(t)kp ≤ kα1(h(t))kp, p∈[1,∞], (2.45) k∂xA(t)k²2+ 2 ε² k∂_x²A(t)k²2 ≤ k∂xα1(h(t))k²2, (2.46)

ε² k∂xA(t)k^∞≤ kα1(h(t))k¹, (2.47)

kB(t)k^p ≤ kΓ(t)k^p, p∈[1,∞], ε≤B(t, x), (2.48) k∂xB(t)k²2+ 2 ε² k∂_x²B(t)k²2 ≤ k∂xΓ(t)k²2, (2.49)

ε² k∂_xB(t)k∞≤ kΓ(t)k1, (2.50)

kp

H(t)∂xΣ(t)k²2 + 2 ε² k∂x(H(t)∂xΣ(t))k²2 ≤ kp

H(t)∂xσ(Γ(t))k²2, (2.51) ε² kH(t)∂xΣ(t)k∞ ≤2kσ(Γ(t))k1. (2.52)

Proof. The first assertion of (2.42) follows from the classical contraction properties of N while the second is a consequence of (2.35), (2.40), and the comparison principle. We next deduce from (2.35) that

k∂_xH(t)k²2+ε² k∂_x²H(t)k²2= Z L

0

∂_xh(t) ∂_xH(t) dx≤ k∂_xH(t)k²2+k∂_xh(t)k²2

2 ,

from which (2.43) follows. Finally, we infer from (2.35) and the positivity of H that −ε² ∂_x²H ≤ h.

For (t, x)∈ J ×(0, L), we integrate the previous inequality first over (0, x) and then over (x, L), and use the homogeneous Neumann boundary conditions to obtain

−ε² ∂xH(t, x)≤ Z x

0

h(t, y) dy≤ kh(t)k¹ and ε² ∂xH(t, x)≤ Z L

x

h(t, y) dy≤ kh(t)k¹. Combining these two inequalities gives (2.44).

Next, the proofs of (2.45)-(2.47) and (2.48)-(2.50) are similar to those of (2.42)-(2.44).

We now turn to Σ and first notice that, since H ≥√

ε by (2.42), the solution Σ to (2.11) belongs to W₂²(0, L). We thus may multiply (2.11) by (−∂x(H∂xΣ)) and argue as in the proof of (2.43) to establish (2.51). Finally, consider (t, x) ∈ J ×(0, L). As in the proof of (2.44), we integrate (2.11) first over (0, x) and then over (x, L), and use the homogeneous Neumann boundary conditions to obtain

−ε² H(t, x)∂xΣ(t, x) ≤ Z x

0

(σ(Γ)−Σ)(t, y)dy≤ kσ(Γ(t))k1+kΣ(t)k1, ε² H(t, x)∂_xΣ(t, x) ≤

Z L x

(σ(Γ)−Σ)(t, y) dy≤ kσ(Γ(t))k1+kΣ(t)k1.

As (2.11) implies thatkΣ(t)k1 ≤ kσ(Γ(t))k1 by classical approximation and monotonicity arguments,

we obtain (2.52).

We next define

Jf := −∂xα1(h) + a2(h)√ H

pha1(h) ∂xΣ, (2.53)

Js := p

α0(h, H) ∂xσ(Γ)−G a2(h) pha1(h)

b2(Γ)

Γ ∂xA , (2.54)

and show the existence of a Liapunov functional for the regularized problem (2.36)-(2.39) inherited from the one of (1.1)-(1.4).

Lemma 2.7. Given t∈ J, we have

L(t) + Z t

0 D(s) ds≤ L(0), (2.55)

with

L(t) :=

Z L 0

G

2 |h(t, x)|²+φ(Γ(t, x))

dx , (2.56)

φ^′′(r) =−σ^′(r)

r ≥0, φ(1) =φ^′(1) = 0, (2.57) 2D(t) := G kJf(t)k²2+kJs(t)k²2+ (η1 −η) kp

H(t)∂xσ(Γ(t))k²2 (2.58) +(1−η1) kp

h(t)∂xσ(Γ(t))k²2+ (1−η)G k∂xα1(h(t))k²2

+2D Z L

0

|∂xσ(Γ(t))|² β₁^′(B(t)) dx .

Observe that the last term in D(t) is well-defined as β₁^′(B)≥σ0ε >0 by (1.5), (2.6), and (2.48).

Proof. It follows from (2.36)-(2.38) that dL

dt = G Z L

0

∂xh −a1(h) ∂xh+ a2(h)√

√ H

h ∂xΣ

! dx +

Z L 0

φ^′′(Γ)∂xΓ

−D β₁^′(Γ)

β₁^′(B) +α0(h, H) Γσ^′(Γ)

∂xΓ dx

−G Z L

0

φ^′′(Γ)∂xΓ a₂(h)b₂(Γ)p

α₀(h, H)

pha1(h) ∂xA dx

= −G

2 kJ_fk²2− 1

2 kJ_sk²2−D Z L

0

|∂xσ(Γ)|²

β₁^′(B) dx+ 1

2 Rf +G 2 Rs, with

R^f :=

Z L 0

G a2(h)²H

ha₁(h) |∂xΣ|²−α0(h, H) |∂xσ(Γ)|²

dx , Rs :=

Z L 0

G a₂(h)² ha1(h)

b₂(Γ)²

Γ² |∂xA|²− |∂xα1(h)|²

dx . On the one hand, it follows from (2.4), (2.5), and (2.51) that

Rf ≤ Z L

0

η H |∂xΣ|² −α0(h, H) |∂xσ(Γ)|² dx

≤ −(η1−η)

√H∂xσ(Γ)

2

2 −(1−η1)

√h∂xσ(Γ)

2 2 . On the other hand, (2.3), (2.4) and (2.46) give

R^s ≤ Z L

0

η |∂xA|²− |∂xα1(h)|²

dx≤ −(1−η) k∂xα1(h)k²2 .

Collecting the above inequalities yields Lemma 2.7 after integration with respect to time.

We next estimate the L2-norm of∂xh. While the previous estimates only depend mildly onε, this will no longer be the case in the remainder of this section. In the following, the constants C, Cj, ... are independent of the free variables. Additional dependence on, say, ε or T > 0, we express explicitly by writingC(ε), C(ε, T), ...

Lemma 2.8. We define the function A¹ by A^′1 =a1 and A¹(0) = 0. For T >0 and t ∈ J ∩[0, T], we have

kh(t)k∞+k∂xA1(h(t))k₂ ≤ C1(ε, T), (2.59) Z t

0 k∂tα1(h(s))k²₂+k∂xh(s)k²∞

ds ≤ C1(ε, T). (2.60)

Proof. Introducing a0(h) :=h and F1 :=−

a₂

√a0

′

(h)∂xh √

H ∂xΣ, F2 := a₂(h) 2√

h

∂_xH

√H ∂xΣ, F3 :=−a₂(h)

√hH ∂x(H∂xΣ) , equation (2.36) reads

∂th−∂_x²A¹(h) =F1+F2+F3. (2.61) Recalling thatα^′₁ =√a1, it follows from (2.61) that

Z L 0

∂th ∂tA¹(h) dx+ Z L

0

∂xA¹(h) ∂t∂xA¹(h) dx= Z L

0

(F1+F2+F3)∂tA¹(h) dx , k∂tα1(h)k²2+1

2 d

dtk∂xA¹(h)k²2 ≤ 1

2 k∂tα1(h)k²2+1 2

Z L 0

a1(h) (F1+F2+F3)² dx , k∂tα1(h)k²2+ d

dtk∂xA1(h)k²2 ≤3

3

X

i=1

pa1(h) Fi

2

2 . (2.62)

To estimate the term involving F1, we write

pa1(h) F1

2 =

a₂

√a0

′

(h) ∂_xA1(h) pa₁(h)

√H ∂xΣ 2

≤

a2

√a0

′

(h) 1

pa1(h) ∞

k∂xA1(h)k₂

√H ∂xΣ ∞ , and observe that (1.6), (2.41), (2.42), and (2.52) ensure that

√H ∂_xΣ

∞≤ kH ∂xΣk∞

ε^1/4 ≤ 2 kσ(Γ)k1

ε^9/4 ≤C(ε) (1 +kΓk1)≤C(ε), while we infer from (2.2) and (2.40) that

a2

√a0

′

(h) 1

pa1(h) ∞

= r3

G

3h²−2√ εh−ε 4h³

∞

≤ C

√ε.

Consequently,

pa1(h)F1

2 ≤C(ε) k∂xA1(h)k2 . (2.63) We next turn to the term involving F₂ and deduce from (2.4) and (2.42) that

pa1(h) F2

2 =

a₂(h) 2p

ha₁(h) a1(h) ∂_xH

H^3/2 H∂xΣ 2

≤

a2(h) 2p

ha1(h) ∞

ka1(h)k2 k∂xHk∞

ε^3/4 kH∂xΣk_∞

≤ η 4Gε^3/2

1/2

ka1(h)k2 k∂xHk∞ kH∂xΣk∞ . Owing to (1.6), (2.41), (2.44), and (2.52), we obtain

pa₁(h) F₂

2 ≤C(ε) ka₁(h)k2 khk¹ ε²

2 kσ(Γ)k1

ε² ≤C(ε)ka₁(h)k2. Since

a1(h) = 4A1(h)

h ≤ 4A1(h)

√ε (2.64)

by (2.2) and (2.40), we end up with

pa1(h) F2

2 ≤C(ε) kA¹(h)k^∞. (2.65)

Finally, by (2.4), (2.42), (2.51), (2.58), and (2.64), we have kp

a1(h)F3k2 =

a2(h) pha1(h)

pa₁(h)

√H ∂x(H∂xΣ) 2

≤ η Gε^1/2

1/2

ka1(h)k^1/2∞ k∂x(H∂xΣ)k2

≤ C(ε)ka1(h)k^1/2∞

√H∂xσ(Γ) 2

ε ,

whence

kp

a1(h) F3k² ≤C(ε)D^1/2 kA¹(h)k^1/2∞ . (2.66) It then follows from (2.62), (2.63), (2.65), and (2.66) that

k∂tα1(h)k²₂ + d

dtk∂xA1(h)k²₂ ≤C(ε) k∂xA1(h)k²₂+kA1(h)k²∞+D kA1(h)k∞

. (2.67)

Owing to (2.2) and A^′1 =a1, we have, for (t, x)∈ J ×(0, L), 0≤LA¹(h(t, x)) ≤ kA¹(h(t))k¹+L^3/2 k∂xA¹(h(t))k2

≤ G

12

1/4 Z L 0

h(t, y)A¹(h(t, y))^3/4 dy+L^3/2 k∂xA¹(h(t))k2

≤ G

12 1/4

khk¹ kA¹(h(t))k^3/4∞ +L^3/2 k∂xA¹(h(t))k2 , and we thus infer from (2.41) and Young’s inequality that

0≤LA1(h(t, x))≤ 3L

4 kA1(h(t))k∞+ Gkh0k⁴1

48L³ +L^3/2 k∂_xA1(h(t))k2 , whence

kA1(h(t))k∞≤C (1 +k∂xA1(h(t))k₂) , t ∈ J . (2.68) Inserting this inequality in (2.67) gives

k∂tα1(h)k²2+ d

dtk∂xA1(h)k²2 ≤C(ε) (1 +D) 1 +k∂xA1(h)k²2

, from which we conclude that, fort∈ J ∩[0, T],

k∂xA1(h(t))k²2+ Z t

0 k∂tα1(h(s))k²2 ds≤ 1 +k∂xA1(h0)k²2

exp

C(ε)

t+ Z t

0 D(s)ds

, hence by (2.55)

k∂xA¹(h(t))k²2+ Z t

0 k∂tα1(h(s))k²2 ds ≤C(ε, T) . (2.69) A first consequence of (2.68) and (2.69) is that (2.59) holds true. Next, recalling (2.38), (2.40), (2.61), (2.63), (2.65), and (2.66), we obtain fort ∈ J ∩[0, T] and x∈(0, L):

|∂_xA1(h(t, x))| =

Z x 0

∂_x²A1(h(t, y)) dy

=

Z x 0

3

X

i=1

F_i−∂_th

!

(t, y) dy

≤ Z L

0

1 pa₁(h)

"

|∂_tα₁(h)|+

3

X

i=1

pa₁(h) |F_i|

# dy

≤ C(ε) k∂tα1(h)k2+

3

X

i=1

pa1(h)Fi

2

!

≤ C(ε) k∂_tα₁(h)k2+k∂_xA1(h)k2+kA1(h)k∞+D^1/2 kA1(h)k^1/2∞

, and we infer from (2.68) and (2.69) that

k∂xA1(h(t))k∞ ≤C(ε, T) 1 +k∂tα1(h(t))k2+D(t)^1/2 .

Using next (2.55) and (2.69) we obtain Z t

0 k∂xA¹(h(s))k²∞ ds ≤C(ε, T)

1 + Z t

0 k∂tα1(h(s))k²2+D(s) ds

≤C(ε, T). Since

|∂xh|= |∂xA1(h)|

a₁(h) ≤ 3

Gε^3/2 k∂xA¹(h)k∞

by (2.2) and (2.40), the estimate (2.60) follows from the above analysis and (2.69).

We now improve the estimates on Γ and begin with an L∞-bound.

Lemma 2.9. Given T >0 and t ∈ J ∩[0, T], we have

kΓ(t)k∞≤C2(ε, T). (2.70)

Proof. Let T >0 and t∈ J ∩[0, T] be given. Define againa0(h) =h, q:=− a2(h)

pa0(h)a1(h)

pα0(h, H) ∂xA and q1 :=∂xq , and the parabolic operator

Pw := ∂tw−∂x

D β₁^′(Γ)

β₁^′(B)−α0(h, H) Γσ^′(Γ)

∂xw

+G q ∂xw+G q1 b2(w), so that (2.37) also reads

PΓ = 0 in J \ {0} ×(0, L). (2.71)

We next observe that q1 =q11+q12+q13 with q11 := −

a2

√a0a1

′

(h) ∂xh p

α0(h, H) ∂xA , q12 := − a2(h)

2p

a0(h)a1(h)

η1∂xH+ (1−η1)∂xh pα0(h, H) ∂xA , q13 := − a₂(h)

pa₀(h)a₁(h)

pα0(h, H) ∂_x²A . By (2.40), (2.42), (2.47), and (2.59), we have

kq11k∞ ≤ 3ε

G

1/2

h−√ ε h³

∞

k∂xhk∞ (η1 kHk∞+ (1−η1)khk∞)^1/2 k∂xAk∞

≤ C(ε) k∂xhk∞ khk^1/2∞

kα1(h)k1

ε² , that is,

kq11k^∞≤C(ε, T) k∂xhk∞ . (2.72)

We next infer from (2.4), (2.40), (2.41), (2.42), (2.44), (2.47), and (2.59) that kq12k^∞ ≤ η

4G

1/2 η1k∂xHk∞+ (1−η1)k∂xhk∞

ε^1/4 k∂xAk∞

≤ C(ε)

kh0k1

ε² +k∂xhk∞

kα1(h)k1

ε² , that is,

kq12k∞≤C(ε, T) (1 +k∂xh(t)k∞) . (2.73) It finally follows from (2.1), (2.4), (2.35), (2.42), (2.45), and (2.59) that

|q13| ≤η G

1/2

khk^1/2∞

|α1(h)−A|

ε² ≤C(ε, T) (kα1(h)k∞+kAk∞)≤C(ε, T)kα1(h)k∞, and thus

kq13k∞≤C(ε, T). (2.74)

Combining (2.72), (2.73), and (2.74), we conclude that

kq1(t)k^∞≤C(ε, T) (1 +k∂xh(t)k∞) , which by (2.60) gives

Z t

0 kq₁(s)k∞ ds ≤C(ε, T), t ∈ J ∩[0, T]. (2.75) Now, let Qbe the solution to the ordinary differential equation

dQ

dt(t)−G kq1(t)k∞ b2(Q(t)) = 0, t∈ J , (2.76) with initial conditionQ(0) :=kΓ0k∞ ≥ε. We clearly have Q(t)≥ε for t∈ J and

PQ(t) =G (kq1(t)k^∞+q1(t, x)) b2(Q(t))≥0, (t, x)∈ J \ {0} ×(0, L). Recalling (2.71), the comparison principle entails that

Γ(t, x)≤Q(t), (t, x)∈ J ×[0, L]. (2.77) Sinceb2(Q)≤Q, we deduce from (2.75), (2.76), and (2.77) that, for T >0 and t∈ J ∩[0, T],

kΓ(t)k∞≤Q(t)≤Q(0) exp

G Z t

0 kq₁(s)k∞ ds

≤C(ε, T),

as expected.

The final step of the proof of Theorem 2.1 is an L2-estimate on ∂xΓ.

Lemma 2.10. For T >0 and t∈ J ∩[0, T], we have

k∂xΓ(t)k₂ ≤C3(ε, T). (2.78)

Proof. Introducing a0(h) =h and γ :=∂xβ1(Γ), it follows from (2.6) and (2.37) that

∂tβ1(Γ)−β₁^′(Γ) ∂x G a2(h)b2(Γ)p

α0(h, H)

pa0(h)a1(h) ∂xA+ D

β₁^′(B) +α0(h, H)

γ

!

= 0. Differentiating with respect tox we obtain

∂tγ−∂x

"

β₁^′(Γ)∂x G a₂(h)b₂(Γ)p

α₀(h, H)

pa0(h)a1(h) ∂xA+ D

β₁^′(B)+α0(h, H)

γ

!#

= 0. Sinceγ(t, x) = 0 for (t, x)∈ J × {0, L} by (2.38), we deduce from the above equation that

1 2

d

dtkγk²2+ Z L

0

D

β₁^′(B)+α0(h, H)

β₁^′(Γ) |∂xγ|² dx=

5

X

i=1

Yi, (2.79)

with

Y1 := − Z L

0

β₁^′(Γ)∂xγ

−D β₁^′′(B)

β₁^′(B)² ∂xB+η1 ∂xH+ (1−η1)∂xh

γ dx , Y2 := −G

Z L 0

β₁^′(Γ) ∂xγ

a2

√a0a1

′

(h) ∂xh b2(Γ)p

α0(h, H) ∂xA dx , Y3 := −G

Z L 0

β₁^′(Γ) ∂xγ a2(h) pa0(h)a1(h)

b^′₂(Γ) β₁^′(Γ) γ p

α0(h, H) ∂xA dx , Y4 := −G

Z L 0

β₁^′(Γ) ∂xγ a2(h)

pa0(h)a1(h) b2(Γ) η1∂xH+ (1−η1) ∂xh 2p

α0(h, H) ∂xA dx , Y5 := −G

Z L 0

β₁^′(Γ) ∂xγ a2(h)

pa0(h)a1(h) b2(Γ)p

α0(h, H) ∂_x²A dx .

We now estimate each of the terms Yi, 1 ≤ i ≤ 5, separately for T > 0 and t ∈ J ∩[0, T]. By (2.41), (2.44), (2.48), (2.50), and (2.70), we have

|Y1| ≤

pβ₁^′(Γ) ∂xγ

2 kβ₁^′(Γ)k^1/2∞

D

β₁^′′(B) β₁^′(B)²

∞

k∂xBk∞+k∂xHk∞+k∂xhk∞

kγk2

≤ C(ε, T)

kΓk1

ε² +khk1

ε² +k∂xhk_∞

kγk2

pβ₁^′(Γ) ∂xγ 2 , that is,

|Y1| ≤ ε 5

pβ₁^′(Γ) ∂xγ

2

2+C(ε, T) 1 +k∂xhk²∞

kγk²2. (2.80)

It next follows from (2.2), (2.40), (2.42), (2.46), (2.59), and (2.70) that

|Y2| ≤ G

pβ₁^′(Γ) ∂xγ 2 k(p

β₁^′b2)(Γ)k∞

a2

√a0a1

′

(h) ∞

k∂xhk_∞ kα0(h, H)k^1/2∞ k∂xAk₂

≤ C(ε, T)

pβ₁^′(Γ)∂xγ 2

3ε G

1/2

h−√ ε h³

∞

k∂xhk∞ khk^1/2∞ k∂xα1(h)k2

≤ C(ε, T) k∂xhk∞

pβ₁^′(Γ) ∂xγ 2 , so that

|Y₂| ≤ ε 5

pβ₁^′(Γ) ∂_xγ

2

2+C(ε, T) k∂_xhk²∞ , (2.81) while (2.4), (2.40), (2.41), (2.42), (2.47), and (2.59) ensure that

|Y3| ≤ p ηG

pβ₁^′(Γ)∂xγ 2

1 pβ₁^′(Γ)

∞

kγk² kα0(h, H)k^1/2∞ k∂xAk∞

≤ C(ε, T) khk^1/2∞

kα₁(h)k1

ε² kγk2

pβ₁^′(Γ) ∂xγ 2 , whence

|Y3| ≤ ε 5

pβ₁^′(Γ) ∂xγ

2

2+C(ε, T) kγk²2. (2.82) Finally, owing to (2.4), (2.40), (2.41), (2.42), (2.44), (2.46), (2.59), and (2.70), we have

|Y4| ≤ p ηG

pβ₁^′(Γ) ∂xγ 2 k(p

β₁^′b2)(Γ)k^∞ k∂xHk_∞+k∂xhk_∞

ε^1/4 k∂xAk2

≤ C(ε, T)

khk¹

ε² +k∂_xhk∞

k∂_xα₁(h)k2

pβ₁^′(Γ)∂_xγ 2 , that is,

|Y4| ≤ ε 5

pβ₁^′(Γ)∂xγ

2

2+C(ε, T) 1 +k∂xhk²∞

, (2.83)

and

|Y5| ≤ p ηG

pβ₁^′(Γ) ∂xγ 2 k(p

β₁^′b2)(Γ)k∞ kα0(h, H)k^1/2∞

∂²_xA

2

≤ C(ε, T) khk^1/2∞ k∂xα1(h)k2

ε

pβ₁^′(Γ)∂_xγ

2 , (2.84)

that is,

|Y5| ≤ ε 5

pβ₁^′(Γ) ∂xγ

2

2+C(ε, T). (2.85)