Derivation and well-posedness of the extended Green-Naghdi equations for flat bottoms with surface tension

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Green-Naghdi equations for flat bottoms with surface tension

Bashar Khorbatly, Ibtissam Zaiter, Samer Israwi

To cite this version:

Bashar Khorbatly, Ibtissam Zaiter, Samer Israwi. Derivation and well-posedness of the extended

Green-Naghdi equations for flat bottoms with surface tension. Journal of Mathematical Physics,

American Institute of Physics (AIP), 2018, 59 (7), pp.071501. �10.1063/1.5020601�. �hal-02202795�

J. Math. Phys. 59, 071501 (2018); https://doi.org/10.1063/1.5020601 59, 071501

© 2018 Author(s).

Derivation and well-posedness of the extended Green-Naghdi equations for flat bottoms with surface tension

Cite as: J. Math. Phys. 59, 071501 (2018); https://doi.org/10.1063/1.5020601

Submitted: 24 December 2017 . Accepted: 11 June 2018 . Published Online: 02 July 2018 Bashar Khorbatly , Ibtissam Zaiter , and Samer Isrwai

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Derivation and well-posedness of the extended

Green-Naghdi equations for flat bottoms with surface tension

Bashar Khorbatly,^1,2,a)Ibtissam Zaiter,^2,b)and Samer Isrwai^2,c)

1Laboratoire de Math´ematique et Physique Th´eorique, U.F.R Sciences et Techniques Universit´e de Tours, Parc Grandmont, 37200 Tours, France

2Laboratory of Mathematics-EDST, Department of Mathematics, Faculty of Sciences 1, Lebanese University, Beirut, Lebanon

(Received 24 December 2017; accepted 11 June 2018; published online 2 July 2018)

In this paper, we will derive the two-dimensional extended Green-Naghdi system {see Matsuno [Proc. R. Soc. A472, 20160127 (2016)] for determination in a various way} for flat bottoms of order three with respect to the shallowness parameter µ.

Then we consider the 1Dextended Green-Naghdi equations taking into account the effect of small surface tension. We show that the construction of solution with a standard Picard iterative scheme can be accomplished in which the well-posedness inX^s=H^s+2(R)×H^s+2(R) for some s>³₂ of the new extended 1D system for a finite large time existencet=O(_ε¹) is demonstrated.Published by AIP Publishing.

https://doi.org/10.1063/1.5020601

I. INTRODUCTION

The water-wave problem in its simplest form concerns two-dimensional motion of an irrotational and incompressible inviscid liquid with a free surface, acted on only by gravity and surface tension.

Assume that the fluid is of constant density ρand denoted byΩ_t={(X,z)∈R^d ×R,−h₀+b(X)<

z< ζ(t,X)}, the domain (d = 1, 2) of the fluid for each time variable t where the surface of the fluid is a graph parametrized byζ and its bottom is parametrized byh₀ + b(X) independent of time withh₀ the depth. This motion of this fluid is described by the following free surface Euler equations:

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∂_tV+ (V· ∇_X,z)V=−g→− e_z−1

ρ∇_X,zP in (X,z)∈Ω_t,t≥0,

∇_X,z·V=0 in (X,z)∈Ω_t,t≥0,

∇_X,z×V=0 in (X,z)∈Ω_t,t≥0, P−P_atm=σκ(ζ) at z=ζ(t,X),t≥0,

∂_tζ− q

1 +|∇_Xζ|²V·n₊=0 at z=ζ(t,X),t≥0, V ·n−=0 at z=−h₀+b(X),t≥0,

|(X,z)|→∞lim |ζ(X,z)|+|V(t,X,z)|=0 in (X,z)∈Ω_t,t≥0,

(1)

whereV:R+×Ω_t−→R^d ×Ris the fluid velocity,P:R+×Ω_t−→Ris the fluid pressure term at point (X,z)∈Ω_t and instantt ≥0, denoted byP_atm the (constant) atmospheric pressure,−g→−e_z is the gravitational field which is acting vertically downward withggreater than zero and→−e_zis a unit

a)Electronic addresses:[email protected]and[email protected] b)E-mail:[email protected]

c)E-mail:s [email protected]

0022-2488/2018/59(7)/071501/20/$30.00 59, 071501-1 Published by AIP Publishing.

vector in the vertical direction. The outward unit normal vector to the free surface and the outward unit normal to the lower boundary ofΩ_tare respectively denoted by

n₊= 1 p1 +|∇_Xζ|²

∇_Xζ^T, 1T

and n−= 1

p1 +|∇_Xb|²

∇_Xb^T,−1T

.

Denoting by σ >0 is the surface tension coefficient, and κ(ζ)=−∇ · ∇ζ p1 +|∇ζ|²

is the mean curvature of the surface where∇=∇_X=(_∂^∂

x,_∂^∂

y)^T.

The complexity of this problem drove physicists, oceanographers, and mathematicians to derive simpler equations in specific physical regimes (see, for instance, Ref. 13), shallow water models (µ 1), and deep water models (µ 3 1). Many approximate models have thus been derived such as Boussinesq type equations (see Refs.4,8,13, and 21for a justification of this approx- imation) that simulated much of the wave motion issue in coastal engineering which can handle most of wave phenomena occurring in the nearshore areas (like refraction, diffraction, shoaling, frequency dispersion, and nonlinear interaction), but they cannot predict either where and when a wave breaks or, particularly, the hydrodynamic features of a breaking wave. Later on, Serre and Green-Naghdi introduced a higher order model (see Refs.5,10, and13for justification), which has since been widely used in coastal oceangraphy,11,12,20,22 since it considers the dispersive impacts ignored by the nonlinear shallow-water [or Saint-Venant request orderO(µ)] equations, and also it is currently the most well-known model for the numerical simulation of waterfront streams.

The Green-Naghdi equations, (see Refs.11,12, and15) which take into consideration neglected rotational effects (i.e., 0 =∇_X,z ×V), are significant for wind driven waves, waves riding upon a sheared current, waves near a ship, or tsunami waves approaching a shore and ensure the existence of ϕ:R+×Ω_t−→R, the velocity potential flow of the fluid such that∇_X,zϕ=VinΩ_t. This plays a great role in writing the dimensionalized water-waves equations under Bernoulli’s formulation with surface tension,

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µ∂_x²ϕ+µ∂_y²ϕ+∂_z²ϕ=0 at −1 +βb(X)<z< εζ(t,X),

∂_zϕ−µ β∇_Xb· ∇_Xϕ=0 at z=−1 +βb(X),

∂tζ −1

µ(−µε∇_Xζ· ∇_Xϕ+∂zϕ)=0 at z=εζ(t,X),

∂_tϕ+ 1

2(ε|∇_Xϕ|²+ ε

µ(∂_zϕ)²) +ζ=− 1 Bo

κ(ε√ µζ) ε√

µ at z=εζ(t,X),

(2)

where the parameter 0≤ε≤1 is often called a nonlinearity parameter, whileµ≥0 is the shallowness parameter, 0≤β≤1 is the typical amplitude of the bottom deformations (the topography parameter), and Bo is theBond numberwhich measures the ratio of gravity forces over capillary forces, introduced by

ε= a

h₀, µ=h²

0

λ², β=b₀

h₀, and Bo=ρgλ² σ ,

where we know thatais the amplitude of the wave,λis the wave length of the wave,b₀is the order of amplitude of the variations of the bottom topography,h₀is the reference depth,ρis the density of the fluid, andσis the surface tension coefficient.

The dimensionless free surface Bernoulli’s equations (2) can be reduced into a system where all functions are evaluated at the free surface (inR+×R^d), and it is known as the dimensionless version of Zakharov/Criag-Sulem⁹formulation of the water-waves equations with surface tension,

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∂tζ− 1

µG_µ[εζ,βb]ψ=0,

∂_tψ+ζ+ ε

2|∇ψ|²−εµ

µ1G_µ[εζ,βb]ψ+∇(εζ)· ∇ψ2

2(1 +ε²µ|∇ζ|²) =− 1 Bo

κ(ε√ µζ) ε√

µ ,

(3)

whereψ:R+×R^d−→Ris the trace of the velocity potential at the free surface

ψ(t,X)=ϕ t,X,εζ(t,X)=ϕ|z=ε ζ, (4) and the Dirichlet-Neumann operatorG_µ[εζ,βb]·is defined by

G_µ[εζ,βb]ψ=−µ ε∇ζ· ∇ϕ

|z=ε ζ + ∂_zϕ

|z=ε ζ=q 1 +µε²

∇ζ

2 ∂_nϕ

|z=ε ζ, (5) withϕsolving (see Ref.1for accurate analysis) the boundary value problem

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µ∂_x²ϕ+µ∂_y²ϕ+∂_z²ϕ=0 in −1 +βb(X)<z< εζ(t,X),

∂nϕ|z=−1+βb=0, ϕ|z=ε ζ=ψ(t,X),

(6)

where∂_nϕ=n·∇_X,zϕrefers to the upward normal derivative at the bottom.

In the event that no presumption is made on the nonlinearity parameter defined above, a shallow- water asymptotic regime is identified assuming that the water depth is small enough with respect to the wave length. Formally, this regime at second orderO(µ²) leads to a large amplitude model (µ1,εv1) called the Green-Naghdi system. A rigorous justification on the well-posedness of the standard Green-Naghdi equations was given by several works such as Refs.2,3, and10in 1D and 2Dwith flat and non-flat bottoms (β= 0), respectively, where a solution was constructed with a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition, unlike the Nash-Moser scheme made in Ref.14for a 2Dcase. The aim of this paper is to derive the 2Dextended (of orderµ³) Green-Naghdi system for flat bottoms (done by Matsuno in Refs.6and7, applying the method used in Ref.15forO(µ²) approximation with non-flat bottoms) represented by

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∂_tζ+∇ ·(hv)=0,

h+µT[h] +µ²T[h]∂_tv+h∇ζ

+εh(v· ∇)v+εµQ₁[U]v+εµ²Q₂[U]v=O(µ³),

(7)

wherev=(v₁,v₂)^T,U=(ζ,v)^T, andh= 1 +εζ, denoted by T[h]v=−1

3∇(h³∇·v) T[h]v=−1 45∇

∇· h⁵∇(∇·v)

Q₁[U]v=−1 3∇

h³ (v·∇)(∇·v)−(∇·v)² with

Q₂[U]v=−1 45∇f

∇ ·(

h⁵ ∇²(∇ ·v)v−5h⁵(∇ ·v)∇(∇ ·v) +∇h⁵× v× ∇(∇ ·v)) g + 2

45∇

h⁵ ∇(∇ ·v)2 + 1

45∇ ·

h⁵∇(∇ ·v)

∇(∇ ·v) + 1

90h⁵∇ ∇(∇ ·v)2

. This model which takes into account, the dispersive effects neglected by shallow-water where the existence ofµ²terms makes the analysis more difficult. The construction of a solution (1Dcase) with a standard Picard iterative scheme as in Refs.2and10cannot be achieved without considering the effect of small surface tension that smoothes the way in controlling the energy estimates. Our objective here is to demonstrate that it is additionally conceivable to utilize such an iterative scheme to study the well-posedness of the extended 1DGreen-Naghdi equations of order µ³ with surface tension.

A. Organization of the paper

The aim of this paper is to derive and study the extended shallow-water approximation to the full water wave problem, and it is ordered as follows. First of all, in Sec.II, we derive the extended 2DGreen-Naghdi system for flat bottoms with or without surface tension. Then in Sec.III A, some preliminary results are given. The well-posedness of the system is stated in Sec.III Band then proved in Sec.III C.

B. Notation

We denote byC(λ₁, λ₂,. . .) a constant depending on the parameters λ₁, λ₂, . . .andwhose dependence on theλ_jis always assumed to be nondecreasing.

The notationa.bmeans thata≤Cb, for some nonnegative constantCwhose exact expression is of no importance (in particular, it is independent of the small parameters involved).

Letpbe any constant with 1≤p<∞and denoteL^p=L^p(R^d) the space of all Lebesgue-measurable functionsf with the standard norm|f|_L^p= ∫_R^d|f(X)|^pdX1/p<∞.

Whenp= 2, we denote the norm| · |_L2simply by| · |₂. The inner product of any functionsf₁and f₂in the Hilbert spaceL²(R^d) is denoted by (f₁,f₂)=∫_R^df₁(X)f2(X)dX.

The spaceL^∞=L^∞(R^d) consists of all essentially bounded Lebesgue-measurable functionsf with the norm|f|_L∞=ess sup|f(X)|<∞.

We denote by W^1,∞=W^1,∞(R^d)=f ∈L^∞,∇f∈(L^∞)^d endowed with its canonical norm.

For any real constants,H^s=H^s(R^d) denotes the Sobolev space of all tempered distributionsf with the norm|f|_Hs=|Λ^sf|₂<∞, whereΛis the pseudo-differential operatorΛ^s=(1−∂_x²)^s/2.

For any functionsu=u(X,t) andv(X,t) defined onR^d×[0,T) withT >0, we denote the inner product, theL^p-norm and especially theL²-norm, as well as the Sobolev norm, with respect to the spatial variableX, by (u,v) = (u(·,t),v(·,t)),|u|_L^p=|u(·,t)|_L^p,|u|_L2=|u(·,t)|_L2, and|u|_H^s=|u(·,t)|_H^s, respectively.

Let f(X, t) be a vector field defined on R^d × [0,∞) of the independent variable X=(x1,x₂,. . .,x_d)∈R^d; its partial derivative with respect tox_k is denoted by ∂xkf=f_xk for 1≤k

≤d. This allows us to define thegradientoff, and we denote it as∇f=(∂_x1f₁,∂_x2f₂,. . .,∂_xdf_d)∈R^d. Also we calldivergenceoff the scalar denoted∇ ·f=Pd

i=1∂_xif_iand whend= 3, we call thecurlof f the vector denoted∇ ×f =(∂_x2f₃−∂_x3f₂,∂_x3f₁−∂_x1f₃,∂_x1f₂−∂_x2f₁)^T.

For any closed operatorT defined on a Banach spaceY of functions, the commutator [T,f] is defined by [T,f]g=T(fg)fT(g) withf,g, andfgbelonging to the domain ofT.

II. DERIVATION OF THE EXTENDED 2DGREEN-NAGHDI SYSTEM

To derive the Green-Naghdi equations (the 2Dcase), we introduce the depth averaged horizontal velocity,

v(t,X)= 1 h(t,X)

_εζ(t,X)

−1+βb(X)∇ϕ(t,X,z)dz with h(t,X)=1 +εζ(t,X)−βb(X). (8) The first equation of the Green-Naghdi system (7),∂_tζ+∇ ·(hv) = 0 which exactly coincides with the first equation of (3) stems from a clear outcome of Green’s identity or by a straightforward calculation and rearranging terms using (2). Now as in Ref.15, in order to derive the evolution equation onv, the key point is to obtain an asymptotic expansion∇ψwith respect toµand in terms ofvandζ. Since µ1, we look for an asymptotic expansion ofϕunder the form

ϕ_app(t,X,z)=ϕ₀+µϕ₁+µ²ϕ₂+· · ·+µ^Nϕ_N= XN

j=0

µ^jϕ_j. (9)

Plugging expression (9) into the boundary value problem (6) and after dropping all terms of order O(µ^N+1), one gets

∀ j=0, 1,. . .,N ∂_z²ϕ_j=−∂_x²ϕ_j−1−∂_y²ϕ_j−1, (10) with the conventionϕ₁= 0 by definition and the boundary condition

∀ j=0, 1,. . .,N 

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−β∇b∇ϕ_j−1+∂_zϕ_j=0 at z=−1 +βb,

(ϕ_j)|z=ε ζ=δ_0,jψ, (11)

whereδ_0,j= 1 ifj= 0 and zero otherwise. Solving the ordinary differential equation (ODE) (10) with (11) yields the following:

ϕ0(t,X,z)=ψ(t,X), (12)

ϕ₁(t,X,z)=(z−εζ) −1

2(z+εζ)−1 +βb∇ ·(∇ψ) +β(z−εζ)∇b· ∇ψ. (13) It is sufficient to have onlyϕ₀,ϕ₁in order to derive the equations to order µ². But for obtaining the extended Green-Naghdi system, it is essential to findϕ2. By proceeding in the same way as above and with the help of (12) and (13), one gets after denoting thatw=∇ψ

ϕ₂(t,X,z)=(z−εζ)β∇b·(∇ϕ₁)|_z=−1+βb+1

2 (z+ 1−βb)²−h²(ε∇ζ)−ε∇ζ+ 2(β∇b)∇ ·w

−2f1

2 (z+ 1−βb)²−h²h(ε∇ζ) +1 2

1

3(z−εζ)³−(z−εζ)h² (β∇b)g

∇(∇ ·w)

−f1

2 (z+ 1−βb)²−h²h∇ ·(ε∇ζ) +1 2

1

3(z−εζ)³−(z−εζ)h²∇ ·(β∇b)g

∇ ·w +f 1

24 z⁴−(εζ)⁴−1

6(−1 +βb)³(z−εζ)−(εζ)²

4 (z+ 1−βb)²−h²

−1 2

1

3(z−εζ)³−h²(z−εζ)

(−1 +βb)g

∇ ·

∇ ∇ ·w + (z+ 1−βb)²−h²(ε∇ζ)∇ β∇b·w

+ 1

2 (z+ 1−βb)²−h²∇ ·(ε∇ζ)β∇b·w

−1 2

1

3(z−εζ)³−(z−εζ)h²∇ · ∇(β∇b·w) ,

the polynomial of order 4 inz. So the horizontal component of the velocity in the fluid domain is given by

V(t,X,z)=∇ϕ_app=∇ϕ₀(t,X,z) +µ∇ϕ₁(t,X,z) +µ²∇ϕ₂(t,X,z) +O(µ³). (14) The averaged velocity is thus given by

v(t,X)=∇ψ+ µ h

_εζ(t,X)

−1+βb(X)∇ϕ₁dz+ µ² h

_εζ(t,X)

−1+βb(X)∇ϕ₂dz+O(µ³). (15) For flat bottoms, we take here and throughout the rest of this paper β= 0.

As in Ref.15, we have _εζ(t,X)

−1

∇ϕ₁dz=T[h]∇ψ=−1

3∇ h³∇ ·(∇ψ)

. (16)

In order to computeJ=∫₋₁^εζ(t,X)∇ϕ₂dz, denote byw=∇ψ(noting thatwis independent ofz),

∇ϕ₂(t,X,z)=(h∇h)h∇ ·(∇h∇ ·w) +1

2 h²−(z+ 1)²∇ h∇ ·(∇h∇ ·w) +h∇h ∇h∇(h∇ ·w)

+ 1

2 h²−(z+ 1)²∇ ∇h∇(h∇ ·w) +f 1

24 z⁴−(εζ)⁴ +1

6(z−εζ)−(εζ)²

4 (z+ 1)²−h² +1

2 1

3(z−εζ)³−h²(z−εζ)g

∇(

∇ ·

∇ ∇ ·w) +f

−1

6(εζ)³∇h−h∇h(z−εζ)− 1 6−1

2h²∇h−(εζ)

2 ∇h (z+ 1)²−h² + (εζ)²

2 h∇h−1

2(z−εζ)²∇hg

∇ · ∇(∇ ·w)

=J₁+J₂+. . .. +J₆.

Now after some calculations, we have the following integrals:

_εζ(t,X)

−1

(J1+J₂)dz=1

3∇ h⁴∇ ·(∇h∇ ·w) , _εζ(t,X)

−1

(J3+J₄)dz= 1

12∇ ∇h⁴∇(h∇ ·w) , _εζ(t,X)

−1

(J5+J₆)dz= 2 15∇

h⁵∇ · ∇(∇ ·w) , with _εζ(t,X)

−1

f 1

24 z⁴−(εζ)⁴ +1

6(z−εζ)−(εζ)²

4 (z+ 1)²−h² +1

2 1

3(z−εζ)³−h²(z−εζ)g dz= 2

15h⁵ and _εζ(t,X)

−1

f−1

6(εζ)³−h(z−εζ)− 1 6−1

2h²−(εζ)

2 (z+1)²−h² +(εζ)²

2 h−1

2(z−εζ)²g

∇h dz= 2 15∇h⁵. Hence, we get

J= _εζ(t,X)

−1

∇ϕ₂dz= 2 15∇

∇ · h⁵∇(∇ ·w) +1

3∇ h³∇ ·(h∇h)∇ ·w=T⁰[h]w.

Thus, we have

v=∇ψ− µ

hT [h]∇ψ+ µ²

h T⁰[h]∇ψ+O(µ³), (17) but

∇ψ=v+ µ

hT[h]v+ µ² h

fT[h] 1

hT[h]v−T⁰[h]vg

(18) and

T[h] 1

hT[h]v=1 9∇

∇ · h⁵∇(∇ ·v) + 1

3∇ h³∇ ·(h∇h)∇ ·v

. (19)

Therefore, we obtain

∇ψ=v+ µ

hT[h]v+ µ²

h T[h]v+O(µ³), (20) where

T[h]v=−1

3∇(h³∇ ·v) and T[h]v=−1 45∇

∇ · h⁵∇(∇ ·v)

. (21)

Now, in order to derive the extended Green-Naghdi equations for flat bottoms without surface tension (i.e.,σ= 0), we will take the gradient of the second equation of (3), then multiply it byh, and replace

∇ψby its expression (20) and ¹_µG[εζ,βb]ψby∇ ·(hv) =∇h·v +h∇ ·vin the resulting equations.

Moreover, we drop theO(µ³) terms, and we use the following vector triple products and the vector identities:

u×(ν×ω)=(u·ω)ν−(u·ν)ω, (22)

∇ ×(∇G)=0 and ∇ ×(GF)=G∇ ×F+∇G×F, (23) whereGis a differentiable scalar function andu,ν,ω,Fare differentiable vector fields; one gets the same results obtained in Refs.6and7done by Matsuno applying another method,

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∂_tζ+∇ ·(hv)=0,

h+µT [h] +µ²T[h]∂_tv+h∇ζ

+εh(v· ∇)v+εµQ₁[U]v+εµ²Q₂[U]v=O(µ³),

(24)

wherev=(v₁,v₂)^T,U= (ζ,v)^T, andh(t,X) = 1 +εζ(t,X), denoted by T[h]v=−1

3∇(h³∇·v) T[h]v=−1 45∇

∇· h⁵∇(∇·v)

Q₁[U]v=−1 3∇

h³ (v·∇)(∇·v)−(∇·v)² , with

Q₂[U]v=−1 45∇f

∇ ·(

h⁵ ∇²(∇ ·v)v−5h⁵(∇ ·v)∇(∇ ·v) +∇h⁵× v× ∇(∇ ·v)) g + 2

45∇

h⁵ ∇(∇ ·v)2 + 1

45∇ ·

h⁵∇(∇ ·v)

∇(∇ ·v) + 1

90h⁵∇ ∇(∇ ·v)2

, where the expression ofQ₂introduces the Laplacian operator∇²=∇ · ∇=∆.

In the presence of surface tension (σ,0), different strategies exist to deal with it in the water- wave problem such as Refs.9and16–18; the main contrast in our work is that the gradient of the capillary term− 1

Bo κ(ε√

µζ) ε√

µ multiplied by h must be added to the right-hand side of the second equation in (24).

Let us define therescaled Bond numberbo instead of theclassical Bond numberBo, as follows:

bo=µBo= ρgh²₀ σ >0,

whereh₀denotes the reference depth,ρdenotes the positive constant density of the fluid,gdenotes the acceleration of gravity, andσ >0 denotes the surface tension coefficient so that Bo¹=µbo¹= O(µ), and the capillary term that should be added becomes

− 1

Boh∇(κ(ε√ µζ) ε√

µ )= 1

boµh∇

∇ · ∇ζ

− 1

2boε²µ²h∇

∇ · |∇ζ|²∇ζ

+O(ε⁴µ³). (25)

III. WELL-POSEDNESS OF THE EXTENDED 1DGREEN-NAGHDI SYSTEM WITH SURFACE TENSION

For one-dimensional (d= 1) surfaces, the Green-Naghdi system (24) with surface tension can be rearranged after a few calculations, and considering (25), one may write

















∂_tζ+∂_x(hv)=0,

h+µT[h] +µ²T[h]∂tv+h∂xζ +εhv∂xv +εµQ₁[U]v+εµ²Q₂[U]v= 1

boµhζ_xxx+ε²µ² 1

boT[U]ζ_x+O(µ³),

(26)

whereU= (ζ,v)^Tandh(t,x) = 1 +εζ(t,x), denoted by T[h]v=−1

3∂_x(h³∂_xv), T[h]v=−1

45∂_x² h⁵∂_x²v

, Q₁[U]v=−1 3∂_x

h³ vv_xx−v_x² , and

Q₂[U]v=−1 45∂_x(

∂_x h⁵(vv_xxx−5v_xv_xx)−3h⁵(v_xx)²)

, T[U]ζ_x=−1

2h∂_x² ζ_x²ζ_x . Now, adding and subtractingεµT[h](vv_x) andεµ²T[h](vv_x) in the second equation of (26), one gets



















∂_tζ+∂_x(hv)=0,

h+µT[h] +µ²T[h] ∂_tv+εvv_x+h∂_xζ+ 2

3εµ∂_x h³v_x² + 1

45εµ²∂_x8∂_x h⁵v_xv_xx

+ 3h⁵v_xx²)

= 1

boµhζ_xxx+ε²µ² 1

boT[U]ζ_x+O(µ³).

(27)

Now, denote by=+ 2µ²T[h]=h+ µT [h] + µ²T[h] [i.e., set±µ²T[h](∂_tv +εvv_x) in (27)2] and note that from (26)2, [i.e., applying a BBM trick (Benjamin-Bona-Mahony²³)], we get the following approximated equation

∂_tv+εv∂_xv=−∂_xζ+O(µ). (28) Furthermore, substitute (28) in the acquired new term 2µ²T[h](∂_tv+εvv_x) to get−2µ²T[h]ζ_x, then after some computations [needed especially for a suitable specification of a new operatorJ_bointro- duced in (39)] we obtain the one-dimensional extended Green-Naghdi system with surface tension rewritten into



















∂_tζ+∂_x(hv)=0,

= ∂_tv+εvv_x

+h∂_xζ− 1

boµhζ_xxx+ 2

45µ²h∂_x² h⁴ζ_xxx

+µ²(I1[h]ζ_x+I₂[h]ζ_x)

−ε²µ² 1

boT[U]ζ_x+εµQ₁[U]v_x+εµ²Q₂[U]v_x+εµ²Q₃[U]v_x=O(µ³), (29) whereU= (ζ,v)^T,h(t,x) = 1 +εζ(t,x), and

T[h]w=−1

3∂_x(h³∂_xw), T[h]w=−1

45∂_x² h⁵∂_x²w

, T[U]f=−1

2h∂_x² ζ_x²f

, (30)

Q₁[U]f =2

3∂_x h³v_xf

, Q₂[U]f= 8

45∂_x² h⁵v_xxf

, Q₃[U]f= 1

15∂_x h⁵v_xx∂_xf

, (31)

I₁[h]f = 4

45h_x∂_x(h⁴∂_x²f) and I₂[h]f= 2

45h⁴h_xx∂_x²f (32) Remark 1. The interest of formulation(27)is that all the fifth order derivatives of vare factorized in h+µT[h] +µ²T[h] ∂_tv+εvv_x which is of notable help. However, the benefit of formulation (29)is in replacing h+µT[h] +µ²T[h]by a new operator=so that the coercivity condition of the bilinear form is satisfied when applying a Lax-Millgram theorem for the proof of the invertibility of

=(see Lemma 1).

Remark 2. The reason for considering the effect of surface tension(see Ref.1for a brief physical relevance)is due to the betterment needed on the natural associated energy norm| · |_Y^s [i.e.,when no suface tension(bo⁻¹=0)]defined by

|(ζ,v)|²_Ys=|ζ|_H²s+µ²|ζ_xx|²_Hs+|v|_H²s+µ|v_x|²_Hs+µ²|v_xx|_H²s.

Because the norm | · |_Y^s is not adequate for the proof of the energy estimate (see for instance the control of the term A₂ + A₃),the addition of the quantities in(25)permits the definition of a new energy norm,

|(ζ,v)|²_Xs=|ζ|_H²s+ µ

bo|ζ_x|_H²s+µ²|ζ_xx|²_Hs+|v|_H²s+µ|v_x|²_Hs+µ²|v_xx|_H²s.

The second term in| · |_X^sis absent from the natural energy| · |_Y^s,and this will be a fundamental term for allowing the control of several inconvenient terms as we are going to figure it out in Sec.III B.

Moreover, the spaceζ∈H^s(R);|ζ|²_Hs+µ²|ζ_xx|_H²s<∞ is not equivalent to the Sobolev space H^s+2(R), which required in defining the energy space X^sof our problem (see Definition 1).

A. Preliminary results

Under the nonzero depth-condition,

∃ h_min>0, inf

x∈R

h≥h_minwhereh(t,x)=1 +εζ(t,x), (33) which is a fundamental condition for the extended Green-Naghdi system to be physically legitimate, and it says that the water bottom is constantly limited from underneath by a nonnegative constant.

We introduce the operator

==h+µT[h]−µ²T[h]=h−1

3µ∂_x(h³∂_x·) + 1

45µ²∂_x² h⁵∂_x²·

, (34)

which plays an important role in the energy estimate and the local well-posedness of the extended Green-Naghdi system. We shall give an essential invertibility result on=and specify some properties on its inverse=⁻¹,which are explained in the following lemmas.

Lemma 1. Assume thatζ(t,·)∈L^∞(R)is a differentiable scalar function under condition(33).

Then, the operator

=:H_µ²(R)−→L²(R) is well defined, one-to-one and onto.

Proof. The proof of the invertibility of=is a direct application on the Lax-Millgram theorem.

We introduce first the spaceH_µ²(R) endowed with the norm| · |_µdefined as

H_µ²(R)=v∈H²(R);|v|²_µ=|v|²₂+µ|v_x|²₂+µ²|v_xx|₂²<∞ , (35) where| · |_µis equivalent to| · |_H2 but not uniformly with respect toµ.

Letf ∈L²(R). Consider the weak problem









Find v∈H_µ²(R) such that a(v,u)=L(u) ∀u∈H_µ²(R), with

a(v,u)=(=v,u)=(hv,u) +1

3µ(h³v_x,u_x) + 1

45µ²(h⁵v_xx,u_xx), L(u)=(f,u).

It is easy to see thataandLare continuous onH_µ²(R)×H_µ²(R) andH_µ²(R), respectively. In addition, ais coercive onH_µ²(R)×H_µ²(R) using (33) with

a(v,v)≥min h_min,1 3h³_min, 1

45h⁵_min|v|²_µ. (36)

Therefore by Lax-Millgram theorem, for everyf ∈L²(R), there exists a uniquev∈H_µ²(R) such that for allu∈H_µ²(R), we have

a(v,u)=(=v,u)=L(u)=(f,u).

Hence the result.

The following lemma gives functional properties to the operator=⁻¹.

Lemma 2. Let t₀>₂¹andζ∈H^t0+1(R)be such that(33)is satisfied. Then, we have the following:

(i) ∀0≤s≤t₀+ 1,

|=⁻¹f|_Hs+√

µ|∂_x=⁻¹f|_Hs+µ|∂_x²=⁻¹f|_Hs≤C 1

h_min,|h−1|_Ht0 +1

|f|_Hs. (ii) ∀0≤s≤t₀+ 1,

√µ|=⁻¹∂_xf|_Hs+µ|∂_x=⁻¹∂_xf|_Hs+µ√

µ|∂_x²=⁻¹∂_xf|_Hs≤C 1

h_min,|h−1|_Ht0 +1

|f|_Hs, and

µ|=⁻¹∂_x²f|_Hs+µ√

µ|∂_x=⁻¹∂_x²f|_Hs+µ²|∂_x²=⁻¹∂_x²f|_Hs≤C 1

h_min,|h−1|_Ht0 +1

|f|_Hs. (iii) ∀s≥t₀+ 1,

k=⁻¹k_Hs(R)→H^s(R)+√

µk=⁻¹∂_xk_Hs(R)→H^s(R)+µk=⁻¹∂_x²k_Hs(R)→H^s(R)≤C_s, with

µk∂_x=⁻¹∂_xk_Hs(R)→H^s(R)+µ√

µk∂_x²=⁻¹∂_xk_Hs(R)→H^s(R)≤C_s,

and

µ√

µk∂x=⁻¹∂_x²k_Hs(R)→H^s(R)+µ²k∂_x²=⁻¹∂_x²fk_Hs(R)→H^s(R)≤C_s,

where C_sis a constant depending on 1/h_minand|h−1|_H^sand independent of (ε,µ)∈(0, 1)². Proof. The proof is adapted as in Ref.2for 1DGreen-Naghdi equations (µ²order withβ,0).

Assume thatf∈H^s(R) andu==⁻¹f, then=u=f. ApplyΛ^sto both sides, then one can deduce the following:

a(Λ^su,Λ^su)=(Hf,Λ^su) +√

µ(∂_xHg,Λ^su) +µ(∂_x²Hp,Λ^su), whereHf,Hg, andHpare written as follows:

Hf=Λ^sf −[Λ^s,h]u, Hg=1 3

√µ[Λ^s,h³]ux, Hp=−1

45µ[Λ^s,h⁵]uxx. Integrating by parts and using (36), we get

min h_min,1 3h_min³ , 1

45h⁵_min|Λ^su|_µ≤ |Hf|₂+| Hg|₂+|

Hp|₂.

Now, using the Kato-Pance commutator estimate (see Lemma 4.6 of Ref.14), we have

[Λ^s,f]u

².|∇f|_Ht0|u|_Hs−1. (37) One can deduce

|Hf|₂+|Hg|₂+|Hp|₂≤ |f|_H^s+C |h−1|_Ht0 +1

|Λ^s−1u|_µ.

Hence, the inequality (i) holds after continuous induction ons. For the proof of (ii), one has to replace u=√

µ=⁻¹∂_xf andu=µ=⁻¹∂_x²f for a second time. The general strategy is the same as in (i) noticing thatΛ^scommutes with∂_x,∂_x². The only difference is in the expression ofHf,Hg, andHp. In fact, when u=√

µ=⁻¹∂xf, we have the following:

Hf=−[Λ^s,h]u, Hg=Λ^sf +1 3

√µ[Λ^s,h³]ux and Hp=−1

45µ[Λ^s,h⁵]uxx, and whenu=µ=⁻¹∂_x²f, we have the following:

Hf =−[Λ^s,h]u, Hg=1 3

√µ[Λ^s,h³]ux and Hp=Λ^sf − 1

45µ[Λ^s,h⁵]uxx.

In (iii), since s ≥ t₀ + 1, then the Kato-Pance comutator estimate (37) is given by∀f∈H^s(R), u∈H^s−1(R)

[Λ^s,f]u

².|∇f|_Hs−1|u|_Hs−1, (38) and one hasH^s(R),→H^t0+1(R). Now, proceeding in the same way as in (i), we get the same result

with the constantC_sdepending on 1/hminand|h−1|_Hs.

B. Linear analysis

In order to rewrite the extended Green-Naghdi system (for flat bottoms) with surface tension in a condensed form, we introduce a new operatorJ_bo,

U=(ζ,v)^T, J_bo=1− µ

bo∂_x²(·) + 2

45µ²∂_x² h⁴∂_x²·

, h(t,x)=1 +εζ(t,x). (39) The first equation in (29) can be written as

∂_tζ+εv∂_xζ+h∂_xv=0.

For the second equation in (29), applying=⁻¹to both sides, we get

∂tv+εvvx+=⁻¹ hJ_boζx+µ²=⁻¹ I₁[h]ζx+I₂[h]ζx−ε²µ² 1

bo=⁻¹ T[U]ζx

+εµ=⁻¹ Q₁[U]v_x+εµ²=⁻¹ Q₂[U]v_x+εµ²=⁻¹ Q₃[U]v_x=O(µ³).

Hence the extended Green-Naghdi system (β= 0) with surface tension can be written in the form

∂_tU+A[U]∂_xU=0, (40)

where

A[U]= εv h

=⁻¹ hJ_bo·

+µ²I−ε²µ^{2 1}_bo=⁻¹ T[U]·εv+εµ=⁻¹ Q₁[U]· +εµ²Q

!

, (41)

with

I==⁻¹ I₁[h]·

+=⁻¹ I₂[h]·

and Q==⁻¹ Q₂[U]·

+=⁻¹ Q₃[U]·

. (42)

Now, consider the linearized system around some reference stateU=(ζ,v)^T,









∂_tU+A[U]∂_xU=0,

U|_t=0=U₀. (43)

The proof of the energy estimate which permits the convergence of an iterative scheme to construct a solution to the extended Green Naghdi system (29) with surface tension for the initial value problem (43) requires one to define theX^sspaces, which are the energy spaces for this problem.

Definition 1. For all s≥0 and T >0,we denote by X^s the vector space H^s+2(R)×H^s+2(R) endowed with the norm

|U|_X²sB|ζ|_H²s+ µ

bo|ζ_x|²_Hs +µ²|ζ_xx|_H²s+|v|²_Hs +µ|v_x|_H²s+µ²|v_xx|²_Hs

while X_T^s stands for C([0,^T_ε];X^s)endowed with its canonical norm.

First, recall that a pseudo-symmetrizer forA[U] is given by S=*

. ,

Jbo 0 0 = + / -

, (44)

withh=1 +εζ,==h+µT[h]−µ²T[h], andJ

bo=1− µ

bo∂_x²(·) + 2

45µ²∂_x² h⁴∂_x²·

. A natural energy for the initial value problem (43) is suggested to be

E^s(U)²=(Λ^sU,SΛ^sU). (45)

The connection betweenE^s(U) and theX^s-norm is examined using the lemma below.

Lemma 3. Let s≥0andζ∈L^∞(R). Under the depth-condition

∃ h_min>0, inf

x∈R

h≥h_min, h(t,x)=1 +εζ(t,x), (46) E^s(U) is uniformly equivalent to the| · |_Xs-norm with respect to (µ,ε)∈(0,1)²,

E^s(U)≤C 1

h_min,|h|∞|U|_X^s and |U|_X^s≤C h_min, 1 h_min

E^s(U).

Proof. First note that,E^s(U)²=(Λ^sU,SΛ^sU) withSΛ^sU=(J

boΛ^sζ,=Λ^sv). Then we get E^s(U)²=(Λ^sζ,J

boΛ^sζ) + (Λ^sv,=Λ^sv).

Using the expression of=,J

bointegrating by parts, we get E^s(U)²=(Λ^sζ,Λ^sζ) + µ

bo Λ^sζx,Λ^sζx+ 2

45µ² h⁴Λ^sζxx,Λ^sζxx+ (Λ^sv,hΛ^sv) + µ

3(Λ^sv_x,h³Λ^sv_x) + 1

45µ²(Λ^sv_xx,h⁵Λ^sv_xx),