Table Of Content

The confined Muskat problem: differences with the deep water regime Diego Córdoba Gazolaz1,4, Rafael Granero-Belinchón2,4 and Rafael Orive Illera3,4,5 September 10, 2012 2 1 0 2 Abstract p e We study the evolution of the interface given by two incompressible fluids with different S densities in the porous strip R×[−l,l]. This problem is known as the Muskat problem and 7 is analogousto the two phase Hele-Shaw cell. The main goalofthis paper is to compare the qualitative properties between the model when the fluids move without boundaries and the ] P modelwhenthefluidsareconfined. Wefindthat,inaprecisesense,theboundariesdecrease A the diffusion rate and the system becomes more singular. . h t Keywords: Darcy’slaw,Hele-Shawcell,Muskatproblem,maximumprinciple,well-posedness, a m blow-up, ill-posedness. Acknowledgments: The authors are supported by the Grants MTM2011-26696 and SEV- [ 2011-0087 from Ministerio de Ciencia e Innovación (MICINN). Diego Co´rdoba was partially 1 supported by StG-203138CDSIF of the ERC. Rafael Granero-Belinchoń is grateful to A. Castro v 5 and F. Gancedo for their helpful comments during the preparation of this work. 7 5 1 1 Introduction . 9 0 In this paper we study the evolution of the interface between two different incompressible fluids 2 1 with the same viscosity in a flat two-dimensional strip. This problem has an interest because it : is a model of an aquifer or an oil well, see [18]. In this phenomena, the velocity of a fluid in a v i porous medium satisfies Darcy’s law X r µ a v = −∇p−gρe2, (1) κ where µ is the dynamic viscosity, κ is the permeability of the medium, g is the acceleration due to gravity, ρ is the density of the fluid, p is the pressure of the fluid and v is the incompressible field of velocities, see [3, 19]. Equation (1) has also been considered as a model of the velocity for cells in tumor growth, see for instance [13, 22] and references therein. 1Email: [email protected] 2Email: [email protected] 3Email: [email protected] 4InstitutodeCienciasMatema´ticasCSIC-UAM-UC3M-UCM,ConsejoSuperiordeInvestigacionesCient´ıficas, C/Nicolás Cabrera, 13-15, Campus deCantoblanco, 28049 - Madrid 5DepartamentodeMatema´ticas, Facultad deCiencias, UniversidadAutońomadeMadrid, CampusdeCanto- blanco, 28049 - Madrid 1 1.5 l 1 ρ1 0.5 z(α,t) 0 −0.5 ρ2 −1 −l −1.5 −8 −6 −4 −2 0 2 4 6 8 Figure 1: Physical situation for an interface z(α,t) in the strip R×(−l,l). The motion of a fluid in a two-dimensional porous medium is analogous to the Hele-Shaw cell problem, see [14]. In this case the fluid is trapped between two parallel plates. The mean velocity in the cell is described by 12µ v = −∇p−gρe , b2 2 where b is the (small) distance between the plates. We consider the two-dimensional flat strip S = R×(−l,l)⊂ R2 with l > 0. In this strip we have two immiscible and incompressible fluids with the same viscosity and different densities, ρ1 in S1(t) and ρ2 in S2(t), where Si(t) denotes the domain occupied by the i−th fluid. The curve z(α,t) = {(z (α,t),z (α,t)) : α ∈ R} 1 2 is the interface between the fluids. We suppose that the initial interface f (x) is a graph and 0 |f (x)| ≤ l for all x. The character of being a graph is preserved at least for a short time (see 0 Section 3). The Rayleigh-Taylor condition is defined as RT(α,t) = −(∇p2(z(α,t))−∇p1(z(α,t)))·∂⊥z(α,t). α Due to the incompressiblity of the fluids and using that the curve can be parametrized as a graph, the Rayleigh-Taylor condition reduces to the sign of the jump in the density: RT = g(ρ2 −ρ1) > 0. This condition is satisfied if the denser fluid is below. We consider the velocity field v, the pressure p and the density ρ ρ(t) = ρ1χ +ρ2χ , (2) S1(t) S2(t) in the whole domain S. We also consider the conservation of mass equation, so we have a weak solution to the following system of equations µ v(x,y,t) = −∇p(x,y,t)−gρe in S, t > 0, 2 κ  ∇·v(x,y,t) = 0 in S, t > 0, (3)  ∂ ρ(x,y,t)+v·∇ρ(x,y,t) = 0 in S, t > 0,  t  f(x,0) = f (x) in R, 0     2 with impermeable boundary conditions for the velocity (see Section 2). Wedenotebyv1(x,y,t)thevelocity fieldinS1(t)andbyv2(x,y,t) thevelocity fieldinS2(t). Because of the incompressibility condition, the normal components of the velocities v1,v2 are continuousthroughtheinterface. Moreover,theinterfacemovesalongwiththefluids. Therefore, if initially we have an interface which is the graph of a function, we have the following equation for the interface: ∂ f(x,t)= (−∂ f(x,t),1)·vi(x,f(x,t),t) = 1+(∂ f(x,t))2n·vi, (4) t x x where n denotes the unit normal to the interface. p In each subdomain Si(t) the fluids satisfy Darcy’s law (1), µ vi(x,y,t) = −∇pi(x,y,t)−gρi(0,1) in Si(t), (5) κ and the incompressibility condition ∇·vi(x,y,t) = 0 in Si(t). (6) We define the following dimensionless parameter (see [4] and references therein) kf k 0 L∞ A = . (7) l This parameter is called the nonlinearity (or amplitude) parameter and we have 0 ≤ A ≤1. The case A= 1 is the case where f reaches the boundaries and we call it the large amplitude regime. In [16] they consider a two dimensional dropletin vacuum over a plate driven by surface tension. The case A = 0 is the deep water regime for which the equation reduces to ρ2−ρ1 (∂ f(x)−∂ f(x−η))η x x ∂ f = P.V. dη. (8) t 2π R η2+(f(x)−f(x−η))2 Z It has been shown, for equation (8), local existence in Sobolev spaces when the Rayleigh-Taylor condition holds (see [9]), a maximum principle for the L∞ norm of f and also a maximum principle for k∂ fk (see [10]). For initial data with k∂ f k ≤ 1 follows global existence x L∞ x 0 L∞ of W1,∞ solution (see [7]). For large initial datum there are turning waves, i.e a blow up for k∂ fk (see [6]). For other results see [1, 5, 6, 8, 15, 23]. x L∞ The equation for the evolution of the interface in our bounded domain, which is deduced in Section 2, is ρ2−ρ1 ∂ f(x,t)= P.V. (∂ f (x)−∂ f(x−η))Ξ (x,η,f) t x x 1 8l R Z (cid:20) ρ2−ρ1 +(∂ f(x)+∂ f (x−η)Ξ (x,η,f) dη = A[f](x), (9) x x 2 4l (cid:21) where the singular kernels Ξ and Ξ are defined as 1 2 sinh πη Ξ (x,η) = 2l , (10) 1 cosh πη −cos(π(f(x)−f(x−η))) 2l 2l(cid:0) (cid:1) corresponding to the singular character(cid:0)of t(cid:1)he problem, and sinh πη Ξ (x,η) = 2l , (11) 2 cosh πη +cos(π(f(x)+f(x−η))) 2l 2l(cid:0) (cid:1) (cid:0) (cid:1) 3 which becomes singular when f reaches the boundaries. The text P.V. denotes principal value. As for the whole plane case (see [7, 11]) the spatial operator A[f](x) can be written as an x-derivative. Indeed, tan π f(x)−f(x−η) 2l 2l 2 A[f](x) = P.V. ∂ arctan dη π R x  (cid:16)tanh π η (cid:17) Z 2l2 2l   (cid:0) (cid:1)π f(x)+f(x−η) π η + P.V. ∂ arctan tan tanh dη (12) x π R 2l 2 2l2 Z (cid:18) (cid:18) (cid:18) (cid:19) (cid:16) (cid:17)(cid:19)(cid:19) and we conclude the mean conservation f(x,t)dx = f (x)dx. (13) 0 R R Z Z When we do not parametrize the curve as a graph, i.e., we consider z(α) = (z (α),z (α)), 1 2 we obtain the equation ρ2−ρ1 (∂ z(α)−∂ z(η))sinh(z (α)−z (η)) α α 1 1 ∂ z = P.V. t 4π ZR(cid:20)cosh(z1(α)−z1(η))−cos(z2(α)−z2(η)) (∂ z (α)−∂ z (η),∂ z (α)+∂ z (η))sinh(z (α)−z (η)) α 1 α 1 α 2 α 2 1 1 + dη. (14) cosh(z (α)−z (η))+cos(z (α)+z (η)) 1 1 2 2 (cid:21) As our interface moves in a bounded medium the correct space to consider is Hs = Hs(R)∩{f : kfk < l}. l L∞ Thedensityρdefinedasin(2)isaweaksolutionoftheconservation ofmassequationpresent in (3) if and only if the interface verifies the equation (4) (see Proposition 1 in Section 2 below). It also follows (see 21) that if we take the limit A → 0 we recover the equation (8) (see [9]). In a recent work [12], J. Escher and B-V.Matioc studied the problem (5), (6) in the case with different viscosities and surface tension in a periodic (in x) domain when 0 < A < 1. They obtained an abstract evolution equation for the interface and showed well-posedness in the classical sense when the Rayleigh-Taylor condition is satisfied and the interface is in a neighbourhood of the zero function in certain Hölder spaces. They consider the problem as a problem in two coupled domains. The domains are coupled by the interface and by the Laplace- Young condition p2(x,f(x,t),t)−p1(x,f(x,t),t) = γκ[f], where κ[f] denotes the curvature of the interface f(x,t) and γ denotes the surface tension coefficient. In Section 3 we study the similarities between the case A = 0 and 0 < A < 1; first, we prove local well-posedness in Sobolev spaces (see Section 3.1) and instant analyticity in a growing complex strip when the Rayleigh-Taylor condition is achieved (see Section 3.2). The last similarity studied in Section 3.3 is that for arbitrary initial curves which are analytic there is an unique local solution, which is analytic, both forward and backward in time. We remark that for this result the Rayleigh-Taylor condition is not needed. The proofs follows the steps of the paper [6]. Here we show that the contribution from the boundary does not affect the a priori estimates from [6]. Themainpurposeofthisworkistostudythedifferencesbetweenthecasewithinfinitedepth and the case with bounded medium. This is done in Section 4, where we study some qualitative 4 properties of the solution. We prove the maximum principle for kf(t)k and also for k∂ fk L∞ x L∞ by studying the evolution of the maximum or the minimum values. The ODEs coming from these analysis have local and non-local terms and the main dificulty is to compare these two different kind of terms in order to ensure the decay. In particular, we prove the following decay estimate for kf(t)k in a confined medium: L∞ ddtkf(t)kL∞ ≤ −c(kf0kL1,kf0kL∞,ρ2,ρ1,l)e−lkπfk(ft0)kkLL∞1 . (15) As a corollary we conclude that the unique one-signed, integrable, stationary solution is the rest state. Let us observe that the natural boundary condition for the velocity, v ·n = 0, imposes that if our initial interface is close enough to the boundary the evolution of the maximum is very slow. Due to this fact, we obtain the slow decay inequality (15). We show that if the initial data is in a region depending on kf k and k∂ f k then we 0 L∞ x 0 L∞ have a maximum principle for k∂ fk in a confined medium. We consider a smooth initial x L∞ data f in the Rayleigh-Taylor stable regime such that the following conditions holds: 0 k∂ f k ≤ 1, (16) x 0 L∞ πkf k π 0 L∞ tan ≤k∂ f k tanh , (17) x 0 L∞ 2l 4l (cid:18) (cid:19) (cid:16) (cid:17) and π π π3 k∂ f k +|2(cos −2)sec4 |k∂ f k3 x 0 L∞ 2l 4l x 0 L∞ 8l3 (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17) tan π k∂xf0kL∞ 1+k∂ f k k∂ f k + 2l 2 x 0 L∞ x 0 L∞ (cid:16)tanh(4πl) (cid:17)!! π2 × 6tanh π 4l2 4l π π +4tan kf(cid:0)0kL(cid:1)∞ −4k∂xf0kL∞cos kf0kL∞ ≤ 0 (18) 2l l (cid:16) (cid:17) (cid:16) (cid:17) Then k∂ f(t)k ≤ k∂ f k . (19) x L∞ x 0 L∞ Moreover, if (x(l),y(l)) is the solution of the system tan πx −ytanh π = 0 2l 4l  y+(cid:0)|2((cid:1)cos π −(cid:0)2)s(cid:1)ec4 π |y3 1+y y+ttaann(h(2πl4πy2l))!! π 5 (20)  2l 4l 6tanh(4πl) 2l (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:1) +4(cid:0)tan(cid:1) πx −4ycos πx = 0, 2l l   andwe have that k∂ f k < y(l) and kf k < x(l) we have (cid:0)that(cid:1) (cid:0) (cid:1) x 0 L∞ 0 L∞ k∂ fk ≤ 1. x L∞ The effect of the boundaries is very important at this level, and we obtain a region (the yellow one in Figure 2) where we do not have maximum principle for k∂ f k but we have an x 0 uniform bound k∂ f(t)k ≤ 1 ∀t≥ 0. The green region is the region where we have maximum x L∞ principle for the derivative. Due to the term coming from the effect of the boundaries, Ξ , 2 the conditions that we obtain are much more restrictives than in the deep water regime (the case with infinite depth). The previous result gives us conditions on the smallness of A and 5 1 0.9 0.8 0.7 0.6 Maximum ∂|| f||∞x0L0.5 Principle 0.4 0.3 0.2 Uniform 0.1 bound 00 0.5 1 1.5 ||f0||L∞ Figure 2: Different regions in (kf k ,k∂ f k ) for the behaviour of k∂ fk when π = 2l. 0 L∞ x 0 L∞ x L∞ k∂ f k (which, for fixed amplitude, can be understood as the ’wavelength’ of the wave) so, x 0 L∞ roughly speaking the Theorem says that if we are in the long wave regime (small amplitude and large wavelenght) then there is no turning effect, i.e. there are no shocks. We remark that if we take the limit A → 0 we recover the result for the deep water regime contained in [10] We study the formation of singularities in Section 4.2. The singularity is a blow up of k∂ f(t)k . Physically this result means that there are waves such that they ’turn over’. x L∞ Moreover,wecomparethisresultwiththeresultforthedeepwaterregime(see[6]). Inparticular, we prove the following turning effect in a confined medium: There exist initial data z (α) = 0 (z (α),z (α)) such that in finite time the solution of (14) achieves the unstable regime only 1 2 when the depth is finite. If the depth is infinite the same curves are depleted. In Section 4.3 we show an ill-posedness result in Sobolev spaces. The key point of this result is that we do not need global existence for some class of solutions to prove the result (compare with [9] and [23]). Remark 1 In order to simplify the notation we take µ/κ = g = 1 and we sometimes suppress the dependence on t. We denote v the component i−th of the vector v. We remark that vi is i the velocity field in Si(t). We write n for the unitary normal to the curve Γ vector and n¯ for the non-unitary normal vector. We denote ρ¯= ρ2−ρ1. 4l 2 The equation for the internal wave In this section we obtain the equation for the interface z(α,t) in an explicit formula. First we have to add impermeable boundary conditions for v, i.e. v(x,±l,t)·n = 0. Using the incompressibility condition we have that there exists a scalar function Ψ such that v = ∇⊥Ψ. The function Ψ is the stream function. Then ∆Ψ = −curl(0,ρ) = ω where the vorticity is supported on the curve ω(x,y) = ̟(α)δ((x,y)−z(α,t)), with amplitude ̟(α) = −(ρ2−ρ1)∂ z (α). α 2 6 InthisdomainweneedtoobtaintheBiot-Savart law. TheGreenfunctionfortheequation∆u= f in the strip R×(0,2l) (with homogeneous Dirichlet conditions) is given by the convolution with the kernel ∞ 1 G(x,y,µ,ν) = log (x−µ)2+(y−(4nl+ν))2 2π n=X−∞(cid:20) (cid:16)p (cid:17) −log (x−µ)2+(y−(4nl−ν))2 . (cid:21) (cid:16)p (cid:17) The Biot-Savart law in this strip is given by the kernel 1 ∞ (Γ+)⊥ (Γ−)⊥ BS(x,y,µ,ν) = ∇⊥ G(x,y,µ,ν) = n − n , x,y 2π |Γ+|2 |Γ−|2 n=−∞(cid:20) n n (cid:21) X where Γ+ = (x−µ,y−(4nl+ν)), Γ− = (x−µ,y−(4nl−ν)). n n It is useful to consider complex variables notation. Then ∞ 1 1 1 BS(x,y,µ,ν) = − . 2πi Γ+ Γ− n=−∞(cid:20) n n (cid:21) X Fixed n, we compute the following 1 1 2(x−µ+i(y−ν)) + = , Γ+ Γ+ (x−µ+i(y−ν))2+(4nl)2 n −n 1 1 2(x−µ+i(y+ν)) + = . Γ− Γ− (x−µ+i(y+ν))2+(4nl)2 n −n We change variables (y−l = y,ν −l = ν) to recover the initial strip S = R×(−l,l), moreover, without lossing generality we take l = π/2. Due to the formula ∞ 1 2z 1 z + = coth , z z2+(2nπ)2 2 2 nX=1 (cid:16) (cid:17) we obtain that the Biot-Savart law in cartesian coordinates is given by 1 sin(y−ν) sin(y+ν) BS(x,y,µ,ν) = − − , 4π cosh(x−µ)−cos(y−ν) cosh(x−µ)+cos(y+ν) (cid:18) sinh(x−µ) sinh(x−µ) − . cosh(x−µ)−cos(y−ν) cosh(x−µ)+cos(y+ν) (cid:19) Using the formula for the vorticity we have that the velocity is 1 −sin(y−z (β)) −sin(y+z (β)) 2 2 v(x,y) = ̟(β) + , 4π ZR (cid:18)cosh(x−z1(β))−cos(y−z2(β)) cosh(x−z1(β))+cos(y+z2(β)) sinh(x−z (β)) sinh(x−z (β)) 1 1 − dβ. cosh(x−z (β))−cos(y−z (β)) cosh(x−z (β))+cos(y+z (β)) 1 2 1 2 (cid:19) We use the identity ∂ log(cosh(z (α)−z (η))±cos(z (α)±z (η))) = 0 η 1 1 2 2 R Z 7 to obtain that the average velocity in the curve is ρ¯ sinh(z (α)−z (η)) 1 1 v(z(α)) = − P.V. ∂ z (η) α 1 (cid:18) 2 ZR (cid:20)cosh(z1(α)−z1(η))−cos(z2(α)−z2(η)) sinh(z (α)−z (η)) 1 1 + dη, cosh(z (α)−z (η))+cos(z (α)+z (η)) 1 1 2 2 (cid:21) ρ¯ sinh(z (α)−z (η)) 1 1 − P.V. ∂ z (η) α 2 2 ZR (cid:20)cosh(z1(α)−z1(η))−cos(z2(α)−z2(η)) sinh(z (α)−z (η)) 1 1 − dη . cosh(z (α)−z (η))+cos(z (α)+z (η)) 1 1 2 2 (cid:21) (cid:19) The interface is convected by this velocity but we can add any velocity in the tangential directionwithoutalteringtheshapeofthecurve. Thetangential velocity inacurveonlychanges the parametrization. We consider then the following equation with the redefined velocity ∂ z(α) = v(z(α))+c(α)∂ z(α), t α where ρ¯ sinh(z (α)−z (η)) 1 1 c(α) = P.V. 2 R cosh(z1(α)−z1(η))−cos(z2(α)−z2(η)) Z sinh(z (α)−z (η)) 1 1 + dη. cosh(z (α)−z (η))+cos(z (α)+z (η)) 1 1 2 2 Following thisapproach weobtain(14). Because ofthatchoice of c(α) weobtain that, if initially the curve can be parametrized as a graph, i.e., z(x,0) = (x,f (x)), we have that the velocity 0 v on the curve is zero, thus our curve is parametrized as a graph for t > 0 and we recover the 1 contour equation (9). Note that when l → ∞ in the equation (9) we recover the equation for the whole plane (8): 1 2 η(∂ f(x)−∂ f(x−η)) x x lim A[f](x) = P.V. dη, (21) l→∞ l π R η2+(f(x)−f(x−η))2 Z where A[f] is the operator defined in (9). Furthermore, we obtain the pressure p (up to a constant) solving the equation −∆p = g∂ ρ, y with Neumann boundary conditions ∂ p| = −gρ1, ∂ p| = gρ2. n y=l n y=−l In this way we obtain v,p satisfying Darcy’s Law and the incompressibility condition. It is easy to check that ρ(x,y,t) is a weak solution of the conservation of mass equation. Definition 1. Let v be an incompressible field of velocities following Darcy’s Law. We define the weak solution of the conservation of mass equation present in (3) as a function satisfying T l ρ(x,y,t)∂ φ(x,y,t)+v(x,y,t)ρ(x,y,t)∇ φ(x,y,t)dydxdt = 0 t x,y Z0 ZRZ−l for all φ∈ C∞(R×(−l,l)×(0,T)). c We conclude this section with the following result. Proposition 1. Let ρ be the function defined in (2). Then ρ is a weak solution of the conservation of mass equation (see Definition 1) if and only if f is a solution of (9). The proofs of these two results are straightforward and, for the sake of brevity, we left them for the interested reader. 8 3 Similar results between the two regimes In this section we show a group of results for 0 < A < 1 similar to those in the regime A = 0. The proofs follow the same ideas but, due to the structure in our equation (9), with a second term coming from the boundaries present in our model, there are some difficulties. We show the well-posedness in Sobolev spaces when the Rayleigh-Taylor condition is satified, i.e. the denser fluidisbelowthelighter one. InthecasewheretheRayleigh-Taylor conditionisnotsatisfiedbut ourinitial data is analytic wealso have awell-posedness resultby means of a Cauchy-Kovalevski Theorem. We also prove the smoothing effect of the spatial operator in (9), i.e. the solution becomes instantly analytic. 3.1 Well-posedness in Sobolev spaces In this section we sketch the proof of local well-posedness in Sobolev spaces in the Rayleigh- Taylor stable case: Without loss of generality we take 2l = π and ρ¯ = 2. We indicate the constants with a dependency on l as c(l). The proof follows the same lines as in [9]. In order to deal with the kernel Ξ , the kernel corresponding to the effect of the boundaries, we define the following 2 energy: E[f](t) = kfk2 (t)+kd[f]k (t), (22) H3 L∞ where d[f]:R2×R+ 7→ R+ is defined as 1 d[f](x,η,t) = . (23) cosh(η)+cos(f(x)+f(x−η)) Thefunction(23)measuresthedistancebetweenf andthetopandfloor±l(recallthat,without losing generality, we are supposing that l = π). In other words, kd[f]k < ∞ implies that 2 L∞ kfk < π. So this is the natural ’energy’ associated to the space H3(R). We obtain ’a priori’ L∞ 2 l energy estimates as in [9]: The integrals corresponding to the kernel Ξ are the more singular 1 terms and can beboundedas in [9] because has a singularity with the same order. The integrals corresponding to the kernel Ξ are harmless and can be bounded using the definition of d[f]. 2 For instance, one of the integrals arising in the study of the third derivative, after an integration by parts, is I = |∂3f(x)|2 + ∂ Ξ (x,η)dηdx = I +I , x x 2 in out ZR ZB(0,1) ZBc(0,1)! and we obtain sin(f(x)+f(x−η))(∂ f(x)+∂ f(x−η)) I ≤ |∂3f(x)|2P.V. sinh(|η|) x x dηdx in ZR x ZB(0,1) (cid:12) (cosh(η)+cos((f(x)+f(x−η))))2 (cid:12) (cid:12) (cid:12) ≤ c(l)kfkC1k∂x3fk2L2kd[f]k2L∞ ≤ c(l)(cid:12)(cid:12)kfk3H3kd[f]k2L∞, (cid:12)(cid:12) sinh(|η|) I ≤ c(l)kfk kfk2 dη ≤ c(l)kfk3 . out C1 H3 (cosh(η)−1)2 H3 ZBc(0,1) With these techniques we obtain d kfk ≤ c(l)(E[f]+1)5. dt H3 9 Inordertouseclassical energymethodswehaveto boundtheevolution of kd[f]k interms L∞ of the energy E[f]. With this method we need a bound on k∂ fk . In order to do this we split t L∞ ∂ f in two terms, one for each kernel t ∂ f = A +A . t 1 2 We give the proof for the A =P.V. (∂ f(x)−∂ f(x−η))Ξ (x,η)dη. 1 x x 1 R Z For the term corresponding to the second kernel, A , the procedure is analogous. 2 We split A in its ’in’ and ’out’ parts, A = Ain +Aout, with 1 1 1 1 Ain ≤ c(l)k∂2fk , 1 x L∞ and Aout ≤ P.V. ∂ f(x)Ξ (x,η)dη + P.V. −∂ f(x−η)Ξ (x,η)dη . 1 x 1 x 1 (cid:13)(cid:13) ZBc(0,1) (cid:13)(cid:13)L∞ (cid:13)(cid:13) ZBc(0,1) (cid:13)(cid:13)L∞ (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) We have that the integral (cid:13) (cid:13) (cid:13) (cid:13) sinh(η) P.V. dη = 0. ZBc(0,1) sinh2 η2 Using this fact and the classical and hyperbolic trig(cid:0)on(cid:1)ometric identities we can write sinh(η) P.V. ∂ f(x)Ξ (x,η)dη = ∂ f(x)P.V. ZBc(0,1) x 1 x ZBc(0,1) 2sinh2 η2 1 (cid:0) (cid:1) 1 · − dη.  sin2((f(x)−f(x−η))/2) 1+(∂ f(x))2 1+ x sinh2(η/2)   We define the function sin2((f(x)−f(x−η))/2) Q(η) = . sinh2(η/2) We observe that Q(0) = lim Q(η) = (∂ f(x))2, and we can apply the Mean Value Theorem η→0 x to this function Q in order to have the cancelation needed at infinity. We obtain P.V. ∂ f(x)Ξ (x,η)dη ≤ c(l)k∂ fk (1+k∂ fk ). x 1 x L∞ x L∞ (cid:13)(cid:13) ZBc(0,1) (cid:13)(cid:13)L∞ (cid:13) (cid:13) (cid:13) (cid:13) To bound (cid:13) (cid:13) P.V. −∂ f(x−η)Ξ (x,η)dη = P.V. ∂ f(x−η)Ξ (x,η)dη x 1 η 1 ZBc(0,1) ZBc(0,1) we integrate by parts. We conclude P.V. −∂ f(x−η)Ξ (x,η)dη ≤ c(l)kfk (1+k∂ fk ). x 1 L∞ x L∞ (cid:13)(cid:13) ZBc(0,1) (cid:13)(cid:13)L∞ (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 10