Global well-posedness and decay for viscous water wave models
aa r X i v : . [ m a t h . A P ] D ec GLOBAL WELL-POSEDNESS AND DECAY OF A VISCOUSWATER WAVE MODEL
RAFAEL GRANERO-BELINCH ´ON AND STEFANO SCROBOGNA
Abstract.
The motion of the free surface of an incompressible fluid isa very active research area. Most of these works examine the case ofan inviscid fluid. However, in several practical applications, there areinstances where the viscous damping needs to be considered. In thispaper we study an asymptotic model for a viscous free boundary flow.In particular, we establish the global well-posedness in Sobolev spacesof a fourth order PDE modelling water waves with viscosity moving indeep water with or without surface tension effects. Furthermore, we alsoprove the decay of these solutions towards the equilibrium state.
Contents
1. Introduction 11.1. Notation and Main Result 32. Proof of Theorem 1 42.1. The linear semigroup 42.2. Decay in the low regularity space 62.3. Boundedness in the high regularity space 82.4. Proof of Theorem 1 11Acknowledgments 12References 12 Introduction
The motion of waves in fluids has been a hot research topic since theXVIIIth century with the works of Laplace and Lagrange. On the one handthere is a large number of papers dealing with the free boundary Euler andNavier-Stokes equations [6, 29]. These are free boundary problems and asa consequence the domain of definition Ω( t ) of the functions (the bulk ofthe fluid) is an unknown of the system that has to be determined from thedynamics (see Figure 1).On the other hand the literature on asymptotic models of such free bound-ary problems is even larger (cf. [30]). These asymptotic models allow to havevery good approximate description of the actual dynamics while simplify-ing the equations under study. In this direction there are many papersdealing with the case of inviscid fluids and, in particular dealing with as-ymptotic models for shallow water waves (see for instance [30] and the ref-erences therein) and models of water waves with small steepness (we refer Key words and phrases.
Water waves, damping, moving interfaces, free-boundaryproblems. Γ( t )Ω( t ) Figure 1.
Scheme of the problemto [1, 3, 32–34] for example). Similar small steepness asymptotics modelscan be derived for other free boundary problems, such as the Muskat prob-lem [8, 15], see [17, 19, 37]. Many of such asymptotic models are used indifferent applications in Coastal Engineering and Physics.Although it is a classical topic, the works studying the case of a viscousfluid are more scarce. The first works studying the case of a viscous waterwave date back to Boussinesq [7], Basset [5] and Lamb [28]. Since thenthere are many other papers studying damped water waves. For instance,we refer to the manuscripts of Kakutani & Matsuuchi [25], Ruvinsky &Freidman [36], Longuet-Higgins [31], Jiang, Ting, Perlin & Schultz [22],Joseph & Wang [23], Wang & Joseph [38] and Wu, Liu & Yue [39].According to the work by Dias, Dyachenko & Zakharov [10], the vis-cous damping of gravity water waves can be described by the following freeboundary problem: ∆ φ = 0 in Ω( t ) , (1a) ρ (cid:18) φ t + 12 |∇ φ | + Gh (cid:19) = − µ∂ φ on Γ( t ) , (1b) h t = ∇ φ · ( − ∂ h,
1) + 2 µρ ∂ h on Γ( t ) , (1c)where h denotes the height of the wave, φ is the velocity potential and G, ρ and µ are the gravity acceleration, density and viscosity of the fluid.Since its appearance, this system was considered by several other authors(see [11–14]). The need for simplified asymptotic models for damped water-waves systems was highlighted at first by Longuet-Higgins, which in [31]stated that For certain applications, however, viscous damping of thewaves is important, and it would be highly convenient to haveequations and boundary conditions of comparable simplicityas for undamped waves.
In this spirit, Kakleas & Nicholls [24] derived a quadratic asymtotic modelwhile Bae, Lin & Shin [4] derived a cubic asymptotic model of (1). Thewell-posedness of this quadratic model was studied by Ambrose, Bona &Nicholls [2] while the well-posedness of the full Dias-Dyachenko-Zakharovwas proved by Ngom & Nicholls in [35] in the case of a nonzero surfacetension and by Granero-Belinch´on & Scrobogna [20] in the case in whichthe surface tension can be zero.
LOBAL WELL-POSEDNESS AND DECAY OF A VISCOUS WATER WAVE MODEL 3
In a series of works [18,21], the authors, starting with the Dias-Dyachenko-Zakharov (1) free boundary problem, derived and studied the followingmodel of viscous water waves f tt + 2 δ Λ f t + Λ f + β Λ f + δ Λ f = − Λ (cid:16) ( H f t ) (cid:17) + ∂ x J H , f K Λ f + β∂ x J H , f K Λ f + δ∂ x J H , H f t K H ∂ x f + δ Λ (cid:0) H f t H ∂ x f (cid:1) − δ∂ x q ∂ x , f y H f t ,f ( x,
0) = f ( x ) ,f t ( x,
0) = f ( x ) , (2)where δ > β ≥ H and Λ denote the Hilbert transform and the square rootof the Laplacian d H f ( k ) = − i sgn( k ) ˆ f ( k ) , c Λ f ( k ) = | k | ˆ f ( k ) , (3)and J A, B K f = A ( Bf ) − B ( Af ) , is the commutator between two operators acting on the function f . In whatfollows we consider ( x, t ) ∈ S × [0 , T ] where S denotes the interval [ − π, π ]with periodic boundary conditions.1.1. Notation and Main Result.
We denote with C any positive constantindependent of any physical parameter of the problem. The explicit valueof C may vary from line to line.We recall the definition of the homogeneous Sobolev spaces of fractionalorder ˙ H s = ˙ H s (cid:0) S (cid:1) = (cid:8) f ∈ L (cid:12)(cid:12) Λ s f ∈ L (cid:9) , for any s ∈ R . It is well known that for zero mean function we have that H s = ˙ H s . As the equation preserves the zero mean property from nowon we will always use the non-homogeneous notation in order to indicate aSobolev space of regularity s . Similarly, we define the homogeneous Wienerspaces ˙ A s = ˙ A s (cid:0) S (cid:1) = n f ∈ L (cid:12)(cid:12)(cid:12) d Λ s f ∈ ℓ o , where ˆ f denotes the Fourier series of f .The main result of this work is the following theorem: Theorem 1.
Let δ > and β ≥ . There exists a c > such that for any ( f , f ) ∈ H × H such that k f k H + k f k H ≤ c , then, there exist a unique global solution ( f, f t ) of (2) stemming from theinitial data ( f , f ) which belongs to the energy space f ∈ C (cid:0) R + ; H (cid:1) ,f t ∈ C (cid:0) R + ; H (cid:1) ∩ L ( R + ; H ) . R. GRANERO-BELINCH ´ON AND S. SCROBOGNA
Furthermore, k f k A + k f t k A ≤ Ce − tδ , k f k H r + k f t k H s ≤ Ce − C ( δ,r,s ) t , ∀ ( r, s ) ∈ [0 , × [0 , . Once the local existence and uniqueness was obtained in [21], we onlyneed to provide with appropriate energy estimates. To do that we are goingto define a modified energy ||| ( f, f t ) ||| T that has two different contributions.On the one hand we consider the low regularity space X where the solutionwill decay while on the other hand we will also regard a high regularity space Y where the solution will only remain bounded. The particular choice of X and Y will be clear below. Then the energy will take the form ||| ( f, f t ) ||| T = sup t ∈ [0 ,T ] (cid:8) e αt k ( f ( • , t ) , f t ( • , t )) k X (cid:9) + k ( f, f t ) k Y for α > . Equipped with this definition of energy, the rest of the paper is devoted toobtain an inequality of the form ||| ( f, f t ) ||| T ≤ C ( f , f ) + P ( ||| ( f, f t ) ||| T ) , for certain polynomial P of degree larger than 1, constant C ( f , f ) thatdepends on the initial data. The previous inequality implies that for smallenough C ( f , f ), the solution satisfies ||| ( f, f t ) ||| T ≤ C ( f , f ) , for all T >
0, then a standard continuation argument allow us to extend thesolution to arbitrary long time intervals.2.
Proof of Theorem 1
According to the result in [21], there is a local in time solution ( f, f t ) forthe problem (2). Let us define the modified energy ||| ( f, f t ) ||| T = e δT max t ′ ∈ [0 ,T ] (cid:8)(cid:13)(cid:13)(cid:0) f (cid:0) t ′ (cid:1) , f t (cid:0) t ′ (cid:1)(cid:1)(cid:13)(cid:13) A (cid:9) + max t ′ ∈ [0 ,T ] (cid:8)(cid:13)(cid:13) f t (cid:0) t ′ (cid:1)(cid:13)(cid:13) H + (cid:13)(cid:13) f (cid:0) t ′ (cid:1)(cid:13)(cid:13) H (cid:9) . The estimates of [21] assures us moreover that the solution exists at least ina time interval [0 , T max ] where T max = T max ( f , f ) is the maximal lifespanof the solution.2.1. The linear semigroup.
We consider the linear nonhomogeneous prob-lem f tt + 2 δ Λ f t + Λ f + β Λ f + δ Λ f = F, (4)where F is a zero mean forcing. Let us denote with u ( x, t ) = (cid:18) f ( x, t ) f t ( x, t ) (cid:19) , u ( x ) = (cid:18) f ( x ) f ( x ) (cid:19) , so that (4) becomes u t + Lu = (cid:18) F (cid:19) , L = (cid:18) −
1Λ + β Λ + δ Λ δ Λ (cid:19) . LOBAL WELL-POSEDNESS AND DECAY OF A VISCOUS WATER WAVE MODEL 5
Applying Duhamel principle we write u = u L + u NL where u L ( t ) = e − tL u , u NL ( t ) = Z t e − ( t − t ′ ) L (cid:18) F ( t ′ ) (cid:19) d t ′ . The eigenvalues of L are the Fourier multipliers λ ± ( n ) = δ | n | ± i r | n | (cid:16) β | n | (cid:17) , so we see that the linear operator L induces both parabolic smoothing effectsand oscillating behavior of the solution. Since the solution has zero mean,we have that λ ± ( n ) = 0. The two ortonormal eigenvectors associated to λ ± ( n ) are e ± ( n ) = 1 q | λ ± ( n ) | (cid:18) − λ ± ( n ) (cid:19) , so that, if we denote D = (cid:18) λ − λ + (cid:19) ,S = (cid:18) − λ − − λ + (cid:19) ,S − = 1 λ − − λ + (cid:18) − λ + − λ − (cid:19) , we have that e − tL = S − e − tD S. With the above considerations we write u L and u NL in terms of f , f and F as ˆ u L ( t ) = 1 λ − − λ + (cid:18) λ − e − tλ + − λ + e − tλ − λ − (cid:0) e − tλ − − e − tλ + (cid:1) (cid:19) ˆ f + 1 λ − − λ + (cid:18) λ + (cid:0) e − tλ + − e − tλ − (cid:1) λ − e − tλ − − λ + e − tλ + (cid:19) ˆ f , ˆ u NL ( t ) = Z t λ − − λ + λ + (cid:16) e − ( t − t ′ ) λ + − e − ( t − t ′ ) λ − (cid:17) λ − e − ( t − t ′ ) λ − − λ + e − ( t − t ′ ) λ + ! ˆ F (cid:0) t ′ (cid:1) d t ′ . We want to obtain now the decay rates of the linear semigroup. Let usat first check the time-decay of u L . We can compute that (cid:12)(cid:12)(cid:12)(cid:12) λ ± λ − − λ + (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ s | n | β ! , (5)since | n | ≥ j = 0 , (cid:12)(cid:12)(cid:12)(cid:12) λ ± ( n ) λ − ( n ) − λ + ( n ) e − tλ ± ( n ) ˆ f j ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − δt δ s | n | β ! (cid:12)(cid:12)(cid:12) ˆ f j ( n ) (cid:12)(cid:12)(cid:12) , which in turn implies that e δT max t ′ ∈ [0 ,T ] (cid:13)(cid:13) u L ( t ′ ) (cid:13)(cid:13) A ≤ C k ( f , f ) k A / . (6) R. GRANERO-BELINCH ´ON AND S. SCROBOGNA
Equivalently, we have that k e − tL k A / A ≤ Ce − δt . (7)2.2. Decay in the low regularity space.
If we write the equation inits mild formulation using Duhamel’s principle, we have that the nonlinearforcing is given by F = X j =1 F j , where F = − Λ (cid:16) ( H f t ) (cid:17) ,F = ∂ x J H , f K Λ f,F = β∂ x J H , f K Λ f,F = δ∂ x J H , H f t K H ∂ x f,F = δ Λ (cid:0) H f t H ∂ x f (cid:1) ,F = δ∂ x q ∂ x , f y H f t . The goal of the present computations is to provide a control of the form k F ( t ) k A / ≤ Ce − δt (1+ q ) ||| ( f, f t ) ||| T , t ∈ [0 , T ] , q > . We are going to use the Sobolev embedding k a k A s ≤ C δ k a k H s +1 / δ ≤ C k a k H s +1 , together with interpolation between Sobolev spaces and the fractional prod-uct rule k ab k A s ≤ C s ( k a k A k b k A s + k a k A s k b k A ) ≤ C s k a k A s k b k A s , to estimate F j . We compute k F k A / ≤ C k f t k A k f t k A / ≤ C k f t k A k f t k H / ≤ C k f t k / A k f t k / H . Using linear interpolation in Wiener spaces k a k A s ≤ C k a k s/rA r k a k − s/rA , we find that k F k A / ≤ k J H , f K Λ f k A / ≤ C ( k f k A / k f k A + k f k A / k f k A ) ≤ C k f k A / k f k A ≤ C k f k H / k f k A ≤ C k f k / H k f k / A . Similarly, k F k A / ≤ C k J H , f K Λ f k A / ≤ C ( k f k A / k f k A + k f k A / k f k A ) LOBAL WELL-POSEDNESS AND DECAY OF A VISCOUS WATER WAVE MODEL 7 ≤ C k f k A / k f k A ≤ C k f k H / k f k A ≤ C k f k / H k f k / A . Similarly, we have that k F k A / ≤ C k J H , H f t K H ∂ x f k A / ≤ C ( k f t k A k f k A / + k f t k A / k f k A ) ≤ C (cid:16) k f t k A k f k H + k f t k / A k f t k / A k f k / A k f k / A (cid:17) ≤ C (cid:16) k f t k A k f k H + k f t k / A k f t k / H k f k / A k f k / H (cid:17) , k F k A / ≤ C (cid:16) k f t k A k f k H + k f t k / A k f t k / H k f k / A k f k / H (cid:17) , Using the commutator structure together with the product rule in Wienerspaces, we estimate k F k A / ≤ C k ∂ x f H f t + 3 ∂ x f ∂ x H f t + ∂ x f ∂ x H f t k A / ≤ C (cid:18) k f k A / k f t k A + k f k A k f t k A / + k f k A / k f t k A + k f k A k f t k A / + k f k A / k f t k A + k f k A k f t k A / (cid:19) . Using interpolation in Wiener spaces and then the Sobolev embedding k f k A / ≤ C k f k H and k f t k A / ≤ C k f t k H , we compute that k F k A / ≤ C (cid:18) k f k / A k f k / H k f t k A + k f k / A k f k / H k f t k / A k f t k / H + k f k / A k f k / H k f t k / A k f t k / H + k f k / A k f k / H k f t k / A k f t k / H + k f k / A k f k / H k f t k / A k f t k / H + k f k / A k f k / H k f t k / A k f t k / H (cid:19) . Let us recall that using Duhamel formulation the solution then can bewritten as u ( x, t ) = e − tL u + Z t e − ( t − t ′ ) L (cid:18) F ( t ′ ) (cid:19) d t ′ and satifies, | ˆ u ( n, t ) | ≤ Ce − δt (1 + p | n | ) (cid:12)(cid:12)(cid:12) ˆ f j ( n ) (cid:12)(cid:12)(cid:12) + C Z t e − ( t − t ′ ) δn p | n | (cid:12)(cid:12)(cid:12) ˆ F (cid:0) n, t ′ (cid:1)(cid:12)(cid:12)(cid:12) d t ′ . Using − (cid:0) t − t ′ (cid:1) δn ≤ − (cid:0) t − t ′ (cid:1) δ ≤ , R. GRANERO-BELINCH ´ON AND S. SCROBOGNA we can estimate k u ( t ) k A ≤ Ce − δt k ( f , f ) k A / + Ce − δt Z t e δt ′ X j =1 k F j ( t ′ ) k A / d t ′ . Recalling the previous estimates for k F j ( t ′ ) k A / and the definition of thenorm ||| ( f, f t ) ||| T , we have that (cid:13)(cid:13) F (cid:0) t ′ (cid:1)(cid:13)(cid:13) A / ≤ Ce − δt ′ (1+73 / ||| ( f, f t ) ||| T , ≤ t ′ ≤ t ≤ T. We conclude that e δt max t ′ ∈ [0 ,t ] (cid:8)(cid:13)(cid:13)(cid:0) f (cid:0) t ′ (cid:1) , f t (cid:0) t ′ (cid:1)(cid:1)(cid:13)(cid:13) A (cid:9) ≤ C k ( f , f ) k A / + C ||| ( f, f t ) ||| T Z t e − (73 / δt ′ d t ′ ≤ C k ( f , f ) k A / + C ||| ( f, f t ) ||| T (8)2.3. Boundedness in the high regularity space.
Similarly as in [21],we test the equation against Λ f t , integrate in S and integrate by partsobtaining the energy balance12 dd t E ( t ) + D (cid:0) t ′ (cid:1) = X i =1 I i ( t ) , (9)with E ( t ) = (cid:13)(cid:13) f t (cid:0) t ′ (cid:1)(cid:13)(cid:13) H + β (cid:13)(cid:13) f (cid:0) t ′ (cid:1)(cid:13)(cid:13) H / + δ (cid:13)(cid:13) f (cid:0) t ′ (cid:1)(cid:13)(cid:13) H + (cid:13)(cid:13) f (cid:0) t ′ (cid:1)(cid:13)(cid:13) H / , D ( t ) = 2 δ k f t ( t ) k H , and I ( t ) = − Z S Λ (cid:16) ( H f t ) (cid:17) Λ f t d xI ( t ) = Z S ∂ x J H , f K Λ f Λ f t d xI ( t ) = β Z S ∂ x J H , f K Λ f Λ f t d xI ( t ) = δ Z S ∂ x J H , H f t K H ∂ x f Λ f t d xI ( t ) = δ Z S Λ (cid:0) H f t H ∂ x f (cid:1) Λ f t d xI ( t ) = − δ Z S ∂ x q ∂ x , f y H f t Λ f t d x. Using the self-adjointness of the operator Λ together with H¨older’s in-equality and the Sobolev embedding k g k L ≤ C k g k H . , we find that I ( t ) = − Z S (cid:16) ( H f t ) (cid:17) Λ f t d x LOBAL WELL-POSEDNESS AND DECAY OF A VISCOUS WATER WAVE MODEL 9 = − Z S (cid:16) ( H f t ) (cid:17) ∂ x Λ f t d x = − Z S ∂ x (cid:16) ( H f t ) (cid:17) Λ f t d xx = − Z S (cid:0) H f t Λ ∂ x f t + 6(Λ ∂ x f t ) + 8Λ f t ∂ x Λ f t (cid:1) Λ f t d x ≤ C k f t k H (cid:0) k f t k H kH f t k L ∞ + k f t k H . + k f t k H k Λ f t k L ∞ (cid:1) ≤ C k f t k H k f t k H k f t k H . ≤ σ k f t k H + C k f t k H k f t k H . , (10)for σ > H s ⊂ H r , r ≤ s, we obtain the estimate I ( t ) ≤ σ k f t k H + C ||| ( f, f t ) ||| T e − ( δ/ t . (11)We recall the following commutator estimate (see equation (1.13) in [9]) (cid:13)(cid:13)(cid:13) ∂ ℓx J H , u K ∂ mx v (cid:13)(cid:13)(cid:13) L p ≤ C (cid:13)(cid:13)(cid:13) ∂ ℓ + mx u (cid:13)(cid:13)(cid:13) L ∞ k v k L p , p ∈ (1 , ∞ ) , ℓ, m ∈ N . (12)Equipped with (12), we can estimate I as follows I ( t ) = Z S Λ ∂ x J H , f K Λ f Λ f t d x ≤ (cid:13)(cid:13) ∂ x J H , f K Λ f (cid:13)(cid:13) L (cid:13)(cid:13) Λ f t (cid:13)(cid:13) L ≤ (cid:13)(cid:13) ∂ x f (cid:13)(cid:13) L ∞ k Λ f k L (cid:13)(cid:13) Λ f t (cid:13)(cid:13) L . As a consequence, by interpolation in Wiener and Sobolev spaces, we havethat I ( t ) ≤ C ||| ( f, f t ) ||| T e − ( δ/ t . (13)Analogously, we find that I ( t ) ≤ (cid:13)(cid:13) ∂ x f (cid:13)(cid:13) L ∞ (cid:13)(cid:13) Λ f (cid:13)(cid:13) L (cid:13)(cid:13) Λ f t (cid:13)(cid:13) L ≤ (cid:13)(cid:13) ∂ x f (cid:13)(cid:13) L ∞ (cid:13)(cid:13) Λ f (cid:13)(cid:13) L + σ (cid:13)(cid:13) Λ f t (cid:13)(cid:13) L ≤ C ||| ( f, f t ) ||| T e − ( δ/ t + σ (cid:13)(cid:13) Λ f t (cid:13)(cid:13) L . (14)We can decompose I ( t ) as follows I ( t ) = δ Z S (cid:2) Λ( H f t Λ ∂ x f ) + ∂ x ( H f t ∂ x f ) (cid:3) Λ f t d x = δ Z S (cid:2) Λ( H f t Λ ∂ x f ) − ∂ x ( H f t Λ f ) (cid:3) Λ f t d x = J + J , with J = δ Z t Z S Λ( H f t Λ ∂ x f )Λ f t d x d t ′ J = − δ Z t Z S ∂ x ( H f t Λ f )Λ f t d x d t ′ . We will use the fractional Leibniz rule (see [16, 26, 27]): k Λ s ( uv ) k L p ≤ C ( k Λ s u k L p k v k L p + k Λ s v k L p k u k L p ) , which holds whenever1 p = 1 p + 1 p = 1 p + 1 p where 1 / < p < ∞ , < p i ≤ ∞ , and s > max { , /p − } . Using the fractional Leibniz rule and the self-adjointness of the operator Λ, we compute J ( t ) = δ Z S Λ ( H f t Λ ∂ x f )Λ f t d x ≤ δ k Λ ( H f t Λ ∂ x f ) k L k Λ f t k L ≤ δC ( k f t k H k f k H + k f t k H k f k H ) k f t k H ≤ δC ( k f t k H k f k H + k f t k H k f k H ) + σ k f t k H ≤ C ||| f ||| T e − ( δ/ t + σ k f t k H . The terms J and I = J can be estimated in a similar way and we findthat I ( t ) + I ( t ) ≤ C ||| f ||| T e − ( δ/ t + σ k f t k H . (15)Now we are left with I . We remark that I ( t ) = − δ Z S ∂ x (cid:2) ∂ x f H f t + 2 ∂ x f Λ f t (cid:3) ∂ x Λ f t d x. Integrating by parts, we find that I ( t ) = δ Z S ∂ x (cid:2) ∂ x f H f t + 2 ∂ x f Λ f t (cid:3) ∂ x Λ f t d x. Hence, using the same ideas as before, we have that I ( t ) ≤ ||| ( f, f t ) ||| T e − ( δ/ t + σ k f t k H + 2 Z S ∂ x f Λ f t ∂ x Λ f t d x. The term J = 2 Z S ∂ x f Λ f t ∂ x Λ f t d x is the highest order term. However, it has an inner commutator structurethat we can exploit as follows: J ( t ) = Z S H ( ∂ x f Λ f t )Λ f t d x − Z S ∂ x f Λ f t H Λ f t d x = Z S J H , ∂ x f K Λ f t Λ f t d x. Then, recalling (12), we conclude that I ( t ) ≤ ||| ( f, f t ) ||| T e − ( δ/ t + σ k f t k H . (16) LOBAL WELL-POSEDNESS AND DECAY OF A VISCOUS WATER WAVE MODEL11
Proof of Theorem 1.
Collecting (11), (13), (14), (15) and (16) andtaking σ small enough, we concludedd t E ( t ) + D (cid:0) t ′ (cid:1) ≤ C ||| ( f, f t ) ||| T e − ( δ/ t + C ||| ( f, f t ) ||| T e − ( δ/ t . (17)Integrating in time and using (8), we conclude the polynomial bound ||| ( f, f t ) ||| T + Z T D (cid:0) t ′ (cid:1) d t ′ ≤ C k ( f , f ) k A / + E (0) + C (cid:16) ||| ( f, f t ) ||| T + ||| ( f, f t ) ||| T (cid:17) , thus, there exists a (fixed, positive) constant 1 < C + such that ||| ( f, f t ) ||| T ≤ C + h ( k f k H + k f k H ) + (cid:16) ||| ( f, f t ) ||| T + ||| ( f, f t ) ||| T (cid:17)i . (18)We observe that, in the previous estimates, we have not used any hypothesison the size of the initial data and the previous bound is valid for everysolution and T ∈ (0 , T max ).We want to prove that, there exists a c > k f k H + k f k H ≤ c , (19)the inequality ||| ( f, f t ) ||| T ≤ C ( k f k H + k f k H ) , holds true for any T > T such that ||| ( f, f t ) ||| T = 34 < . The above inequality allow us to deduce the polynomial bound ||| ( f, f t ) ||| T ≤ C + h ( k f k H + k f k H ) + ||| ( f, f t ) ||| T i , (20)or equivalently,2 C + ||| ( f, f t ) ||| T ≤ (2 C + ) ( k f k H + k f k H ) + (cid:2) C + ||| ( f, f t ) ||| T (cid:3) . (21)Then, without loss of generality we can restrict our analysis to a polynomialof the form ||| ( f, f t ) ||| T ≤ C ( f , f ) + ||| ( f, f t ) ||| T . (22)Now we observe that if C ≪ Q ( y ) = C − y + y has two positive real roots y ± = 1 ± √ − C , moreover if 0 < C ≪ y − = min { y + , y − } = 1 − √ − C ≤ C . Furthermore, analogously as in in [21], we know that the application t ( f, f t ) ||| t is continuous for t ∈ [0 , T max ). This, together with the smallnessin the initial data, implies that ||| ( f, f t ) ||| T ∈ [0 , y − ] . We combine the above deduction with the estimate y − ≤ C and we deducethat ||| ( f, f t ) ||| T ≤ C < , if we take C small enough. This is a contradiction with the definition of T and implies that the solution is global. Acknowledgments
The research of S.S. is supported by the European Research Councilthrough the Starting Grant project H2020-EU.1.1.-639227 FLUID-INTERFACE.R. G-B has been funded by project ”Mathematical Analysis of Fluids andApplications” with reference PID2019-109348GA-I00 and acronym ”MAFyA”funded by the Ministerio de Ciencia, Innovacion y Universidades (MICIU) .
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Departamento de Matem´aticas, Estad´ıstica y Computaci´on, Universidad deCantabria. Avda. Los Castros s/n, Santander, Spain.
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