GLOBAL WEAK SOLUTIONS OF THE SERRE–GREEN–NAGHDI EQUATIONS WITH SURFACE TENSION

. We consider in this paper the Serre–Green–Naghdi equations with surface tension. Smooth solutions of this system conserve an H 1 -equivalent energy. We prove the existence of global weak dissipative solutions for any relatively small-energy initial data. We also prove that the Riemann invariants of the solutions satisfy a one-sided Oleinik inequality.


Introduction
The Euler equations are usually used to describe water waves in oceans and Channels.Due to the difficulties to resolve the Euler equations both numerically and analytically, several simpler approximations have been proposed in the literature for different regimes.In the shallow water regime, the main assumption is on the ratio of the mean water depth h to the wave wave-length ι, the shallowness parameter σ = h2 /ι 2 is considered to be small.Beside the shallowness condition, a restriction on the amplitude of the wave a can be considered assuming that the nonlinearity (or the amplitude) parameter ǫ = a/ h is small.Considering the shallow water regime with the small-amplitude condition [29,35] (σ ≪ 1, ǫ ≪ 1).Many equations have been derived to model the propagation of the 1 waves, such as the Camassa-Holm equation [10], the Korteweg-deVries (KdV) equation [34] and some variants of the Boussinesq equations [8,9,53].Considering shallow water with possibly large-amplitude waves (σ ≪ 1, ǫ ≈ 1), by neglecting the terms of order O(σ) in the water waves equations, Saint-Venant obtained the Nonlinear Shallow Water (or Saint-Venant) equations [52].Smooth solutions of the Saint-Venant equations have a precision of order O(tσ), where t denotes the time [35].In order to obtain a better precision, one can keep the O(σ) terms in the equations and only neglect the O(σ 2 ) terms.This leads to the Serre-Green-Naghdi equations.Those equations were firstly derived by Serre [48], rediscovered independently by Su and Gardner [51] and another time by Green, Laws and Naghdi [23,24].The Serre-Green-Naghdi equations are the most general and most precise, but also the most complicated of the models of shallow water equations presented above.Of course, one can always keep higher order terms in the equation (keeping terms of order O(σ 2 ) for example), this will lead to equations with a better precision, but with higher order derivatives.Those equations are not accurate due to the high order derivative terms which make their numerical resolution much slower.
The influence of the surface tension is generally neglected on water waves problems.However, in certain cases, the effect of the surface tension is appreciable.Indeed, Longuet-Higgins [40] showed that the surface tension is significant in certain localized regions, and cannot be neglected near the sharp crest of the breaking wave.Other experimental studies showed the importance of the surface tension on thin layers [21,43,44].Those experimentations have been done for different fluids including water and mercury.Various mathematical studies of water waves equations with surface tension exist in the literature, we refer to [2,3,4,7,12,13,41,47,49,55].Considering a two-dimensional coordinate system Oxy (Figure 1) and an incompressible fluid layer.Considering the still fluid level at y = 0, the fluid layer is bounded between the flat bottom at y = − h and a free surface y = h(t, x) − h, where h is the total water depth.Assuming long waves in shallow water with possibly large-amplitude.The Serre-Green-Naghdi system (without neglecting the surface tension influence) reads where u denotes the depth-averaged horizontal velocity, g is the gravitational acceleration and γ > 0 is a constant (the ratio of the surface tension coefficient to the density).
The classical Serre-Green-Naghdi equations (without surface tension) are recovered taking γ = 0.The Serre-Green-Naghdi (SGN γ ) equations (1.1) have been derived in [17] as a generalisation of the classical SGN equations (γ = 0).Due to the appearance of time derivatives in (1.1c), it is convenient to apply the inverse of the Sturm-Liouville operator the system (1.1) becomes then (1.3b) When h > 0, the operator L −1 h is well defined and smoothes two derivatives (see Lemma 5.2 below).This is not enough to control the term containing h xx on the right-hand side of (1.3b).To overcome this problem, we use the definition of L h to rewrite the system (1.3) in the equivalent form (1.4b) Smooth solutions of the SGN γ equations (1.4) satisfy the energy equation (see Appendix B) ) where (1.7) Linearising the SGN γ equations (1.4) around the constant state (h, u) = ( h, 0) and looking for travelling waves having the form exp {(kx − ωt) i} we obtain the dispersion relation ω 2 = g hk 2 (1 + γk 2 /g) / 1 + h2 k 2 /3 .Defining the Bond number B = g h2 /γ, the SGN γ equations are linearly dispersive if and only if B = 3.In the dispersionless case (B = 3), the SGN γ equations admit weakly singular peakon travelling wave solutions [19,42].A mathematical study of the Serre-Green-Naghdi equations with or without surface tension have been widely studied in the literature.We refer to [1,28,32,33,35,37] for the case inf h 0 > 0 and to [36] for the shoreline problem (sign(h) = 1 x>x 0 ).In [1,28,37], a proof of the local well-posedness of the SGN equations without surface tension (γ = 0) is given.Kazerani has proved in [32] the existence of global smooth solutions of the SGN equations with viscosity for small initial data.A full justification of the model (1.4) is given in [33,35].By "full justification" we mean local well-posedness of the system and, the solution is close to the solution of the water waves equations with the same initial data.In a recent work [25], we have obtained a precise blow-up criterion of (1.4) (Theorem 2.3 below) and we proved that such scenario occurs for a class of small-energy initial data (Theorem 2.6 below).Then, in general, smooth solutions cannot exist globally in time.This paper investigates the existence of global weak solutions of (1.4) with γ > 0. To the best of the author's knowledge, the existence of global weak solutions for all the different variants of the inviscid Serre-Green-Naghdi equations has not been established before.
Here, the existence of global weak solutions is established by approximating the system (1.4) with another system that admits global smooth solutions.We recover weak solutions of (1.4) by taking the limit.The proof involves several steps.
We consider initial data satisfying E 0 dx < √ gγ h2 , which is propagated due to energy conservation (1.5).Using the fact that the energy is equivalent to (h − h, u) 2 H 1 and a Sobolev-like inequality (essentially H 1 ֒→ L ∞ , see Proposition 2.4 below) we obtain a uniform lower bound of h.This is important for ensuring the invertibility of the operator L h defined in (1.2).Smooth solutions of (1.4) blow-up in finite time due to the presence of quadratic terms in the associated Riccati-type equations.In order to approximate the SGN γ system, we use a cut-off to obtain a linear growth that leads to global smooth solutions (due to Gronwall's inequality).However, cutting-off directly as in [58,59] violates the energy conservation (1.5).The choice the approximated system is crucial and must conserve the properties of the SGN γ system.In Section 4 below, we chose carefuly a suitale approximated system that is globaly well-posed and satisfies the energy equation (5.11).In order to pass to the limit, some uniform estimates are needed.In the previous studies of smooth solutions of the SGN equations, some estimates of the operator L −1 h have been obtained, those estimates usually depend on the L ∞ norm of h x which may blow-up for weak solutions.In Lemma 5.2 below, we present some new estimates of L −1 h depending only on the L ∞ norm of h and 1/h.As in [54,58,59], an L p loc estimate of (h x , u x ) with p < 3 is also needed.In our case and due to the complexity of the SGN γ equations, we have to use a change of coordinates to obtain this estimate (see Lemma 5.6 below).We use then some classical compactness arguments with Young mesures [30] and a generalised compensated compactness result [22] to pass to the limit.We follow in this step the techniques developed in [54] for the Camassa-Holm equation and in [58,59] for the variational wave equation.The structure of the SGN γ system being more complex, we have to handle the weak limit of some nonlinear terms that do not exist in [58,59] (see Lemma 6.4 for example).Finally, the global weak solutions of (1.4) are obtained by taking the limit in the approximated system, and are shown to dissipate the energy and satisfy the one-sided Oleinik inequality (3.6).
The existence of global solutions to the Boussinesq equations [9,53] have been studied in [5,46].Schonbek [46] regularised the conservation of the mass by adding a defusion term, i.e., h t + [hu] x = εh xx , with ε > 0.She proved the global wellposedness of the regularised system, and she obtained global weak solutions of (1.8) by taking ε → 0. In [5], Amick proved that if the initial data, (h 0 , u 0 ), is smooth, then the solution, (h, u), obtained by Schonbek [46] is also smooth and is the unique smooth solution of the Boussinesq equations (1.8).
The SGN γ equations (1.1) can be compared with the dispersionless regularised Saint-Venant (rSV) system presented in [11].The rSV system can be obtained replacing R in (1.1c) by εR rSV with x and ε 0, the classical Saint-Venant system is recovered taking ε = 0. Weakly singular shock profiles of the rSV equations are studied in [45].In [39], Liu et al. proved the local well-posedness of the rSV equations and identified a class of initial data such that the corresponding solutions blow-up in finite time.The rSV system have been generalised recently to obtain a regularisation of any unidimensional barotropic Euler (rE) system [26].
The system (1.1) can also be compared with the modified Serre-Green-Naghdi (mSGN) equations derived in [14] to improve the dispersion relation of the classical SGN system.
The mSGN system presented in [14] can be obtained replacing R in (1.1c) by where β is a positive parameter.The rSV, rE and mSGN conserve H 1 -equivalent energies and have similar properties as the SGN γ system (1.1).One may can obtain the existence of global weak solutions of those equations following the proof given in this paper.
The study of the classical Serre-Green-Naghdi equations is more challenging.Indeed, when γ = 0, the energy (1.6) fails to control the H 1 norm of h − h, then, a lower bound of h cannot be obtained.This bound is crucial to obtaining the blow-up result [25] and the global existence in this paper for γ > 0. To the author's knowledge, the questions of the blow-up of smooth solutions and the existence of global solutions of the SGN equations without surface tension are still open.However, Bae and Granero-Belinchén [6] proved recently that for a class of periodic initial data satisfying inf h 0 = 0, smooth solutions cannot exist globally in time.For this class of initial data, it is not known if smooth solutions exist locally in time, but if they do, a singularity must appear in finite time.
This paper is organised as follows.In Section 2 we present the local well-posedness of (1.4) and some blow-up results.Section 3 is devoted to define weak solutions of (1.4) and to present the main result which is the existence of global dissipative weak solutions.We discuss in Section 4 the properties needed of the approximated system and we propose a suitable choice.Section 5 is devoted to prove the existence of global smooth solutions of the approximated system and to obtain some uniform estimates.We obtain strong precompactness results in Section 6.The existence of the global weak solutions is proved in Section 7. In Appendix A we recall some classical lemmas that are used in this paper.Appendix B is devoted to obtain the energy equations of the approximated system and of (1.4).

Review of previous results
We consider the Serre-Green-Naghdi equations with surface tension in this form (2.1a) with (2.3) The system (2.1) is locally well-posed in the Sobolev space where s 2 is a real number.
Theorem 2.1.Let γ > 0, h > 0 and s 2, then, for any a unique solution of (2.1) such that inf Moreover, the solution satisfies the conservation of the energy Remark 2.2.The solution given in Theorem 2.1 depends continuously on the initial data, i.e., If (h The proof of Theorem 2.1 is classic and omitted in this paper.See [26,28,35,39] and Theorem 3 of [27] for more details.It is clear from Theorem 2.1 that if the solution at time T remains in H s and inf x h(T, x) > 0 then one can extend the interval of existence.This leads to the blow-up criterion where T max is the maximum time existence of the solution.This criterion has been improved in [25] to Theorem 2.3.( [25]) Let T max be the maximum time existence of the solution given by Theorem 2.1, then which is equivalent to the second criterion Noting that the energy conserved in (2.4) is equivalent to the H 1 norm of (h − h, u).Due to the continuous embedding H 1 ֒→ L ∞ , we can obtain a uniform (on time) estimate of (h − h, u) L ∞ , and, if the initial energy is not very large compared to h, we can obtain a lower bound of h.For that purpose, we present the following proposition.
Proposition 2.4.For γ > 0, h > 0, let E be a positive number such that Defining Then, for any Remark 2.5.Taking an initial data satisfying R E 0 dx E, then, due to the energy conservation (2.4) and Proposition 2.4 the depth h cannot vanish.The blow-up criteria given in Theorem 2.3 becomes then Proof of Proposition 2.4.The Young inequality 1 2 a 2 + 1 2 b 2 ±ab implies that which implies that h min h h max .Doing the same estimates with u one obtains 3 h 2 min |u| 2 , the last inequality ends the proof of u min u u max .
As in [25], we can build some initial data with small initial data such that the corresponding solutions blow-up in small time.
Theorem 2.6.( [25]) For any T > 0 and E satisfying (2.5), there exist satisfying R E 0 dx E such that the corresponding solution of (2.1) blows-up at finite time T max T and Ẽ0 dx E such that the corresponding solution of (2.1) blows-up at finite time Tmax T and

Main results
Since smooth solutions fail to exist globally in time, even for arbitrary small-energy initial data, we shall define weak solutions of the SGN γ system (2.1).For that purpose, we define the domain Moreover, (h(t, •) − h, u(t, •)) belongs to D for all t 0 and (h − h, u) ∈ C r (R + , H 1 ).More precisely, for all t 0 0 we have Now we can state the main result of this paper.
• The solution dissipates the energy

.5)
• There exists C > 0 such that the solution satisfies the Oleinik inequality Remark 3.3.The constants C α,Ω and C depend on h, γ, g and R E 0 dx but not on the initial data.
In order to obtain global solutions of (2.1), we use a suitable approximation of the system (2.1) that admits global smooth solutions.Using some compactness arguments and taking the limit we recover a global weak solution of (2.1).In the next section we present the choice of the suitable approximated system.

An approximated system
The blow-up of the solutions given in Theorem 2.6 is due to the Riccati-type equations.In order to prevent the singularities from appearing, we modify slightly the Riccati-type equations.

Riccati-type equations. Defining the Riemann invariants
The system (2.1) can be rewritten as From the definition of L h in (1.2), we obtain that for any smooth function Ψ satisfying Ψ(±∞) = 0.Then, From (1.1c) and (2.1b) we obtain Let the characteristics X x , Y x starting from x defined as the solutions of the ordinary differential equations d dt Differentiation (4.3) with respect to x, and using (4.8) we obtain the Ricatti-type equations where d λ dt , d η dt denote the derivatives along the characteristics with the speed λ, η respectively.We prove below that the term R is bounded.Also, we obtain a bound of the integral of P 2 (respectively Q 2 ) on the characteristics X x (respectively Y x ).Then, the singularities given in Theorem 2.6 appear due to the term P 2 in (4.11a) and/or the term Q 2 in (4.11b).

4.2.
The choice of the approximated system.In order to obtain a system that admits global smooth solutions, we linearise the negative quadratic terms on the right-hand side of (4.11) on the neighbourhood of −∞.For that purpose, let ε > 0 and we define as in [58,59] Noting that (4.11) is like a derivative of (2.1), then, adding terms to (4.11) will involve some primitive terms in (2.1) which are not uniquely defined and cannot vanish at ∞ and −∞.That is why the system (2.1) will not be approximated simply by adding χ ε to (4.11) as in [58,59].
Our goal is to obtain a system on the form where h + , u + are suitable terms to be chosen.As in Section 4.1, we obtain as a linear map.This prevents singularities from appearing in finite time.From (1.6), we have Then the energy equation (1.5) becomes The goal is to find h + and u + such that • The right-hand side of (4.13) is a derivative of some quantity (i.e., [• • • ] x ), which will insure that R E dx is a decreasing function of time.• When P, Q are large, we have P = O(P ) and Q = O(Q).This insures (with Gronwall inequality) that no singularity will appear in finite time.We can write the right-hand side of (4.13) as On another hand we have In next section we prove the global existence of smooth solutions of the approximated system, and we obtain some uniform estimates that do not depend on ε.

Uniform estimates
In this section, we consider γ > 0, h > 0, and Let also j ε be a Friedrichs mollifier, we define h ε which implies that there exists Following the arguments of the previous section (see (4.14) and (4.15)), we consider the system where with G is defined as Differentiation (5.3) with respect to x we obtain Smooth solutions of (5.3) satisfy the energy equation (see Appendix B) where The first result on this section is the global well-posedness of (5.3).
Theorem 5.1.Let h > 0, (h 0 − h, u 0 ) ∈ D and ε ∈ (0, ε 0 ] then there exists a global smooth solution 3) and for all t > 0 we have Moreover, there exist A, B > 0 depending only on h, γ, g and E such that for any t > 0, x 2 ∈ R, and for In order to prove Theorem 5.1, we need to prove the invertibility of the operator L h and to obtain some estimates of its inverse.
Step 0. Let (•, •) be the scalar product in L 2 .Defining the bilinear map a : It is easy to check that a is continuous and coercive.Then, Lax-Milgram theorem insures the existence of a continuous bijective linear operator J : If Ju ∈ L 2 , an integration by parts shows that (h 3 u x ) x = hu − Ju ∈ L 2 and J = L h , this implies that u ∈ H 2 which finishes the proof that L h is an isomorphism from h can be defined as we obtain ψ = h u − 1 3 h 3 u zz . (5.27) The maximum principle insures that u L ∞ C ψ L ∞ which implies with (5.27) that (5.28) Using Landau-Kolmogorov inequality we obtain u z L ∞ C ψ L ∞ .Using again the change of variables (5.26) we get u x L ∞ C ψ L ∞ which completes the proof of (5.14).The estimate (5.15) follows directly from the change of variables (5.26), (5.28) and (5.14).
Differentiating now (5.31) with respect to x, using the definition of L h and replacing h −3 ψ by ψ we obtain Then, using (5.20) we obtain (5.22).
Step 3. It remains only to prove the inequalities (5.24).Since the operator (g is nothing but a convolution with the function G, the result follows directly using the Young inequality. , then, there exists a constant C = C(γ, h, E) > 0 independent on ε and h such that where A ε , B ε , V ε 1 and V ε 2 are defined as in (5.4), (5.5), (5.9) and (5.10) by replacing (h ε , u ε ) with (h, u).

Lemma 5.5. [Oleinik inequality]
There exists C > 0 that depends only on γ, h, g and E such that for all (t, x) ∈ (0, ∞) × R and ε ε 0 we have (5.64) Proof.Let D > 0 be a constant such that 2D −1 16h ε D, and A, B > 0 be the constants given in Theorem 5.1.Using Lemma 5.4, we obtain a constant M > 0 depending only on γ, h and E such that The goal is to prove that for all t and x we have Since the proof is the same, we only prove the inequality for P ε .Using the inequality −A ε x P ε 2(A ε ) 2 x + (P ε ) 2 /8 and using (5.7a), we obtain d λ dt (5.66) Let x ∈ R be fixed, we suppose that there exist t 1 > 0 such that P ε (t 1 , X x (t 1 ) = F(t 1 ) and P ε (t, X x (t) F(t) for all t ∈ [t 1 , t 2 ].Since P ε 0 then χ ε (P ε ) = 0. Integrating (5.66) on the characteristics between t 1 and t ∈ [t 1 , t 2 ] we obtain ), then initially we have P ε (0 + , X x (0 + ) < F(0 + ) = ∞.The inequality (5.67) shows that if P ε crosses F at some t 1 > 0, P ε remains always smaller than G for t t 1 .This completes the proof of P ε (t, X x (t)) G(t) for all t > 0. The proof for Q ε can be done similarly.

Precompactness
The goal of this section is to obtain a compactness of the solution.Due to the nonlinear terms of the equations, strong precompactness is needed to pass to the limit ε → 0. The strong precompactness of (h ε ) ε and (u ε ) ε is easy to obtain.However, the strong precompactness of (P ε ) ε and (Q ε ) ε is more challenging.Several lemmas in this section are inspired by [15,54,56,57,58,59].Along this section, Lemma A.2 is used many times without mentioning it.
Lemma 6.1.There exist ) and a subsequence of (h ε , u ε ) ε such that we have the following convergences Proof.From the energy equation (5.12) we have that ( . Then, using (5.3), and (5.59) we obtain that The weak convergence in H 1 ([0, T ] × R) follows directly.Using the inequality with (6.1) we obtain that uniformly on ε.Then, using Theorem 5 in [50] we can deduce that up to a subsequence, (h ε , u ε ) converges uniformly to (h, u) on any compact set of [0, ∞) × R when ε → 0. Now, we establish the weak precompactness of (P ε ) ε and (Q ε ) ε .
Lemma 6.2.There exist a subsequence of {P ε , Q ε } ε denoted also {P ε , Q ε } ε and families of probability Young measures ν 1 t,x , ν 2 t,x on R and µ t,x on R 2 , such that for all functions f, φ ∈ uniformly on any compact set [0, T ] ⊂ [0, ∞), and Moreover, the map belongs to L ∞ (R + , L 1 (R)), and We define which is from (6.4) the weak limit of g(P ε , Q ε ). Proof.
Step 1.The pointwise convergence of (6.2) is a direct corollary of Lemma A.1 with O = R and p = 2 and the energy equation (5.12).The key point to prove the uniform convergence is to show that the map is equicontinuous.Multiplying (5.7a) with f ′ (P ε ) one obtains Multiplying by φ(x) and integrating over [t Using that f ∈ C ∞ c , the the energy equation (5.12), Proposition 2.4, and Lemma 5.4 we find that the map (6.8) is equicontinuous.This finishes the proof of the uniform convergence of (6.2).The same proof can be used for (6.3).Using (A.2) we deduce that the map (6.5) belongs to L ∞ (R + , L 1 (R)).
Step 3. It only remains to prove (6.6), for that purpose, let f ∈ C ∞ c (R), we rewrite (6.9) on the form R), using (5.59) and the energy equation (5.12) we obtain that the remaining terms of the right-hand side of (6.11) are uniformly bounded in L 1 loc ((0, ∞) × R).Then, due to Lemma A.3 they are relatively compact in H −1 loc ((0, ∞) × R).Doing the same we can prove that for all Then, using Lemma A.6 (a generalised compensated compactness result) we obtain when ε → 0, ( where f (P ), g(Q) is the weak limit of (f (P ε ), g(Q ε )) defined in (6.7).Then, for any ϕ ∈ C ∞ c ((0, ∞) × R) , we have which implies (6.6).The proof of Lemma 6.2 is completed.Using (4.4), Lemma 6.2, (5.3a), (5.63) and Lemma 6.1 we can obtain the identities Now, we present some technical lemmas that are needed to obtain the strong precompactness of (P ε ) ε and (Q ε ) ε .
Lemma 6.6.The measures ν 1 , ν 2 given in Lemma 6.2 are Dirac measures, and .38)Proof.Since the proof is the same, we present here only the proof of ν 1 t,x (ξ) = δ P (t,x) (ξ).Note that if P 2 = P 2 then R P − ξ 2 dν 1 t,x (ξ) = 0 which implies that supp(ν 1 t,x ) = P .Since ν 1 t,x is a probability measure, necessarily ν 1 t,x = δ P .It remains then only to prove that P 2 = P 2 .The goal is to obtain an evolutionary inequality of P 2 − P 2 , then, since it is equal to zero initially, we prove that it remains zero for all time.The proof is given in several steps.

The global weak solutions
We use in this section the precompactness results given in the previous section to prove that the limit (h, u) given in Lemma 6.1 is a weak solution of (2.1).