Global existence for the confined Muskat problem

In this paper we show global existence of Lipschitz continuous solution for the stable Muskat problem with finite depth (confined) and initial data satisfying some smallness conditions relating the amplitude, the slope and the depth. The cornerstone of the argument is that, for these \emph{small} initial data, both the amplitude and the slope remain uniformly bounded for all positive times. We notice that, for some of these solutions, the slope can grow but it remains bounded. This is very different from the infinite deep case, where the slope of the solutions satisfy a maximum principle. Our work generalizes a previous result where the depth is infinite.


Introduction.
In this paper we study the dynamics of two different incompressible fluids with the same viscosity in a bounded porous medium. This is known as the confined Muskat problem. For this problem we show that there are global in time Lipschitz continuous solutions corresponding to initial data that fulfill some conditions related to the amplitude, slope, and depth. This problem is of practical importance because it is used as a model for a geothermal reservoir (see [6] and references therein) or as a model of an aquifer or an oil well (see [22]). The velocity of a fluid flowing in a porous medium satisfies Darcy's law (see [2,22,23]) where μ is the dynamic viscosity, κ is the permeability of the medium, g is the acceleration due to gravity, ρ( x) is the density of the fluid, p( x) is the pressure of the fluid, and v( x) is the incompressible velocity field. To simplify the notation we assume g = μ/κ = 1. The motion of a fluid in a two-dimensional porous medium is analogous to the Hele-Shaw cell problem (see [7,9,10,16,18] and the references therein). Let us consider the spatial domain S = R × (−l, l) for 0 < l. We assume impermeable boundary conditions for the velocity in the walls. In this domain we have two immiscible and incompressible fluids with the same viscosity and different densities; ρ 1 fills the upper subdomain and ρ 2 fills the lower subdomain (see Figure 1). The graph f (x, t) is the interface between the fluids.
It is well known that the system is in the (Rayleigh-Taylor) stable regime if the denser fluid is below the lighter one in every point x, i.e., ρ 2 > ρ 1 . Conversely, the system is in the unstable regime if there is at least a point x where the denser fluid is above the lighter one.
If the fluids fill the whole plane the contour equation satisfies (see [11]) where PV means principal value. For this equation the authors show the existence of a classical solution locally in time (see [11] and also [1,14,15,19]) in the Rayleigh-Taylor stable regime, and maximum principles for f (t) L ∞ (R) and ∂ x f (t) L ∞ (R) (see [12]). Moreover, in [4,5] the authors show the existence of turning waves and finite time singularities. In [8] the authors show an energy balance for the L 2 norm and some results concerning the global existence of solutions corresponding to "small" initial data. Furthermore, they show that if initially ∂ x f 0 L ∞ (R) < 1, then there is a global Lipschitz solution and if the initial data have small H 3 norm then there is a global classical solution.
The case where the fluid domain is the strip S = R × (−l, l), with 0 < l, has been studied in [3,13,14,15,17]. In this domain the equation for the interface is For (1.3) the authors in [13] obtain local existence of the classical solution when the system starts its evolution in the stable regime and the initial interface does not reach the walls. The authors construct initial data such that ∂ x f L ∞ (R) blows up in finite time. The authors also study the effect of the boundaries on the evolution of the interface, obtaining the maximum principle and a decay estimate for f (t) L ∞ (R) and the maximum principle for ∂ x f (t) L ∞ (R) for initial data satisfying the following hypotheses: < ∂ x f 0 L ∞ (R) tanh π 4l , (1.5) Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php and These hypotheses are smallness conditions relating ∂ x f 0 L ∞ (R) , f 0 L ∞ (R) , and the depth. We define (x(l), y(l)) as the solution of the system 2l − y tanh π 4l = 0, y + |2 cos π 2l − 2 sec 4 π 4l |y 3 1+y y+ x − 4y cos π l x = 0. Then, for initial data satisfying the authors in [13] show that These inequalities define a region where the slope of the solution can grow but it is bounded uniformly in time. This region only appears in the finite depth case.
In this paper the question of global existence of a weak solution (in the sense of Definition 1) for (1.3) in the stable regime is adressed. In particular we show the following theorem.
Moreover, if the initial data satisfy (1.4), (1.5), and (1.6) the solution fulfills the bounds while, if the initial data satisfy (1.8), the solution satisfies the bounds This result excludes the formation of cusps (blow up of the first and second derivatives) and turning waves for these initial data, remaining open to the existence (or nonexistence) of corners (blow up of the curvature with finite first derivative) during the evolution. Notice that in the limit l → ∞ we recover the result contained in [8]. In this paper and the works [3,13,17,20] the effect of the boundaries over the evolution of the internal wave in a flow in porous media has been addressed. When these results for the confined case are compared with the known results in the case where the depth is infinite (see [5,8,11,12,24]) three main differences appear: Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1. the decay of the maximum amplitude is slower in the confined case; 2. there are smooth curves with finite energy that turn over in the confined case but do not show this behavior when the fluids fill the whole plane; 3. to avoid the turning effect in the confined case you need to have smallness conditions in f 0 L ∞ (R) and ∂ x f 0 L ∞ (R) . However, in the unconfined case, only the condition in the slope is required. Moreover, in the confined case a new region without turning effect appears: a region without a maximum principle for the slope but with a uniform bound. In both cases (the region with the maximum principle and the region with the uniform bound), Theorem 1 ensures the existence of a global Lipschitz continuous solution. Keeping these results in mind, there are some questions that remain open. For instance, the existence of a wave whose maximum slope grows but remains uniformly bounded, or the existence of a wave with small slope such that, due to the distance to the boundaries, its slope grows and the existence (or nonexistence) of corner-like singularities when the initial data considered are small in W 1,∞ (R).
The proof of Theorem 1 is achieved using some lemmas and propositions. First, we define "ad hoc" diffusive operators and the regularized system (see section 2). For this regularized system, we show some a priori bounds for the amplitude and the slope. With these "a priori" bounds we show global existence of the H 3 solution (see section 3). Then, we obtain the weak solution to (1.3), f , as the limit of the regularized solutions (see sections 4 and 5).
Remark 1. In the rest of the paper we take π/2l = 1 and ρ 2 − ρ 1 = 4π and we drop in the notation the t dependence. We write c for a universal constant that can change from one line to another. We denote B(y, r) = [y − r, y + r].
2. The regularized system. In this section we define the regularized system and obtain some useful a priori bounds for the amplitude and the slope. To clarify the exposition we write f (x, t) for the solution of the regularized system.

Motivation and methodology.
We remark that the term is not if the curve does not reach the boundaries. In order to remove the singularity while preserving the inner structure, we put a term | tanh η 2 | for 0 < < 1/10 in both kernels. We define /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php To pass to the limit we use compactness coming from a uniform bound in Thus, we need to obtain a priori bounds for the amplitude and the slope. We define α i , i = 1, 2, 3, 4, positive constants that will be fixed below depending only on the initial datum considered. Taking derivatives in Ξ i , we obtain some terms with positive contribution. So, we attach some diffusive operators to the regularized system. Given a smooth function φ, we define We notice that, if the depth is not l = π/2, the previous operators should be rescaled and we write the subscript l to keep this dependence in mind. These operators are finite depth versions of the classical Λ α = (−Δ) α/2 . Roughly speaking, there are three different types of extra terms appearing in the derivatives of (2.1) and (2.2) that we need to control to obtain the a priori bound for the slope.
1. There are terms which have an integrable singularity and they appear multiplied by . In order to handle these terms we add − α 2 Λ 1− These terms go to zero due to the convergence of the operators but they are not multiplied by . In order to handle these terms we add −( 3. To absorb the nonsingular terms we add − √ α 1 f (x). We notice that, as < 1/10, the square root converges to zero less than linearly. This factor will be used because the contribution of some terms is O( a ) with 1/2 < a < 1. Once the a priori bounds are achieved, we should prove global solvability in H 3 for the regularized system. To get this bound we add ∂ 2 x f (x). We also regularize the initial datum. We take J ∈ C ∞ c (R), J ≥ 0, and J L 1 = 1, a symmetric mollifier, and define J (x) = J (x/ )/ . Given f 0 ∈ W 1,∞ (R) we define the initial datum for the regularized system as Putting all together, we define the regularized system (2.5) where α i are universal constants that will be fixed below depending only on the initial datum f 0 . We remark that f 0 ∈ H k (R) for all k ≥ 0. Notice that, due to the continuity of f 0 , uniformly on any compact set in R.
We use some properties of the operators Λ 1− l . For the reader's convenience, we collect them in the following lemma.
Lemma 1. For the operators Λ 1− l (see (2.3)), the following properties hold: let φ be a Schwartz function. Then, they converge acting on φ as goes to zero: 4. let φ be a Schwartz function. Then, the derivative can be written in two different forms as dη.
Proof. The proof of the first two statements follows from (2.3). For the proof of the third part we recall some useful facts: if |y| ≥ δ > 0, due to the mean value theorem, we get Now the proof follows in a straightforward way. For the last statement we use the cancellation coming from the principal value to define Using the uniform convergence of the derivative, we conclude the result.

2.2.
Maximum principle for f . In this section we prove an a priori bound for f . To simplify notation we define /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (1), define f 0 as in (2.4), and let f be the classical solution of (2.5) corresponding to the initial datum f 0 . Then f verifies Moreover, if f 0 has a sign then this sign is preserved during the evolution of f .
Proof. Changing variables and taking the derivative we obtain that (2.5) is equivalent to For notational convenience we use the notation σ = π 2 − f (x t ) and we define .
Using the definition ofθ and classical trigonometric identities we have . Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Putting together all the terms in Π , we obtain where in the last step we use the definition (2.4). In order to prove that the initial sign propagates we observe that if f 0 is positive (respectively, negative) the same remains valid for f 0 . Assume now that f 0 ≥ 0 and suppose that the line y = 0 is reached (if this line is not reached at any time t we are done). We write f ( We have tan(θ) > 0 and Π ≥ 0. Integrating in time we conclude the result.

Maximum principle
for ∂ x f . In this section we prove an a priori bound for ∂ x f . We define where θ andθ are defined in (2.8) and x t is a critical point for ∂ x f (x). We will use some bounds for μ 1 and, for the reader's convenience, we collect them in the following lemma.
Notice that we can take 0 < < 1/10 small enough to ensure that f (x, 0) defined in (2.4) also fulfills the hypotheses (1.4), (1.5), and (1.6). From (2.9), taking one derivative and using Lemma 1, we get where I 1 is the integral corresponding to Ξ 1 , I 2 is the integral corresponding to Ξ 2 , and This extra term appears from the regularization present in both Ξ i .
We have and The second term is given by We compute The second term is given by (2.22) Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php We need to obtain the local decay ∂ Assuming the classical solvability for (2.5) with an initial datum f 0 fulfilling the hypotheses (1.4), (1.5), and (1. 6) we have that f (x, δ) also fulfills (1.4), (1.5), and and ∂ x θ > 0. The linear terms in (2.14) have the appropriate sign and they will be used to control the the positive contributions of the nonlinear terms. We need to prove that ∂ t ∂ x f (x δ ) < 0. For the sake of simplicity, we split the proof of this inequality into different lemmas.
Proof. Using the linear term Λ 1− l to control (2.16), we have Then, the term (2.17) is The term (2.18) is These kinds of terms will be absorbed by α 1 . We have to deal with I 1 . We start with the term corresponding to Γ 2 2 in (2.20). We write Since 0 < δ 1 is small enough to ensure that the hypotheses (1.4), (1.5), and (1.6) hold at time δ, we have that, if |η| > 1, The term B 1 is not singular and can be bounded using (2.11) and (2.23): We compute and Using the mean value theorem, we bound the inner term D 1 as Due to (2.23), the outer term is Putting all together, we obtain /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Then, using the diffusion given by Λ 1− l to control C 2 , we get Due to |η/ sinh(η)| < 1 and 0 < < 1/10, some terms have the appropriate sign: and thus we can neglect their contribution. Furthermore, we have Taking α 2 /2 > 1 and using the mean value theorem, we get Combining these terms we conclude this result.
The term corresponding to Γ 1 2 in (2.20) is Lemma 5. If α 3 > 1, we have /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Proof. The proof follows the same ideas as in Lemma 4. We are done with Γ 1 2 ; thus, using the previous bound for Γ 2 2 , we are done with Γ 2 in (2.20). The terms in Γ 1 are not multiplied by and we have to obtain this decay from the integral. We write

Lemma 6. We have
Proof. We have The term B 5 is not singular and can be bounded using (2.6) and (2.7) as follows: We can bound B 6 in the same way, and To bound C 6 we need to use the diffusion coming from Λ l − Λ 1− l . Notice that, according to Lemma 1, we have and, when evaluating at the point where ∂ x φ(x) reaches its maximum, the first two terms are positive and they can be neglected. We get /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php where in the last step we have used the previous splitting in B(0, ) and R − B(0, ), (2.6), and (2.7). This concludes the result. Now that we have finished with Γ 2 1 , the term with Γ 1 1 is We have the following lemma.
Proof. The proof is similar to the proof of Lemma 6 and, for the sake of brevity, omit it.
Proof. Using classical trigonometric identities we can write and Therefore, as in [13], the sign of A 8 is the same as the sign of /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php then we can ensure that this contribution is negative. From (2.23) we get Using the cancellation when μ 1 (δ) = ∂ x f (x δ ), we obtain We consider the cases given by the sign and the size of μ 1 (δ).
Therefore, we get B 11 < 0 and we can neglect it.
3. Case μ 1 (δ) < −∂ x f (x δ ). We remark that in this case we have The last term is now positive due to the definition of ∂ x f (x δ ). Then, in this case, we have and we can neglect its contribution. Using Taylor's theorem in (2.26) we obtain the bound (2.13) and (2.25). Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php We are done with I 1 in (2.15) and now we move on to I 2 . These terms are easier because the integrals are not singular. With the same ideas as before we can bound the term involving Ω 2 .
We have the following result concerning the evolution of the slope. Proposition 2. Let f 0 ∈ W 1,∞ (R) be the initial datum in (1) satisfying (1.4), (1.5), and (1.6), define f 0 as in (2.4), and let f be the classical solution of (2.5) corresponding to the initial datum f 0 . Then f verifies Proof. For the sake of simplicity we split the proof in different steps.
Step 1 (local decay). Combining B 11 in (2.24) and A 9 in Lemma 10, and using the bounds (2.25) and (2.27) and the hypothesis (1.6) we obtain We take α 4 = 2 sec 2 ( f 0 L ∞ (R) ), α 3 = 2, α 2 = 3(1 + sec 2 ( f 0 L ∞ (R) )). Since we have a term √ and 0 < < 1/10, we can compare the bounds in Lemmas 3-11 with is chosen big enough. The universal constant c in all these bounds can be c = 1000. We have shown that for every 0 < δ 1 small enough, there is local-in-time decay. As δ is positive and arbitrary, we have Step 2 (from local decay to an uniform bound). Then, in the worst case, we have These inequalities ensure that the hypotheses (1.4), (1.5), and (1.6) hold at time t = t * and ∂ x f (t) L ∞ (R) decays again.
Step 3 (the case where f (x t ) = min x ∂ x f (x, t)). This case follows the same ideas, and we conclude, thus, the result.
Proposition 3. Let f 0 ∈ W 1,∞ (R) be the initial datum in (1) satisfying (1.8) and define f 0 as in (2.4). Let f be the classical solution of (2.5) corresponding to the initial datum f 0 . Then, f verifies Proof. The region delimited by (x(l), y(l)) is below the region with maximum principle (see [13]). Then, in the worst case, at some t * > 0 we have that ( and ∂ x f (t) L ∞ (R) ≤ 1. Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php If the initial datum satisfies (1.8), by Propositions 1 and 3, the solution to the regularized system again satisfies the bounds (3.1). Then we have the following proposition.
Proposition 4. Let f 0 ∈ W 1,∞ (R) be the initial datum in (1) satisfying (1.4), (1.5), and (1.6) or (1.8) by itself, and define f 0 as in (2.4). Then for every > 0 and Proof. We have to bound the L 2 norm of the function and its third derivative. We split the proof in different steps.
Step 1 (the function). We have Using (2.3) we get and we obtain that the contribution of the linear terms is negative. The nonlinear term Ξ 1 defined in (2.1) is Using the cancellation coming from the principal value we have Inserting (2.11) and (2.10) in the expression for A 1 we obtain The second term in I 2 is | tanh(η/2)| tanh(η/2) dηdx. Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Using the Cauchy-Schwarz inequality, the equality ∂ x f (x − η) = −∂ η f (x − η), and integrating by parts we get To finish with the L 2 norm we have to deal with I 3 . We have whereθ is defined in (2.8). Using the same ideas as in I 2 and we conclude the bound Putting all these bounds together we get Step 2 (the third derivative). To study the L 2 norm of the third derivative, we compute The term I 4 is positive due to Lemma 1: The nonlinear terms related to θ are The term A 3 is not singular if > 0 and can be bounded using Hölder and Nirenberg interpolation inequalities. For the sake of brevity, we write some terms detailedly, the rest being analogous to them. We have A 3 = B 1 + B 2 + lower order terms. Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php we obtain The second term is and using the classical interpolation inequality We split the term A 4 as follows: These terms are not singular because of the domain of integration. We have to deal with the integrability at infinity in η. We compute dηdx.
The integrability at infinity is obtained using (2.6) and (2.7). We only bound the more singular terms in B 3 and B 4 . The most singular term in B 3 is dηdx. Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Using (2.6), (2.7), and (2.10), we obtain Analogously, the more singular term in B 4 is Using the same bounds as in C 1 , we get Using classical trigonometric identities, we obtain And the most singular term in B 5 is Using the cancellation of the principal value integral we obtain sin 2 (θ) sinh(η) 2 sinh 4 (η/2) 1 + sin 2 (θ) thus, Integrating by parts in D 2 , we obtain the required decay at infinity and we conclude Putting all together, we get The nonlinear terms related toθ are We observe that, due to 1/10 > > 0 and f 0 L ∞ (R) < π/2, this integral is not singular. Thus the inner part A 5 can be bounded following the same ideas as for A 3 . Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php , and every T > 0. Fixing t, due to the uniform bound in W 1,∞ (R) and the Ascoli-Arzela theorem we have that, up to a subsequence, f (t) → f (t) uniformly on any bounded interval I ⊂ R. Moreover, for all N , we have In order to prove this uniform convergence on compact sets we use the spaces and results contained in [8].
Proposition 5. Let f be the limit of the regularized solutions f . Then f is a weak solution of (1.3). Downloaded 05/21 /19 to 193.144.198.194. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php The only thing to check is the convergence of I 2 . Due to the compactness of the support of φ, we have Since we have (up to a subsequence) that f → f uniformly on compact sets (see Lemma 12), the uniform convergence | tanh(η/2)| → 1 if |η| > δ, and the continuity of all the functions in this integral, the limit in and the integral commute and we get We conclude the proof of Theorem 1 by taking δ 1 and N 1 to control the tails and then we send → 0.