On an Asymptotic Model for Free Boundary Darcy Flow in Porous Media

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. M\mathrm{A}\mathrm{T}\mathrm{H}. A\mathrm{N}\mathrm{A}\mathrm{L}.©2020 Society for Industrial and Applied Mathematics Vol. 52, No. 5, pp. 4937--4970 ON AN ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW IN POROUS MEDIA\ast RAFAEL GRANERO-BELINCH\' ON\dagger \mathrm{A}\mathrm{N}\mathrm{D} STEFANO SCROBOGNA\ddagger Abstract. We provide a rigorous mathematical study of an asymptotic model describing Darcy flow with free boundary in a small amplitude/large wavelength approximation. In particular, we prove several well-posedness results in critical spaces. Furthermore, we also study how the solution decays towards the flat equilibrium. Key words. Muskat problem, moving interfaces, free boundary problems AMS subject classifications. 35455, 35B41, 92C17 DOI. 10.1137/19M126623X 1. Introduction. Since the pioneering works of Boussinesq [8], the derivation and study of asymptotic models for free boundary flows (usually, the water waves problem) is a hot research area (the interested reader can refer to [43]). In this paper we study an asymptotic model for the intrusion of water into oil sand. This is known as the Muskat problem [44]. The study of the (full) Muskat problem has received a lot of attention in the last years with several very recent advancements on the problem (cf. [1, 2, 4, 5, 31, 45]), and, as a consequence, there is a large literature available (we refer to [36] for a recent survey of the available results). More precisely, this work considers the following one-dimensional equation: \Biggl\{ \partial tf= - \nu \Lambda 3f - \Lambda f+\partial x\bigl( J\scrH , fK\bigl( \nu \Lambda 3f+ \Lambda f\bigr) \bigr) , f| t=0 =f0, (1.1) where JA, BK=AB - BA denotes the commutator between Aand B, the Bond number \nu \geq 0 represents the ratio between capillarity and gravitational effects, and the Hilbert transform \scrH and the Calder\'on operator \Lambda are defined as the following Fourier multiplier operators: \widehat \scrH f(n) = - isgn(n)\^ f(n),\widehat \Lambda f(n) = | n| \^ f(n).(1.2) Equation (1.1) was derived by the authors as an asymptotic model for the Darcy flow in porous media under the assumption that the amplitude over the wavelength, a quotient known as steepness, is small [37] (see also [14, 38, 39, 41]). In the gravity driven case (when \nu = 0), (1.1) reads as \ast Received by the editors June 5, 2019; accepted for publication (in revised form) August 6, 2020; published electronically October 14, 2020. https://doi.org/10.1137/19M126623X Funding: The work of the first author was supported by the Ministerio de Ciencia, Innovacion y Universidades MICIU through the project ``Mathematical Analysis of Fluids and Applications,"" grant PID2019-109348GA-I00 and acronym ``MAFyA."" The work of the second author was supported by the Basque Government through the BERC 2018-2021 program and the Spanish Ministry of Economy and Competitiveness (MINECO) through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project MTM2017-82184-R funded by AEI/FEDER, UE, and acronym ``DESFLU."" \dagger Departamento de Matem\'aticas, Estad\'{\i}stica y Computaci\'on, Universidad de Cantabria, Santander, Spain ([email protected]). \ddagger Departamento de An\'alisis Matem\'atico \& IMUS, Sevilla, 41012, Spain (scrob[email protected]). 4937 Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4938 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA \Biggl\{ \partial tf+ \Lambda f=\partial xJ\scrH , fK\Lambda f, f| t=0 =f0. (1.3) Although (1.1) and (1.3) seem semilinear fourth and second order nonlocal PDEs, respectively (due to terms like f\partial 4 xfand f\partial 2 xf), the commutator structure of the nonlinearity implies that they are a quasilinear third and a fully nonlinear first order nonlocal PDE, respectively (see Lemma 3.1 below). This feature is key in the analysis of the Dirichlet-to-Neumann operator (cf. [3, Lemma A.1.11] and the pioneer works by Craig [27, 28, 29]). There are several motivations to study asymptotic models of free boundary Darcy flow. One of them arises from computational reasons. The idea is, then, to simulate the asymptotic model to obtain an accurate description of the full problem at a lower computational cost. In that regard, let us briefly emphasize that the one-phase Muskat problem reveals itself as somehow harder (computationally speaking) than the twophase Muskat problem. Let us try to briefly explain the reasons for this ``paradoxical"" fact. In the case when there are two fluids, the gravity driven Muskat problem reads as the following single nonlocal PDE [18]: \partial tf= p.v. \int \BbbR (\partial xf(x) - \partial xf(x - y)) y (f(x) - f(x - y))2+y2dy. Simplified models for this case were provided (following heuristic ideas) by C\'ordoba, Gancedo, Orive [20] (see also [42]). Remarkably, when the one-phase Muskat problem is considered, the previous PDE has to be modified, and one is forced to study the following system of a nonlocal PDE and an integral equation [17]: \partial tf(x) = p.v. \int \BbbR \varpi (x - y)y y2+ (f(x) - f(x - y))2dy - \partial xf(x)p.v. \int \BbbR \varpi (x - y)f(x) - f(x - y) y2+ (f(x) - f(x - y))2dy, - \partial xf(x) = p.v. \int \BbbR \varpi (\beta )\scrB (x, f(x), \beta , f(\beta ))d\beta \cdot (1, \partial xf(x)) + \varpi (x) 2, where \scrB denotes the kernel of \nabla \bot \Delta - 1, i.e., \scrB (x1, x2, y1, y2) = \biggl( - x2 - y2 (x2 - y2)2+ (x1 - y1)2,x1 - y1 (x2 - y2)2+ (x1 - y1)2\biggr) . Thus, to write the amplitude of the vorticity in terms of the interface, one needs to invert an operator as in C\'ordoba, C\'ordoba, and Gancedo [17]. This is, mathematically and computationally, a challenging issue. Another reason is the possibility of finding new finite time singularity scenarios. In particular, for the two-phase Muskat problem, Castro et al. [10] proved the existence of turning waves, i.e., interfaces that can be parametrized as a smooth graphs at time t= 0 but that become smooth curves that cannot be parametrized as graphs after a finite time (see also [7, 21, 22, 23, 33]). We observe that these turning waves are interfaces such that there exist 0 < T1<\infty and lim sup t\rightarrow T1\| \partial xf(t)\| L\infty =\infty . In the case of the one-phase Muskat problem, Castro et al. [11] proved the existence of curves that self-intersect in finite time in what is called a splash singularity (see Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4939 also [24, 26, 30, 32]. It remains as an interesting open problem whether the Muskat problem can have a cusp singularity, i.e., a singularity where the slope and curvature of the interface blows up while the interface remains a graph (see [9, 13, 15, 16, 34, 40] for global existence results). In that regard, we are optimistic enough to think that such a scenario should be simpler to prove (or discard) in an asymptotic model rather than the full problem. Taking this into consideration, the results of this paper then give conditions that excludes finite time blow ups of turning type. 1.1. Functional spaces. The space domain considered in the present article is the one-dimensional torus, i.e., \BbbT =\BbbR \big/ 2\pi \BbbZ .The domain \BbbT can also be understood as the interval [ - \pi , \pi ] endowed with periodic boundary conditions. Let f(x) denote aL2function on \BbbT . Then, its Fourier transform is an \ell 2(\BbbZ ) sequence defined as \^ f(n) = \^ fn=1 \surd 2\pi \int \BbbT f(x)e - ix ndx for any n\in \BbbZ , with inverse Fourier transform f(x) = 1 \surd 2\pi \sum n\in \BbbZ \^ f(n)eix n. Then, we define the L2-based (homogeneous) Sobolev spaces \. H\alpha (\BbbT ) \. H\alpha (\BbbT ) = \Biggl\{ u(x) such that \| u\| 2 \. H\alpha (\BbbT ):= \sum k| k| 2\alpha | \^u(k)| 2<\infty \Biggr\} . In the present work we will need to define Sobolev spaces with fractional derivatives and integrability indexes different from two, i.e., spaces \. Ws,p (\BbbT ) for s\in \BbbR and p\in [1,\infty ). We provide here a characterization of such spaces using using the theory of Littlewood-Paley (see Appendix A). If we consider the dyadic block \bigtriangleup qwe can hence define the seminorm \| u\| \. Ws,p(\BbbT )=\left( \sum q\in \BbbZ 2pqs \| \bigtriangleup qu\| p Lp(\BbbT )\right) 1/p . It is well-known that \| u\| \. Ws,2(\BbbT )\sim \| u\| \. Hs(\BbbT ), in the sense that there exists a positive absolute constant C >0 such that 1 C\| u\| \. Hs(\BbbT )\leqslant \| u\| \. Ws,2(\BbbT )\leqslant C\| u\| \. Hs(\BbbT ). Similarly, we define the (homogeneous) Wiener spaces \. A\alpha (\BbbT ) and Wiener spaces with weight as \. A\alpha (\BbbT ) = \Biggl\{ u(x) such that \| u\| \. A\alpha (\BbbT ):= \sum k| k| \alpha | \^u(k)| <\infty \Biggr\} , \. A\alpha \nu (\BbbT ) = \Biggl\{ u(x) such that \| u\| \. A\alpha \nu (\BbbT ):= \sum k| k| \alpha e\nu | k| | \^u(k)| <\infty \Biggr\} , respectively. Obviously, \. A\alpha =\. A\alpha 0, and the functions in As \nu are analytic in the complex strip Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4940 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA \BbbS \nu =\{ x+iy, | y| < \nu \} . We observe that (1.1) conserves the average of a solution f. Thus, without losing generality, we can assume this average to be zero. Thus, from this point onwards we identify the spaces \. Hs(\BbbT ) and Hs(\BbbT ), \. As(\BbbT ) and As(\BbbT ), and \. As \nu (\BbbT ) and As \nu (\BbbT ). Then, the previously defined seminorms when restricted to average-free functions are in fact genuine norms. Let us notice that the following embedding, valid for mean-free distributions on the one-dimensional torus \. H1/2(\BbbT )\lhook \rightarrow Lp(\BbbT )\forall p\in [1,\infty ), and in particular the following inequality hold true for any u\in \. H1/2\cap Lp: \| u\| Lp\leqslant C\surd p\| u\| \. H1/2. We also recall that Hs(\BbbT ) for s > 1/2 embeds continuously in L\infty (\BbbT ) with bound \| u\| L\infty \leqslant C \eta \| u\| \. H1 2+\eta .(1.4) 1.2. A word on scaling. We observe that, when \nu = 0, (1.1) is left invariant by the scaling f\lambda (x, t) = 1 \lambda f(\lambda x, \lambda t).(1.5) Remarkably, this scaling is the same as in the full Muskat problem (see [25]). Then, we note that the following spaces are critical for this scaling: L\infty \Bigl( 0, T;\. A1(\BbbT )\Bigr) , L\infty \Bigl( 0, T;\. H3/2(\BbbT )\Bigr) , L\infty \Bigl( 0, T;\. W1,\infty (\BbbT )\Bigr) . Let us now consider (1.1) for \nu > 0. In this setting (1.1) reads, when expanded, as \partial tf+\nu \Lambda 3f+ \Lambda f=\nu \Bigl[ \Lambda \bigl( f\Lambda 3f\bigr) - \partial x\bigl( f \partial 3 xf\bigr) \Bigr] +\Bigl[ \Lambda (f\Lambda f) + \partial x(f \partial xf)\Bigr] .(1.6) There is no scale invariance satisfied by (1.6). Despite this fact we can rewrite (1.6) as 1 2\partial tf+\nu \Lambda 3f - \nu \Lambda \bigl( f\Lambda 3f\bigr) +\nu \partial x\bigl( f \partial 3 xf\bigr) = - 1 2\partial tf - \Lambda f+ \Lambda (f\Lambda f) + \partial x(f \partial xf) (1.7) and then study separately the homogeneity of the left- and right-hand side of (1.7). Concerning the right-hand side the deductions of above still hold, and it is invariant with respect to the scaling (1.5). The left-hand side of (1.7) is invariant with respect to the scaling f\lambda (x, t) = 1 \lambda f\bigl( \lambda x, \lambda 3t\bigr) .(1.8) We remark that the space L\infty t(\BbbR +) is invariant with respect to both the dilations t\mapsto \rightarrow \lambda t and t\mapsto \rightarrow \lambda 3twhich characterize, respectively, (1.5) and (1.8). Then, the spaces L\infty \Bigl( 0, T;\. H3/2(\BbbT )\Bigr) , L\infty \Bigl( 0, T;\. A1(\BbbT )\Bigr) are, in a certain sense relatively to scale invariance, reasonable critical spaces to consider for (1.6). Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4941 2. Main results and discussion. 2.1. Results for the gravity driven case (1.3). Our first result proves the existence of solution for analytic initial of arbitrary size. Theorem 2.1. Let f0\in A1 1(\BbbT ); there exist a short enough time T, 0< T \leq 1 4\| f0\| A1 1 , and a unique mild solution to (1.3) stemming from f0which moreover belongs to the space f\in L\infty \bigl( 0, T;A1 1\bigr) \cap C\bigl( [0, T], A1 0.5\bigr) . This result is a sort of Cauchy--Kovalevsky theorem. However, we emphasize that the solution is less regular in time (merely L\infty instead of continuous) but maintains its original strip of analyticity. Furthermore, we are able to prove a result establishing the decay of certain norms of the solution. Proposition 2.2. Assume that the solution to (1.3) constructed in Theorem 2.1 satisfies sup 0\leq t\leq T\| f(t)\| \. A1\leqslant 1; then sup 0\leq t\leq T max yf(y, t)\leqslant max yf0(y), inf 0\leq t\leq Tmin yf(y, t)\geqslant min yf0(y), and, as a consequence, sup 0\leq t\leq T\| f(t)\| L\infty \leqslant \| f0\| L\infty . This result seems the analogue of the maximum principles known for the full Muskat problem [5, 19, 23]. In particular, when compared to the results in the previous papers, it seems that the model equation (1.3) is less stable than the full Muskat problem. The fact that the maximum principle holds if and only if \Lambda fis small (see below for the proof) may seem somehow surprising. However, we should recall that the model studied in this work is only valid when the steepness parameter (i.e., a quantity akin to \| f\| \. A1) is small. In other words, the range of validity of the PDE under consideration as a model of the full Muskat problem is related to the smallness condition guaranteeing the decay stated in the previous proposition. In this sense, since the range of validity of the model imposes restrictions in the slope of f, there is no contradiction between the conditional decay of the PDE under consideration and the decay with no extra assumptions of the full Muskat problem. Once the local solution for analytic data is known, we turn our attention to the global solution for initial data satisfying certain size restrictions in critical spaces. Then, our well-posedness result for Wiener class initial data reads as follows. Theorem 2.3. Let f0\in A1be the initial data for (1.3). Assume that \| f0\| \. A1<1/2; then the Cauchy problem (1.3) is globally well-posed and admits a unique solution f\in L\infty \bigl( \BbbR +;A1(\BbbT )\bigr) \cap \scrM \bigl( \BbbR +;A2(\BbbT )\bigr) , Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4942 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA verifying \| f(t)\| A1\leqslant \| f0\| A1 (2.1) and \| f(t)\| A0\leqslant \| f0\| A0e - (1 - 2\| f0\| A1)t. We observe that these are a global existence and decay for initial data in a critical space. The next result we prove in the present manuscript is a global existence result for initial data in H3 2\cap H3 2+\varepsilon , \varepsilon > 0; we suppose the initial data to be small in H3 2 and of arbitrary size in H3 2+\varepsilon . Very recently D. Cordoba and O. Lazar proved in [25] that it possible to construct global solutions for the two-dimensional (full) Muskat problem when the initial data is small in the space H3/2and also belongs to H5/2. The following theorem establishes a similar result for the evolution equation (1.3). Theorem 2.4. Let us suppose f0\in H3 2\cap H3 2+\varepsilon , \varepsilon > 0. There exists a constant C > 0independent of \varepsilon and f0such that, if f0satisfies \| f0\| H3 2<min \left\{ 1,\varepsilon 2 C\Bigl( \| f0\| H3 2+\varepsilon log \| f0\| H3 2+\varepsilon + 1\Bigr) 2\right\} , the Cauchy problem (1.3) is globally well-posed and admits a unique solution fsuch that for any T > 0 f\in C0,1 2 - \vargamma \Bigl( [0, T) ; H1+\vargamma (\varepsilon +1)\Bigr) \cap L\infty \Bigl( [0, T ) ; H3 2+\varepsilon \Bigr) \cap L2\bigl( [0, T) ; H2+\varepsilon \bigr) , 1 2 (\varepsilon + 1) \leqslant \vargamma < 1 2. In other words, the initial data must be small in H3/2even for large \epsilon . Furthermore, if the initial data is large in H3/2+\varepsilon , then the size condition for the initial data in H3/2is even more restrictive. The methodology used in order to prove Theorem 2.4 differs completely from the one used in [25], since (1.3) presents a nonlinearity of polynomial type we will be able to use tools characteristic of the paradifferential calculus, we refer the reader to [6, Chapter 2]; we will hence decompose the nonlinearity in an infinite sum of elementary packets on which we will be able to highlight some nontrivial regularizing properties of (1.3). 2.2. Results for the gravity-capillary driven case (1.1). We are able to extend Theorem 2.3 to the case with surface tension. Theorem 2.5. Let \nu > 0be a fixed parameter and f0\in A1be the initial data for (1.1). Assume that \| f0\| \. A1<1/2; then the Cauchy problem (1.1) is globally well-posed and admits a unique solution f\in L\infty \bigl( \BbbR +;A1(\BbbT )\bigr) \cap \scrM \bigl( \BbbR +;A4(\BbbT )\bigr) , verifying \| f(t)\| A1\leqslant \| f0\| A1 and \| f(t)\| A0\leqslant \| f0\| A0e - (1 - 2\| f0\| A1)(1+\nu )t. Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4943 Remark 2.6. We observe that the size condition is \nu -independent. The following result is a result analogous to the one stated in Theorem 2.4 for the system (1.1) when \nu > 0. Theorem 2.7. Let us suppose that f0is a zero mean function on \BbbT ,f0\in H2(\BbbT ). There exists a C > 0such that if \| f0\| H2(\BbbT )\leqslant 1 Cmin \Bigl\{ 1, \nu - 1 4\Bigr\} ,(2.2) then there exists a unique solution f\in \scrC (\BbbR +;H2)\cap L2(\BbbR +;H7 2)of the Cauchy problem (1.1) such that for any t > 0 \| f(t)\| 2 H2(\BbbT )+\int t 0\biggl[ \nu \bigm\| \bigm\| \bigm\| \Lambda 3/2f(s)\bigm\| \bigm\| \bigm\| 2 H2(\BbbT )+\bigm\| \bigm\| \bigm\| \Lambda 1/2f(s)\bigm\| \bigm\| \bigm\| 2 H2(\BbbT )\biggr] ds\leqslant \| f0\| 2 H2(\BbbT ). 2.3. Discussion. In this paper we prove several well-posedness results for an asymptotic model of free boundary Darcy flow. Most of them are global existence results in scale invariant spaces. In that regard, our results should be understood as nonlinear stability results rather than linear stability (even if some size conditions are imposed on the initial data). These results exclude the existence of turning wave singularities but leave open the door to cusp singularities. The occurrence of such behavior will be the object of future research. 3. Gravity driven system (1.3). 3.1. Proof of Theorem 2.1. The cascade of linear equations. We look for a solution of the form f(x, t) = \lambda \infty \sum \ell =0 \lambda \ell f(\ell )(x, t), where \lambda =\lambda (f0) will be chosen below. Then, the existence of solution is reduced to the summability of a series where each term satisfies a linear problem. A similar idea can be tracked back to the works of Oseen [46] (see also [14]). Indeed, matching the appropriate terms, we find that f(\ell )satisfies \partial tf(\ell )+ \Lambda f(\ell )= \ell - 1 \sum j=0 \Lambda \Bigl( f(j)\Lambda f(\ell - j - 1)\Bigr) +\partial x\Bigl( f(j)\partial xf(\ell - j - 1)\Bigr) with initial data f(\ell )(x, 0) = 0 if \ell \not = 0 and f(0)(x, 0) = f0 \lambda otherwise. Then f(0) =e - t\Lambda f0 \lambda , and we can solve recursively for the other f(\ell ). In particular, using that the solution of \partial tu(x, t)+\Lambda u(x, t) = F(x, t), u(x, t) = g(x) is given by \^u(k, t) = e - t| k| \^g(k) + \int t 0 e - (t - s)| k| \^ F(k, s)ds, Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4944 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA we find that \widehat f(\ell )(k, t) = \ell - 1 \sum j=0 \infty \sum n= - \infty \int t 0 e - (t - s)| k| \Bigl[ | k| \Bigl( \widehat f(j)(n, s)| k - n| \widehat f(\ell - j - 1)(k - n, s)\Bigr) +ik \Bigl( \widehat f(j)(n, s)i(k - n)\widehat f(\ell - j - 1)(k - n, s)\Bigr) \Bigr] . Thus, using | k| | k - n| - k(k - n) = | k| | k - n| (1 - sgnksgn (k - n)) \leqslant 2| k| | k - n| , we have that \widehat \partial x(J\scrH , aK\Lambda b) = \sum n \^a(k - n)\^ b(n) (| k| | k - n| - k(k - n)) \leqslant 2\sum n\bigm| \bigm| \bigm| \^a(k - n)\^ b(n)| n| | k - n| \bigm| \bigm| \bigm| .(3.1) Thus, we obtain that \bigm| \bigm| \bigm| \widehat f(\ell )(k, t)\bigm| \bigm| \bigm| \leq 2 \ell - 1 \sum j=0 \infty \sum n= - \infty \int t 0 e - (t - s)| k| \bigm| \bigm| \bigm| \widehat f(j)(n, s)\bigm| \bigm| \bigm| | n| | k - n| \bigm| \bigm| \bigm| \widehat f(\ell - j - 1)(k - n, s)\bigm| \bigm| \bigm| ds. Approximate problems. We now fix 1 \ll Rand consider the partial sum fR(x, t) = \lambda R \sum \ell =0 \lambda \ell f(\ell )(x, t). Then, fRsatisfies \partial tfR+ \Lambda fR= R \sum \ell =0 \lambda \ell +1 \ell - 1 \sum j=0 \Lambda \Bigl( f(j)\Lambda f(\ell - j - 1)\Bigr) +\partial x\Bigl( f(j)\partial xf(\ell - j - 1)\Bigr) = R \sum \ell =1 \ell - 1 \sum j=0 \Lambda \Bigl( \lambda j+1f(j)\lambda \ell - j - 1+1\Lambda f(\ell - j - 1)\Bigr) +\partial x\Bigl( \lambda j+1f(j)\lambda \ell - j - 1+1\partial xf(\ell - j - 1)\Bigr) . Nonlinear estimates. We estimate \lambda \bigm\| \bigm\| \bigm\| f(0)\bigm\| \bigm\| \bigm\| A1 1\leq \| f0\| A1 1.(3.2) Define \nu (\ell ) = R - \ell + 1. Applying Tonelli's theorem, we find that \| f(\ell )\| A1 \nu (\ell ) \leq 2 \ell - 1 \sum j=0 \infty \sum k= - \infty \infty \sum n= - \infty \int t 0 e - (t - s)| k| | k| e\nu (\ell )| k| \bigm| \bigm| \bigm| \widehat f(j)(n, s)\bigm| \bigm| \bigm| | n| | k - n| \bigm| \bigm| \bigm| \widehat f(\ell - j - 1)(k - n, s)\bigm| \bigm| \bigm| ds \leq 2 \ell - 1 \sum j=0 \infty \sum k= - \infty \infty \sum n= - \infty \int t 0 e(\nu (\ell )+1)| k| \bigm| \bigm| \bigm| \widehat f(j)(n, s)\bigm| \bigm| \bigm| | n| | k - n| \bigm| \bigm| \bigm| \widehat f(\ell - j - 1)(k - n, s)\bigm| \bigm| \bigm| ds, Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4945 where we have used | k| \leq e| k| . Now we observe that, since 0 \leq j\leq \ell - 1 and 0 \leq \ell - j - 1\leq \ell - 1, \nu (\ell ) + 1 = R - (\ell - 1) + 1 \leq R - j+ 1 = \nu (j) and \nu (\ell ) + 1 = R - (\ell - 1) + 1 \leq R - (\ell - j - 1) + 1 = \nu (\ell - j - 1). Thus, \bigm\| \bigm\| \bigm\| f(\ell )(t)\bigm\| \bigm\| \bigm\| A1 \nu (\ell ) \leq 2 \ell - 1 \sum j=0 \infty \sum n,k= - \infty \int t 0 e\nu (j)| n| e\nu (\ell - j - 1)| k - n| \bigm| \bigm| \bigm| \widehat f(j)(n, s)\bigm| \bigm| \bigm| | n| | k - n| \bigm| \bigm| \bigm| \widehat f(\ell - j - 1)(k - n, s)\bigm| \bigm| \bigm| ds \leq 2 \ell - 1 \sum j=0 \int t 0\bigm\| \bigm\| \bigm\| f(j)(s)\bigm\| \bigm\| \bigm\| A1 \nu (j)\bigm\| \bigm\| \bigm\| f(\ell - j - 1)(s)\bigm\| \bigm\| \bigm\| A1 \nu (\ell - j - 1) ds. We define the numbers A\ell = 2 \bigm\| \bigm\| \bigm\| f(\ell )(t)\bigm\| \bigm\| \bigm\| A1 \nu (\ell ) ,A0= 1. These numbers satisfy A\ell \leq \ell - 1 \sum j=0 \int t 0 AjA\ell - j - 1ds. Then, we prove by induction that A\ell \leq C\ell t\ell , where C\ell are the Catalan numbers. It is known that the Catalan numbers grow like 4\ell . We also observe that \nu (\ell )\geq 1 for \ell \leq R. As a consequence, we have the bound \| fR(t)\| A1 1\leq \lambda \bigm\| \bigm\| \bigm\| f(0)\bigm\| \bigm\| \bigm\| A1 1 +\lambda R \sum \ell =1 \lambda \ell \bigm\| \bigm\| \bigm\| f(\ell )(t)\bigm\| \bigm\| \bigm\| A1 1 \leq \lambda \bigm\| \bigm\| \bigm\| f(0)\bigm\| \bigm\| \bigm\| A1 1 +\lambda R \sum \ell =1 \lambda \ell \bigm\| \bigm\| \bigm\| f(\ell )(t)\bigm\| \bigm\| \bigm\| A1 \nu (\ell ) \leq \lambda \bigm\| \bigm\| \bigm\| f(0)\bigm\| \bigm\| \bigm\| A1 1 +\lambda 2 R \sum \ell =1 \lambda \ell A\ell \leq \lambda \bigm\| \bigm\| \bigm\| f(0)\bigm\| \bigm\| \bigm\| A1 1 +\lambda 2 R \sum \ell =1 \lambda \ell C\ell t\ell \leq \| f0\| A1 1+\lambda 2 \infty \sum \ell =1 (4\lambda t)\ell , Downloaded 10/15/20 to 193.144.183.85. 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Unauthorized reproduction of this article is prohibited. 4952 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA 1 2 d dt\| f(t)\| 2 L2(\BbbT )+\bigm\| \bigm\| \bigm\| \Lambda 1/2f(t)\bigm\| \bigm\| \bigm\| 2 L2(\BbbT )=\int \BbbT f\Bigl( (\Lambda f)2 - (\partial xf)2\Bigr) dx, =\int \BbbT f(\scrH - 1) \partial xf(\scrH + 1) \partial xfdx, =\scrN (f, f, f), (3.8) where \scrN (g1, g2, g3) = \int \BbbT g1(\scrH - 1) \partial xg2(\scrH + 1) \partial xg3dx.(3.9) In order to conclude the inequality (3.7), we have to prove that the following inequality holds true: | \scrN (g, h, h)| \leqslant C\| g\| H3/2\| h\| 2 H1/2.(3.10) Indeed, if we denote as (\^gn)n\in \BbbZ ,(\^ hn)n\in \BbbZ the Fourier transform of gand h, respectively, and use the Plancherel theorem, we can rewrite \scrN (g, h, h) as \scrN (g, h, h) = \sum (n,k,m)\in \BbbZ 3 k+m=n \^gn( - isgnm - 1) (im)\^ hm( - isgnk+ 1) (ik)\^ hk =\sum (n,k,m)\in \BbbZ 3 k+m=n \^gnmk \^ hm\^ hk(1 + sgnmsgnk). We observe that, if the addends are nonzero, kand nmust verify 0\leqslant k\leqslant nor n\leqslant k\leqslant 0. In other words, | k| \leqslant | n| . Furthermore, using the previous cancellation to find a symmetry in the series, we find that \scrN (g, h, h)=4 \sum (n,k)\in \BbbZ 2 0\leqslant k\leqslant n (n - k)kRe \Bigl( \^gn\^ hn - k\^ hk\Bigr) .(3.11) Thus, | \scrN (g, h, h)| \leqslant 4\sum (n,k)\in \BbbZ 2 0\leqslant k\leqslant n | n - k| | k| | \^gn| \bigm| \bigm| \bigm| \^ hn - k\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \^ hk\bigm| \bigm| \bigm| , but since 0 \leqslant k\leqslant nwe deduce that (n - k)k\leqslant n3/2(n - k)1/4k1/4, from where, using the \ell 2 - \ell 2Cauchy--Schwarz inequality, we can obtain the following: | \scrN (g, h, h)| (3.12) \leqslant 4\Biggl( \sum n n3| \^gn| 2\Biggr) 1/2\left( \sum n\bigm| \bigm| \bigm| \bigm| \bigm| \sum k (n - k)1/4\bigm| \bigm| \bigm| \^ hn - k\bigm| \bigm| \bigm| k1/4\bigm| \bigm| \bigm| \^ hk\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 2\right) 1/2 . Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4953 Let us define the following distribution: H(x) = \scrF - 1\Bigl( \Bigl( \bigm| \bigm| \bigm| \^ hn\bigm| \bigm| \bigm| \Bigr) n\Bigr) (x) = 1 2\pi \sum n\in \BbbN \bigm| \bigm| \bigm| \^ hn\bigm| \bigm| \bigm| einx. Using the auxiliary distribution Hwe can rewrite (3.12) as | \scrN (g, h, h)| \leqslant 4\| g\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/4H\Lambda 1/4H\bigm\| \bigm\| \bigm\| L2.(3.13) Using the H\"older inequality and Sobolev embeddings we deduce that \bigm\| \bigm\| \bigm\| \Lambda 1/4H\Lambda 1/4H\bigm\| \bigm\| \bigm\| L2\leqslant \| H\| 2 \. W1 4,4\leqslant C\| H\| 2 H1/2, but \| H\| 2 H1/2=\sum n| n| \bigm| \bigm| \bigm| \^ Hn\bigm| \bigm| \bigm| 2=\sum n| n| \bigm| \bigm| \bigm| \^ hn\bigm| \bigm| \bigm| 2=\| h\| 2 H1/2, whence we deduce from (3.13) that | \scrN (g, h, h)| \leqslant C\| g\| H3/2\| h\| 2 H1/2. As a consequence, we find that (3.8) now reads 1 2 d dt\| f(t)\| 2 L2(\BbbT )+\bigm\| \bigm\| \bigm\| \Lambda 1/2f(t)\bigm\| \bigm\| \bigm\| 2 L2(\BbbT )\leq C\| f\| H3/2\| f\| 2 H1/2, (3.14) from where we find the desired inequality (3.7). We need now to find the appropriate bound for the higher order energy given by the H3/2norm. Remark 3.2. It is known that the L2energy is a Lyapunov functional for the one-phase Muskat problem (cf. [2, 5]), but for the truncated equation (1.3) such conservation is no longer valid. Let us in fact consider the analytic function \phi (x) = 1 \surd 2\pi \sum n\in \BbbZ e - | n| eixn; we use the formulation (3.11) in order to argue that \int \BbbT \phi \Bigl( (\Lambda \phi )2 - (\partial x\phi )2\Bigr) dx= 4 \sum (n,k)\in \BbbZ 2 0\leqslant k\leqslant n (n - k)k e - 2n = 4 \sum n\geqslant 0 e - 2n\biggl[ n2(n+ 1) 2 - n(n+ 1) (2n+ 1) 6\biggr] = 2 \sum n\geqslant 0 ne - 2n\biggl( n2 - 1 3\biggr) \in (0,\infty ), i.e., (1.3) does not conserve the L2energy even for analytic functions. Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4954 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA Step 1.2: Dyadic estimates. Let us apply the truncation operator \bigtriangleup qto (1.3). We then multiply the resulting equation for \bigtriangleup qfand integrate in space to obtain 1 2 d dt\| \bigtriangleup qf\| 2 L2+\bigm\| \bigm\| \bigm\| \bigtriangleup q\Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 L2=Aq - Bq (3.15) with Aq=\int \bigtriangleup q(f\Lambda f)\bigtriangleup q\Lambda fdx, Bq=\int \bigtriangleup q(f \partial xf)\bigtriangleup q\partial xfdx. Using now Bony decomposition (A.2), we can split the previous terms as follows: Aq=TA 1,q +TA 2,q +TA 3,q +RA q, Bq=TB 1,q +TB 2,q +TB 3,q +RB q, where TA 1,q =\int Sq - 1f(\bigtriangleup q\Lambda f)2dx, TB 1,q =\int Sq - 1f(\bigtriangleup q\partial xf)2dx, TA 2,q =\sum | q - q\prime | \leqslant 4\int [\bigtriangleup q, Sq\prime - 1f]\bigtriangleup q\prime \Lambda f\bigtriangleup q\Lambda fdx, TB 2,q =\sum | q - q\prime | \leqslant 4\int [\bigtriangleup q, Sq\prime - 1f]\bigtriangleup q\prime \partial xf\bigtriangleup q\partial xfdx, TA 3,q =\sum | q - q\prime | \leqslant 4\int (Sq\prime - 1 - Sq)f\bigtriangleup q\bigtriangleup q\prime \Lambda f\bigtriangleup q\Lambda fdx, TB 3,q =\sum | q - q\prime | \leqslant 4\int (Sq\prime - 1 - Sq)f\bigtriangleup q\bigtriangleup q\prime \partial xf\bigtriangleup q\partial xfdx, RA q=\sum q\prime >q - 4\int \bigtriangleup q(\bigtriangleup q\prime - 1f Sq\prime +2\Lambda f)\bigtriangleup q\Lambda fdx, RB q=\sum q\prime >q - 4\int \bigtriangleup q(\bigtriangleup q\prime - 1f Sq\prime +2\partial xf)\bigtriangleup q\partial xfdx. We have to estimate these eight terms. The terms TA 1and TB 1are the more singular. We will make use of the commutation properties (3.10) to estimate these terms. First of all we remark that TA 1,q - TB 1,q =\scrN (Sq - 1f, \bigtriangleup qf, \bigtriangleup qf), where the trilinear operator \scrN is defined in (3.9). We can hence use inequality (3.10) and (A.3) in order to deduce \bigm| \bigm| TA 1,q - TB 1,q\bigm| \bigm| \leqslant C\| f\| H3/2\bigm\| \bigm\| \bigm\| \bigtriangleup q\Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 L2 \leqslant Cbq2 - 2qs \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs (3.16) for any s\in \BbbR . Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4955 Next we handle the remainder terms RA q, RB q. Indeed, we have that RA q=\sum q\prime >q - 4\int \bigtriangleup q(\bigtriangleup q\prime - 1f Sq\prime +2\Lambda f)\bigtriangleup q\Lambda fdx \leqslant \sum q\prime >q - 4\| \bigtriangleup q\prime - 1f\| L 2p p - 2\| Sq\prime +2\Lambda f\| Lp\| \bigtriangleup q\Lambda f\| L2. (3.17) Since H1 2(\BbbT )\lhook \rightarrow Lp(\BbbT ) for any p\in [2,\infty ) we can say that \| Sq\prime +2\Lambda f\| Lp\lesssim \surd p\| f\| H3/2.(3.18) Moreover due to the localization in the Fourier space and the fact that q\prime > q - 4 we deduce \| \bigtriangleup q\prime - 1f\| L 2p p - 2\| \bigtriangleup q\Lambda f\| L2\lesssim \| \bigtriangleup q\prime - 1f\| L 2p p - 22q/2\bigm\| \bigm\| \bigm\| \bigtriangleup q\Lambda 1/2f\bigm\| \bigm\| \bigm\| L2 \lesssim 2q\prime /2\| \bigtriangleup q\prime - 1f\| L 2p p - 2\bigm\| \bigm\| \bigm\| \bigtriangleup q\Lambda 1/2f\bigm\| \bigm\| \bigm\| L2 \lesssim \bigm\| \bigm\| \bigm\| \bigtriangleup q\prime - 1\Lambda 1/2f\bigm\| \bigm\| \bigm\| L 2p p - 2\bigm\| \bigm\| \bigm\| \bigtriangleup q\Lambda 1/2f\bigm\| \bigm\| \bigm\| L2. We use (A.3), (3.18), and a summation in q\prime in order to transform (3.17) into RA q\lesssim bq2 - 2qs\surd p\| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| Hs\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| \. Ws, 2p p - 2.(3.19) Using the Sobolev inequality (1.4), we can also obtain the following estimate for the term RA q: RA q=\sum q\prime >q - 4\| Sq\prime +2\Lambda f\| L\infty \bigm\| \bigm\| \bigm\| \bigtriangleup q\prime - 1\Lambda 1/2f\bigm\| \bigm\| \bigm\| L2\bigm\| \bigm\| \bigm\| \bigtriangleup q\Lambda 1/2f\bigm\| \bigm\| \bigm\| L2 \lesssim bq2 - 2qs 1 \delta \| f\| H3 2+\delta \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs (3.20) for any 0 < \delta \ll 1. A very similar procedure allows us to deduce the following bound: RB q\lesssim bq2 - 2qs\surd p\| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| Hs\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| \. Ws, 2p p - 2,(3.21) RB q\lesssim bq2 - 2qs 1 \delta \| f\| H3 2+\delta \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs.(3.22) We turn now our attention to the element TA 2. Using Lemma A.1 we obtain \bigm| \bigm| TA 2,q\bigm| \bigm| \leq \sum | q - q\prime | \leqslant 4\| [\bigtriangleup q, Sq\prime - 1f]\bigtriangleup q\prime \Lambda f\| L2\| \bigtriangleup q\Lambda f\| L2 \leq \sum | q - q\prime | \leqslant 4 2 - q\| \bigtriangleup q\Lambda f\| L2\| \Lambda f\| Lp\| \bigtriangleup q\prime \Lambda f\| L 2p p - 2 \lesssim \sum | q - q\prime | \leqslant 4\bigm\| \bigm\| \bigm\| \bigtriangleup q\Lambda 1/2f\bigm\| \bigm\| \bigm\| L2\| \Lambda f\| Lp\bigm\| \bigm\| \bigm\| \bigtriangleup q\prime \Lambda 1/2f\bigm\| \bigm\| \bigm\| L 2p p - 2 \lesssim bq2 - 2qs\surd p\| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| Hs\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| \. Ws, 2p p - 2. (3.23) As a consequence, we can conclude the following bound: Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4956 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA \bigm| \bigm| TA 2,q\bigm| \bigm| \lesssim bq2 - 2qs 1 \delta \| f\| H3 2+\delta \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs.(3.24) Similarly, we can deduce the bounds \bigm| \bigm| TB 2,q\bigm| \bigm| \lesssim bq2 - 2qs\surd p\| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| Hs\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| \. Ws, 2p p - 2,(3.25) \bigm| \bigm| TB 2,q\bigm| \bigm| \lesssim bq2 - 2qs 1 \delta \| f\| H3 2+\delta \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs.(3.26) The estimates \bigm| \bigm| TA 3,q\bigm| \bigm| +\bigm| \bigm| TB 3,q\bigm| \bigm| \lesssim bq2 - 2qs \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs (3.27) can be easily obtained since the terms composing the elements TA 3,q, T B 3,q are all localized in dyadic annuli, and hence these terms are more regular than the ones estimated above. We note that 1 2 d dt\| \bigtriangleup qf\| 2 L2+\bigm\| \bigm\| \bigm\| \bigtriangleup q\Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 L2\leqslant Cbq2 - 2qs\Biggl[ \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs +\surd p\| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| Hs\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| \. Ws, 2p p - 2\Biggr] , where we have used (3.16), (3.19), (3.21), (3.23), (3.25), and (3.27). Now, multiplying the above equation for 22qs and summing in q\in \BbbZ , we obtain (3.28) 1 2 d dt\| f\| 2 Hs+\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs\leqslant C\Biggl[ \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs +\surd p\| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| Hs\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| \. Ws, 2p p - 2\Biggr] . Using (3.16), (3.20), (3.22), (3.24), (3.26), and (3.27) instead we deduce 1 2 d dt\| f\| 2 Hs+\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs\leqslant C\biggl[ \| f\| H3/2+1 \delta \| f\| H3 2+\delta \biggr] \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 Hs.(3.29) Step 1.3: \bfitH \bfthree /\bftwo estimates. We can now simply consider the estimate (3.28) in which we set s= 3/2 and p=p\varepsilon =1 \varepsilon in order to deduce 1 2 d dt\| f\| 2 H3/2+\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3/2\leqslant C\Biggl( \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3/2 +1 \surd \varepsilon \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| \. W3 2,2p p - 2\Biggr) . Next we remark that H3 2+\varepsilon \lhook \rightarrow \. W3 2,2p p - 2, whence Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4957 (3.30) 1 2 d dt\| f\| 2 H3/2+\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3/2\leqslant C\Biggl( \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3/2 +1 \surd \varepsilon \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| H3 2+\varepsilon \Biggr) . Step 1.4: \bfitH \bfthree \bftwo +\bfitvarepsilon estimates. We invoke the estimate (3.29) with s=3 2+\varepsilon and 0< \delta \ll \varepsilon : 1 2 d dt\| f\| 2 H3 2+\varepsilon +\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon \leqslant C\biggl( \| f\| H3/2+1 \delta \| f\| H3 2+\delta \biggr) \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon . Using interpolation between Sobolev spaces we deduce \| f\| H3 2+\delta \leqslant \| f\| 1 - \delta \varepsilon H3/2\| f\| \delta \varepsilon H3 2+\varepsilon , from where we obtain that 1 2 d dt\| f\| 2 H3 2+\varepsilon +\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon \leqslant C\biggl( \| f\| H3/2+1 \delta \| f\| 1 - \delta \varepsilon H3/2\| f\| \delta \varepsilon H3 2+\varepsilon \biggr) \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon . (3.31) Step 1.5: Closing the estimates. In this section we conclude the high order energy estimates. We sum up (3.30) and (3.31) to find that 1 2 d dt\Bigl( \| f\| 2 H3/2+\| f\| 2 H3 2+\varepsilon \Bigr) +\biggl( \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3/2+\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon \biggr) (3.32) \leqslant C\biggl( 1 + 1 \surd \varepsilon \biggr) \| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3/2 +C\biggl( \biggl( 1 + 1 \surd \varepsilon \biggr) \| f\| H3/2+1 \delta \| f\| 1 - \delta \varepsilon H3/2\| f\| \delta \varepsilon H3 2+\varepsilon \biggr) \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon . We define \beta =\beta (f0) = \| f0\| H3 2+\varepsilon \| f0\| H3 2 >1. We want now to absorb the contribution of 1 \delta \| f\| 1 - \delta \varepsilon H3/2\| f\| \delta \varepsilon H3 2+\varepsilon \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon appearing in the right-hand side of (3.32) in the parabolic term in the left-hand side of the same estimate. In order to do so we must split the proof according to the explicit value of \beta . Case 1: \bfitbeta \leqslant 2. In this setting we fix \delta =\varepsilon /2; hence \beta \delta /\varepsilon /\delta \leqslant 23/2/\varepsilon . Thus 1 \delta \| f\| 1 - \delta \varepsilon H3/2\| f\| \delta \varepsilon H3 2+\varepsilon =1 \delta \| f\| H3 2\beta \delta /\varepsilon \leqslant 23/2 \varepsilon \| f\| H3 2.(3.33) So, if the initial data is small enough in H3/2, then (3.33) is as small as needed in order to absorb the desired term. Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4958 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA Case 2: \bfitbeta > 2. In this setting we can define \delta =\delta (f0) = \varepsilon log\beta 2< \varepsilon . The value of \delta has been defined so that \beta \delta /\varepsilon = 2. We compute 1 \delta \| f0\| 1 - \delta \varepsilon H3/2\| f0\| \delta \varepsilon H3 2+\varepsilon =1 \delta \| f0\| H3 2\beta \delta /\varepsilon =2 \varepsilon log\beta 2\| f0\| H3 2. We want to make sure that 2C \varepsilon log\beta 2\| f0\| H3 2<1/2. Thus, simplifying the previous expression, we obtain that our goal is to prove that the following inequality holds: \| f0\| H3 2log \Biggl( \| f0\| H3 2+\varepsilon \| f0\| H3 2\Biggr) =\| f0\| H3 2\Bigl( log \| f0\| H3 2+\varepsilon - log \| f0\| H3 2\Bigr) <\varepsilon log 2 4C. We will now use the algebraic inequality - xlog x\leq \surd x, x \in [0,1] to obtain \| f0\| H3 2log \Biggl( \| f0\| H3 2+\varepsilon \| f0\| H3 2\Biggr) =\| f0\| H3 2\Bigl( log \| f0\| H3 2+\varepsilon - log \| f0\| H3 2\Bigr) \leq \| f0\| 1/2 H3 2\Bigl( \| f0\| 1/2 H3 2+\varepsilon log \| f0\| H3 2+\varepsilon + 1\Bigr) . It is now clear that, under the assumptions of the statement, we have that \| f0\| 1/2 H3 2\Bigl( \| f0\| 1/2 H3 2+\varepsilon log \| f0\| H3 2+\varepsilon + 1\Bigr) <\varepsilon log 2 4C. (3.34) In other words, we can now collect the estimates (3.33) and (3.34) to obtain that if the initial data satisfies the condition \| f0\| H3 2<min \left\{ 1,\varepsilon 2log22 16C2\Bigl( \| f0\| H3 2+\varepsilon log \| f0\| H3 2+\varepsilon + 1\Bigr) 2\right\} , then 1 \delta \| f\| 1 - \delta \varepsilon H3/2\| f\| \delta \varepsilon H3 2+\varepsilon \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon <1 2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon . Thus, due to the previous energy estimates, we can absorb the contribution from the nonlinear terms and conclude that \| f(t)\| 2 H3/2+\| f(t)\| 2 H3 2+\varepsilon +\int t 0\biggl( \bigm\| \bigm\| \bigm\| \Lambda 1/2f(s)\bigm\| \bigm\| \bigm\| 2 H3/2+\bigm\| \bigm\| \bigm\| \Lambda 1/2f(s)\bigm\| \bigm\| \bigm\| 2 H3 2+\varepsilon \biggr) ds(3.35) \leqslant \| f0\| 2 H3/2+\| f0\| 2 H3 2+\varepsilon . The above equation implies, hence, that smooth solutions stemming from small H3/2 initial data are nonlinearly stable in H3 2\cap H3 2+\varepsilon . Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4959 Step 1.6: Estimates for the time derivative. The purpose of this section is to find estimates for the time derivative of the solution. In particular, we want to obtain the following: \int t 0\| \partial tf(s)\| 2 H1ds(3.36) \leqslant C\Biggl\{ \Bigl( \| f0\| 2 H3/2+\| f0\| 2 H3 2+\varepsilon \Bigr) \biggl[ 1 + \Bigl( \| f0\| 2 H3/2+\| f0\| 2 H3 2+\varepsilon \Bigr) \biggl( 1 + 1 \surd \varepsilon \biggr) \biggr] +\| f0\| 3 H3/2+\| f0\| 3 H3 2+\varepsilon \Biggr\} . We multiply (1.3) by \partial t\Lambda 2fand integrate in \BbbT \times (0, T) to obtain \| \partial t\Lambda f\| 2 L2 TL2+\Bigl( \| f(t)\| 2 H3 2 - \| f0\| 2 H3 2\Bigr) =I1+I2,(3.37) where I1=\int T 0\int \BbbT \Lambda (f\Lambda f)\partial t\Lambda 2fdxds and I2=\int T 0\int \BbbT \partial x(f\partial xf)\partial t\Lambda 2fdxds. Using the identity \Lambda 2= - \partial 2 xand integrating by parts we can deduce the following equalities: I1=\int T 0\int \BbbT \partial xf\Lambda f \partial t\Lambda \partial xfdxds+\int T 0\int \BbbT f\Lambda \partial xf \partial t\Lambda \partial xfdxds =I1,1+I1,2, I2= - \int T 0\int \BbbT (\partial xf)2\partial t\partial 2 xfdxds - \int T 0\int \BbbT f\partial 2 xf\partial t\partial 2 xfdxds =I2,1+I2,2. We hence obtain that I1,2+I2,2=1 2\int T 0\int \BbbT f\partial t\Bigl[ (\Lambda \partial xf)2 - \bigl( \partial 2 xf\bigr) 2\Bigr] dxds =1 2\scrN (f, \partial xf, \partial xf)\bigm| \bigm| \bigm| \bigm| T 0 - 1 2\int T 0\scrN (\partial tf, \partial xf, \partial xf) ds =J1+J2. The term J1can be estimated using (3.10), and we deduce J1\lesssim \| f(T)\| 3 H3/2+\| f0\| 3 H3/2.(3.38) The term J2can be estimated in a similar way. In particular, we obtain for any \epsilon > 0 Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4960 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA J2\lesssim \int T 0\| \partial tf\| H1\bigm\| \bigm\| \bigm\| \Lambda 3/2f\bigm\| \bigm\| \bigm\| 2 L4ds \lesssim \int T 0\| \partial tf\| H1\| f\| H3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| H3/2ds \leqslant \epsilon \| \partial tf\| 2 L2 TH1+C \epsilon \| f\| 2 L\infty TH3/2\bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 L2 TH3/2. (3.39) We have to estimate the term I1,1+I2,1. We find that I1,1=\int T 0\int \BbbT \partial xf\Lambda f \partial t\Lambda \partial xfdxds \sim \int T 0\int \BbbT \Lambda 2f\Lambda f \partial t\Lambda fdxds \lesssim \bigm\| \bigm\| \Lambda 2f\bigm\| \bigm\| L2 TL 2p\varepsilon p\varepsilon - 2\| \Lambda f\| L\infty TLp\varepsilon \| \partial t\Lambda f\| L2 TL2 \lesssim 1 \surd \varepsilon \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| L2 TH3 2+\varepsilon \| f\| L\infty TH3/2\| \partial tf\| L2 TH1 \leqslant \eta \| \partial tf\| 2 L2 TH1+C \eta \surd \varepsilon \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 L2 TH3 2+\varepsilon \| f\| 2 L\infty TH3/2, (3.40) where \eta > 0 is arbitrary. The term I2,1is completely analogous to I1,1and can be handled in a similar way, and we obtain the estimate I2,1\leqslant \eta \| \partial tf\| 2 L2 TH1+C \eta \surd \varepsilon \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 L2 TH3 2+\varepsilon \| f\| 2 L\infty TH3/2. Using the estimates (3.38), (3.39), (3.40) in (3.37) we obtain that (1 - 2\eta )\| \partial tf\| 2 L2 TH1\leqslant C\Biggl[ \| f0\| 2 H3 2 - \| f(T)\| 2 H3 2+\| f(T)\| 3 H3/2+\| f0\| 3 H3/2 +1 \eta \biggl( \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 L2 TH3/2+1 \surd \varepsilon \bigm\| \bigm\| \bigm\| \Lambda 1/2f\bigm\| \bigm\| \bigm\| 2 L2 TH3 2+\varepsilon \biggr) \| f\| 2 L\infty TH3/2\Biggr] . We fix \eta = 1/4 and use (3.35) in order to deduce the bound \| \partial tf\| 2 L2 TH1 2 \leqslant C\Biggl\{ \Bigl( \| f0\| 2 H3/2+\| f0\| 2 H3 2+\varepsilon \Bigr) \biggl[ 1 + \Bigl( \| f0\| 2 H3/2+\| f0\| 2 H3 2+\varepsilon \Bigr) \biggl( 1 + 1 \surd \varepsilon \biggr) \biggr] +\| f0\| 3 H3/2+\| f0\| 3 H3 2+\varepsilon \Biggr\} . Step 2: Approximated solutions and passing to the limit. The a priori estimates provided before the proof of Theorem 2.4 become a standard approximation argument which we outline here. Let us define the truncation operator \scrJ nu(x) = 1 \surd 2\pi \sum | k| \leqslant n \^ukei k\cdot x=\scrF - 1\bigl( \bigl( 1\{ | k| \leqslant n\} (k) \^uk\bigr) k\bigr) .(3.41) Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. ASYMPTOTIC MODEL FOR FREE BOUNDARY DARCY FLOW 4961 We apply this truncation operator to the initial data f0. Then, this truncated initial data is analytic, and, invoking Theorem 2.1, we obtain the existence of smooth (in time and space) local solution approximate solutions. We can use the previous estimates (3.35) to deduce that the local approximate solutions are actually globally defined as H3/2+\varepsilon functions. The sequence of approximate solutions (fn)nis bounded uniformly in nand Tin the spaces L\infty \Bigl( [0, T) ; H3 2\cap H3 2+\varepsilon \Bigr) \cap L2\bigl( [0, T) ; H2\cap H2+\varepsilon \bigr) , so there exists a weak limit f\in L\infty \Bigl( [0, T) ; H3 2+\varepsilon \Bigr) \cap L2\bigl( 0, T;H2+\varepsilon \bigr) . Furthermore, estimate (3.36) ensures that the sequence of approximate solutions (fn)n is uniformly bounded in H1\bigl( [0, T) ; H1\bigr) , from where, using interpolation, we deduce that (fn)nis uniformly bounded in H1 - \vargamma \Bigl( [0, T) ; H1+\vargamma (\varepsilon +1)\Bigr) , \vargamma \in [0,1] . Setting \vargamma \in \biggl[ 1 2 (\varepsilon + 1) ,1 2\biggr) ,(3.42) we deduce that (fn)nis uniformly bounded in C0,1 2 - \vargamma \Bigl( [0, T) ; H1+\vargamma (\varepsilon +1)\Bigr) . We obtain that (fn)nis equicontinuous-in-time. We can hence apply Ascoli--Arzel\'a theorem in order to deduce that there exists an f\in C0,1 2 - \vargamma \bigl( [0, T) ; H1+\vargamma (\varepsilon +1)\bigr) such that, taking a subsequence if necessary, fn\rightarrow fin C0,1 2 - \vargamma \bigl( [0, T) ; H1+\vargamma (\varepsilon +1)\bigr) as long as \vargamma satisfies (3.42). It is a classical argument to prove that fsolves (1.3) and that it is unique. 4. Gravity-capillarity driven system (1.1). 4.1. Proof of Theorem 2.5. Step 1: A priori estimates. We only prove the a priori estimates, since the approximation procedure is standard and it can be done using a Galerkin scheme. We have that d dt\| f(t)\| \. A1+\nu \| f(t)\| A4+\| f(t)\| A2\leqslant \bigm\| \bigm\| \partial 2 x\bigl( J\scrH , fK(\nu \Lambda 3f+ \Lambda f)\bigr) \bigm\| \bigm\| A0. As before, \bigm\| \bigm\| \partial 2 x(J\scrH , fK\Lambda f)\bigm\| \bigm\| A0\leqslant 2\| f\| A1\| f\| A2. In Fourier variables, we have that \widehat \partial 2 x(J\scrH , fK\Lambda 3f) = \widehat \partial x\Lambda (f\Lambda 3f) - \partial 2 x(f\partial 3 xf) =\^ f(k - m)\^ f(m)p(k, m) Downloaded 10/15/20 to 193.144.183.85. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 4968 RAFAEL GRANERO-BELINCH\' ON AND STEFANO SCROBOGNA In the paper [12] J.-Y. Chemin and N. Lerner introduced the following decomposition which will be very useful in our context: \bigtriangleup q(uv) = Sq - 1u\bigtriangleup qv(A.2) +\sum | q - q\prime | \leqslant 4\{ [\bigtriangleup q, Sq\prime - 1u]\bigtriangleup q\prime v+ (Sq\prime - 1u - Squ)\bigtriangleup q\bigtriangleup q\prime v\} +\sum q\prime >q - 4\bigtriangleup q(Sq\prime +2v\bigtriangleup q\prime u), where the commutator [\bigtriangleup q, a]bis defined as [\bigtriangleup q, a]b=\bigtriangleup q(ab) - a\bigtriangleup qb. There is an interesting relation of regularity between dyadic blocks and full function in the Sobolev spaces, i.e., \| \bigtriangleup qf\| Lp(\BbbT )\leqslant Cc(p) q2 - qs \| f\| Ws,p(\BbbT ) (A.3) with \| \{ c(p) q\} q\in \BbbZ \| \ell p(\BbbZ )\equiv 1; if p= 2 we denote \{ c(2) q\} q=\{ cq\} qfor simplicity. In the same way we denote as bqa sequence in \ell 1(\BbbZ ) such that \sum q| bq| \leqslant 1. Finally we state a lemma that shows that the commutator with the dyadic block in the vertical frequencies is a regularizing operator. The proof of this lemma can be found in [6, Lemma 2.97, page 110] for the whole space \BbbR d; the proof for the periodic space \BbbT dis analogous. Lemma A.1. Let \BbbT dbe a d-dimensional torus and p, r, s real positive numbers such that p, r, s \in [1,\infty ]and 1 p=1 r+1 s. There exists a constant Csuch that for all vector fields uand von \BbbT dwe have the inequality \| [\bigtriangleup q, u]v\| Lp\leqslant C2 - q\| \nabla u\| Lr\| v\| Ls. 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