Global regularity for 2D Muskat equations with finite slope

Transcription

1 Global regularity for 2D Muskat equations with finite slope Peter Constantin, Francisco Ganceo, oman Shvykoy, an Vla Vicol ABSTACT. We consier the 2D Muskat equation for the interface between two constant ensity fluis in an incompressible porous meium, with velocity given by Darcy s law. We establish that as long as the slope of the interface between the two fluis remains boune an uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bouns for nonlocal operators. These are use to euce that as long as the slope of the interface remains uniformly boune, the curvature remains boune. The nonlinear bouns then allow us to obtain local existence for arbitrarily large initial ata in the class W 2,p, < p. We provie furthermore a global regularity result for small initial ata: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time. July 6, 205 CONTENTS. Introuction 2. Preliminaries 5 3. Nonlinear lower bouns 9 4. Bouns for the nonlinear terms 2 5. Proof of Theorem.2, part (i, Blowup Criterium for the Curvature 9 6. Proof of Theorem., Local Existence Proof of Theorem.3, Global Existence for Small Datum Proof of Theorem.2, part (ii, Blowup Criterium for Smooth Solutions Proof of Theorem.4, Uniqueness 29 Appenix A. aemacher Theorem 30 eferences 3. Introuction In this paper we stuy solutions of the one imensional nonlinear nonlocal equation δ α f (x, tα t f(x, t = (δ α f(x, t 2 α, (. + α2 where we have enote by δ α f(x, t = f(x, t f(x α, t the α-finite ifference of f at fixe time t 0, an by f (x, t = x f(x, t the partial erivative of f with respect to the position x. 200 Mathematics Subject Classification. 76S05,35Q35. Key wors an phrases. porous meium, Darcy s law, Muskat problem, maximum principle.

2 2 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL.. Physical origins of the problem. Equation (. is erive from a two imensional problem, where y = f(x, t escribes a graph (in coorinates whereby gravity acts in the y irection which represents the interface separating two immiscible fluis of ifferent constant ensities. The flow is incompressible, an the fluis permeate a porous meium. The 2D Darcy s law [Dar56] µ u(x, y, t = p(x, y, t (0, gρ(x, y, t k is use to moel the balance of gravity, internal pressure forces an velocity (replacing acceleration. Here u an p are the velocity an pressure of the flui, which are functions of time t 0 an position (x, y 2. The physical constants of the moel are viscosity µ, permeability k, an gravity g. For simplicity we fix these physical parameters to equal. We pay special attention to the ensity function ρ, which is given by a step function representing the jump in the fluis ensities. Taking the size of this jump equal to 2π (enser flui below, we arrive at (. (see [CG07] for a etaile erivation. This is the classical Muskat problem [Mus34], which has been broaly stuie [Bea72] ue to its physical relevance, wie applicability, an mathematical analogy to a completely ifferent phenomenon: the interface ynamics of fluis insie Hele-Shaw cells [ST58]. The Muskat problem has been wiely stuie, its multiple features being taken uner consieration. It can incorporate capillary forces that eal with surface tension effects [CP93, C93, KM5], ensity jumps [CG07], viscosity [SCH04], an permeability iscontinuities [BCGB4]. The ynamics is ifferent for the one flui case (with a ry region [CCFG3a], two fluis [SCH04], an multi-phase flows [CG0]. Moreover, one may consier bounary effects [GB4] an stuy the interface in the three imensional case [A07, CCG3b]..2. Summary of known results. The basic mathematical questions regaring (. are the existence an regularity of solutions. The Muskat problem can be ill-pose, ue to existence of unstable configurations [Ott99, Szé2]. This feature can be unerstoo in the contour ynamics escription via the ayleigh- Taylor conition, which involves the geometry of the ensity jump or the ynamics of fluis with ifferent viscosities. Essentially, the unstable situation occurs when a enser flui is above a less ense flui, or when a less viscous flui pulls a more viscous one [A07]. Surface tension effects regularizes the equation [ES97, A4], so that the system is well-pose without satisfying the ayleigh-taylor conition, but there are still instabilities in this case [GHS07, EM]. In the stable cases it is possible to fin global-in-time solutions for small initial ata [SCH04, CG07, CGBS4] ue to the parabolicity of the equation. This smallness can be measure in ifferent regularity classes an sizes [CCGS3, CCG + 3a, GB4, BSW4, CGBS4]. Low regular global existence results are obtaine for Lipschitz weak solutions in the case of small slopes in [CCGS3, CCG + 3a]. A very interesting phenomenon for the Muskat problem is the evelopment of finite time singularities starting from stable initial ata. One possible singularity formation happens by the breakown of the ayleigh-taylor conition in finite time [CCF + 2]. In this case the interface cannot be parameterize by a graph of a function any longer, i.e. there exists a time t > 0 when solutions of equation (. satisfy lim f (t L =. t t After that time the free bounary evolves in a way that prouces a region with enser flui above less ense flui. For short time the interface is still regular (although it is not a graph ue to the fact that the parabolicity of the system gives instant analyticity. This sequence of events is terme wave-turning because it is a blow up of the graph-parameterization. Wave-turning has been proven to arise in more general cases with ifferent geometries [BCGB4, GSGB4]. The regularity of the interface is lost at some time t 2 > t after wave-turning, an the interface ceases to belong to C 4. Thus, starting from a stable regime, the Muskat solution enters an unstable situation, an then the regularity breakown occurs [CCFG3a]. This phenomenon is far from trivial, as some unstable solutions can become stable, an again reach unstable regime [CGSZ].

3 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 3 A ifferent type of singularity which may evelop in solutions of the Muskat problem is given by the occurrence of a finite time self-intersection of the free bounary. If the collapse is at a point, while the free bounary remains regular, then the singularity is terme a splash. If the collapse is along a curve, while the free bounary remains regular, then it is terme a splat (or squirt singularity. Both kins of collisions have been rule out in the case of a stable ensity jump in [CG0, GS3]. In the one flui case, there exists a finite time splash blow up [CCFG3b] in the stable regime, but the splat singularity is not possible [CPC4]..3. Main results. The main purpose of this work is to evelop conitional regularity results (blow up criteria for the Muskat problem involving the bouneness of the first erivative of the interface. Our motivation is both to complement an to aie the theory of singularities for the Muskat equation by obtaining sharp regularity results in terms of purely geometric quantities such as the slope. The bouns of nonlinear terms use to obtain the conitional regularity results also allow us to obtain several existence results in low regularity regimes. Specifically, we prove local-in-time well-poseness an global well-poseness for Muskat solutions in classes of functions of boune slope an L p (with p (, ] integrability of curvature. ecently, results involving solely the secon erivative of f appeare in the work [CGBS4], where the authors explore the PDE structure of the original Darcy s law in the bulk rather than the corresponing integral equation for contour ynamics (.. They evelop an H 2 well-poseness theory for the equation in the case of ensity an viscosity jumps. Both the local an global existence results are obtaine uner smallness assumptions for the interface: H 3/2+ɛ for the local results an H 2 for the global results. These smallness conitions imply f C β for some β > 0, requiring thus smallness of the Höler continuity of the slope. The optimal well-poseness theory (local, conitional, or global for the Muskat equation shoul (conjecturally involve only assumptions of uniform bouneness of the slope f. To the best of our knowlege, this remains an outstaning open problem. The ifficulty in reaching a W, regularity theory for f may be seen in at least two ways: When consiering the evolution of f, linearize aroun the steay flat solution, the resulting equation has L as a scaling invariant norm. Therefore, for large slopes, obtaining higher regularity of the solution (require in orer to obtain uniqueness involves solving a large ata critical nonlinear nonlocal problem. Of course, the aitional complication is that, ue to the funamentally nonlinear nature of the equation, in the large slope regime the linearization aroun the flat solution rapily ceases to be useful, an new severe ifficulties arise. The velocity v transporting f, f, an f (cf. (2.2, (2.5, an (2.6 below is obtaine by computing a Calerón commutator applie to the ientity (cf. (2.3 below. However, when f is merely boune, while the Calerón commutator maps L p L p for p (,, the enpoint L inequality fails [MS3]. Moreover, since we are in D the x-erivative of v enters the estimates (we cannot use incompressibility, an this term (cf. (2. yiels a Calerón commutator applie to δ α f (x, t/α. Thus, to boun this requires estimates on the maximal curvature, not slope. This is the reason why up to ate the existing continuation criteria [CG07, CGBOI4] require that the solution lies in C 2,ɛ for some ɛ > 0. In this paper we move closer to the conjecture critical f W, well-poseness framework. The main new iea is that the maximal curvature can be controlle by either assuming uniform continuity of the slope f (cf. part (i of Theorem.2, or by assuming that the initial slope f 0 is sufficiently small (cf. part (ii of Theorem.3. We moreover show that uner either of these conitions one can also control f L p for all p (,. In the regime p (, 2 not even the local existence of solutions was previously known. The important novelty of our large ata result is that the maximal slope (an the maximal initial curvature are allowe to be arbitrarily large, as long as the slope obeys any uniform moulus of continuity. Our uniform C 0 assumption on f is a local smallness of variation assumption instea of a global smallness assumption of f in L. The moulus of continuity of f is not require to vanish at the origin at any

4 4 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL particular rate, an it can be even weaker than a Dini moulus (an thus the Calerón commutator in v an the nonlinear terms woul not necessarily yiel boune functions. egaring our small ata result, we raw the attention to the fact that the smallness is assume only on the initial slope, not on the W 2,p norm. The main tools use in our analysis are various nonlinear maximum principles for the evolution of f, in the spirit of the ones previously evelope in [CV2, CTV5] for the critical SQG equation. The robustness of the pointwise an integral lower bouns available for the nonlocal operator L f efine in (2.8 below allows us to treat all values of p (, ] an enables us to analyze the long time behavior of the curvature, cf. (.5. The enpoint case W 2,, which scales as W,, remains however open. Next we state our results more precisely. First we establish a low regularity local-existence result for (. with initial atum that has integrable, respectively boune, secon erivatives. THEOEM. (Local-existence in W 2,p. Assume the initial atum has finite energy an finite slope, that is f 0 L 2 W,. Let p (, ], an aitionally assume that f 0 W 2,p. Then there exists a time T = T ( f 0 W 2,p W, > 0 an a unique solution f L (0, T ; W 2,p C(0, T ; L 2 W, of the initial value problem (. with initial atum f(x, 0 = f 0 (x. The main result of this paper is: THEOEM.2 (egularity criterion in terms of continuity of slope. Consier f 0 H k for k 3 an assume that f is boune on [0, T ], i.e. that sup f (t L B < (.2 t [0,T ] for some B > 0. Furthermore, assume that f is uniformly continuous on [0, T ]. That is, assume there exits a continuous function ρ: [0, [0,, that is non-ecreasing, boune, with ρ(0 = 0, such that f obeys the moulus of continuity ρ, i.e. that δ α f (x, t ρ( α (.3 for any x, α an t [0, T ]. Then the following conclusions hol: (i sup t [0,T ] f (t L < C( f 0 L, B, ρ. (ii The solution stays regular on [0, T ] an f C([0, T ]; H k L 2 (0, T ; H k+ 2. We note that in Theorem.2 we o not require the moulus of continuity ρ to vanish at 0 at any specific rate. For example, it can be guarantee by a conition f L (0, T ; C β for β > 0. The remarkable feature of conclusion (i, however, is that the control on f is furnishe by the initial ata only. If however the initial slope f is small enough, then we can show that the conition f L p can be propagate without any further assumptions on the continuity of the slope. An that way we can obtain a global existence in the corresponing class. THEOEM.3 (Small slope implies global existence. Consier initial atum f 0 L 2, with maximal slope that obeys f 0 L B C, (.4 for a sufficiently large universal constant C >. Let p (, ]. If we aitionally have f 0 Lp, then the local in time solution obtaine in Theorem. is in fact global, an we have f (t L p f 0 L p for all t > 0. Furthermore, we have that (i for p = the curvature asymptotically vanishes as f (t L + f 0 L f 0 L 00B t (.5

5 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 5 for all t 0, (ii for p [2, the solution obeys the L p energy inequality t f (t p L + p2 p + B 2 0 for all t 0, (iii for p (, 2 the solution obeys f (s p/2 2 Ḣ /2 s + f (t p L p + CB( + B 2 200B( + B 2 for all t > 0, for some sufficiently large constant C > 0 that may epen on p. t 0 t 0 f (s p+ L p+ s f (0 p L p (.6 f (s p+ L p+ s f (0 p L p (.7 We note that the assumption (.4 together with the maximum principle for f establishe in [CG09, Section 5] show that (.4 hols at all times t > 0. Now we see that bouns establishe in (i, (ii, (iii also prove maximum principles at the level of the curvature. We complement existence result with the following uniqueness statement which in part shows that solutions of Theorems.,.2,.3 are unique. THEOEM.4 (Uniqueness. Let f 0 L, an assume that f is a solution to (. on time interval [0, T ] with the following properties: sup t [0,T ] f (t L < ; f(x, t 0 as x, for all t T ; t f(x, t exists for all (x, t an t f L ( [0,T ] < ; f is uniformly continuous on [0, T ], i.e. (.3 hols for some moulus of continuity ρ. Then f is the unique solution with f 0 (x = f(x, 0 satisfying all the liste properties. We note that the only essential assumption in the list of Theorem.4 is the same uniform continuity from Theorem.2. The remaining assumptions are present to justify the application of the aemacher Theorem, see the Appenix, an we believe these aitional assumptions may be avoie. For instance, a sufficient conition which ensures that t f(x, t exists for all (x, t is that f L t Wx 2,p for some p >, or that the moulus ρ obeys the Dini conition. We also note that in the course of proving Theorems. an.3 we follow the stanar strategy: we obtain all necessary a priori estimates an construct solutions by passing to the limit in the regularize system as elaborate for instance in [CCG + 3a]. To avoi reunancy, the proofs in this paper only consist of these a priori estimates. 2. Preliminaries For the remainer of the manuscript we shall use the following notation for finite-ifference quotients: α f(x = δ αf(x α for x an α \ {0}. After integration by parts (. becomes t f(x, t = P.V. f (x, tα δ α f(x, t (δ α f(x, t 2 + α 2 α (2. or equivalently, δ α f(x, t t f(x, t + v(x, t x f(x, t + P.V. (δ α f(x, t 2 α = 0 (2.2 + α2 where we have efine the velocity fiel v(x, t = P.V. α (δ α f(x, t 2 α = P.V. + α2 α ( α f(x, t 2 + α. (2.3 In particular, from (2.2 it is immeiate that the maximum principle for the global boun on f hols (see [CG09]. Moreover it is known that the maximum principle also hols for the energy (see [CCG + 3a], i.e. f(t L p f 0 L p (2.4

6 6 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL hols for p = 2 an p =. 2.. Equation for the first erivative. The equation obeye by f (x, t is t (f + v(x, t x (f δ α f (x, t + P.V. (δ α f(x, t 2 + α 2 α = 2 (δ α f(x, t αf (x, t(δ α f(x, t(δ α f (x, t ((δ α f(x, t 2 + α 2 2 α. ( Equation for the secon erivative. The equation obeye by the secon erivative is t (f + v(x x (f δ α f (x +P.V. (δ α f(x 2 + α 2 α (δ α f (x αf (x δ α f(x, tδ α f (x, t = 4 P.V. ((δ α f(x 2 + α 2 2 α (δ α f(x αf (x(δ α f (x P.V. ((δ α f(x 2 + α 2 2 α (δ α f(x αf (x(δ α f(x(δ α f (x + 2 P.V. ((δ α f(x 2 + α 2 2 α (δ α f(x αf (x(δ α f(x 2 (δ α f (x 2 8 P.V. ((δ α f(x 2 + α 2 3 α We now pointwise multiply (2.6 by f (x, t an obtain =: T + T 2 + T 3 + T 4. (2.6 ( t + v x + L f f (x, t 2 + D f [f ](x, t = 2f (x, t (T + T 2 + T 3 + T 4 (2.7 where we have enote δ α g(x L f [g](x = P.V. (δ α f(x 2 α (2.8 + α2 an (δ α g(x 2 D f [g](x = P.V. (δ α f(x 2 α (2.9 + α2 for any smooth function g, an T,..., T 4 are as given by (2.6. It will be useful to also consier (2.7 where the transport term is written in ivergence form, namely, ( t + L f f 2 + x (v f 2 + D f [f ] = 2f (T + T 2 + T 3 + T 4 + f 2 T 5 (2.0 where we have enote T 5 (x, t = x v(x, t = 2P.V. δ α f (x, t α f(x, t α 2 ( + ( α f(x, t 2 α. (2. 2 Note that when f c, where c is a constant, the above operators become δ α g(x L c [g](x = P.V. α 2 α =: Λg(x (δ α g(x 2 D c [g](x = P.V. α 2 α =: D[g](x. (2.2 We further note that D[g](xx = g 2 Ḣ /2 (2.3

7 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 7 an for all x. D f [g](x + f 2 L D[g](x ( Evolution of the pth power of the secon erivative, for p 2. When p [2,, we consier the function ϕ p (r = r p/2, an we multiply (2.7 by ϕ p( f (x, t 2. The convexity of ϕ p (which hols if p 2 ensures that an we thus obtain ϕ p( f (x, t 2 L f [ f (x, t 2 ] L f [ϕ p ( f (x, t 2 ] ( t + v x + L f f (x, t p + p f (x, t p 2 D f [f ](x, t p f (x, t p T + T 2 + T 3 + T 4. (2.5 2 Moreover, we may again write this in ivergence form as ( t + L f f p + x (v f p + p f p 2 D f [f ] p f p T + T 2 + T 3 + T 4 + f p T 5 (2.6 2 where T 5 is as efine in ( Evolution of the pth power of the secon erivative, for p (, 2. When p (, 2, the above trick oes not work, since the function ϕ p is in this case concave. Instea, we consier the function which is convex, an obeys ψ p (r = r p ψ p (s ψ p (r + ψ p(r(r s = s p r p + p r p (r s p(p r { (r s 2, 2 r 2 p for all r, s. Thus, for p (, 2 we are lea to consier the nonlocal operators D p f [g](x = g(x g(x α p α 2 + (δ α f(x 2 α an D p [g](x = g(x g(x α p α. α 2 2 r s r s p 4, s 2 r. (2.7 Note that for p = 2, we have D 2 = D, an that for g L C the above integrals are well-efine (even without a principal value, since p >. Using (2.7 we may show that upon multiplying (2.6 by ψ p(f (x, t = p f (x, t p /f (x, t, that or in ivergence form, ( t + v(x, t x + L f f (x, t p + ( t + L f f p + x (v f p + in analogy with (2.5 an (2.6. p(p 4 { 2Df [f } ](x, t min f (x, t 2 p, Dp f [f ](x, t p f (x, t p T (x, t T 4 (x, t, (2.8 p(p 4 { 2Df [f } ] min f 2 p, Dp f [f ] p f p T T 4 + f p T 5, (2.9

8 8 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL 2.5. Bouns for Taylor expansions. The finite ifference quotient may be boune irectly as for any α, x. We note that we may expan The boun [f ](x, α = α f(x f (x = α α f(x B (2.20 x [f ](x, α 2B x α (f (z f (xz. is immeiate ue to f L B, but if aitionally f has a moulus of continuity ρ, then we have the improve boun α f(x f (x ρ( α. (2.2 Without loss of generality we assume ρ is not linear near the origin, since then the conclusion an the assumption of the theorem are ientical. It will be convenient to enote an [f ](x, α := α f (x f (x = α x x α 2 [f ](x, α := α f(x f (x + α 2 f (x = [f ](x, α + α 2 f (x = α x x α z x (f (z f (xz (2.22 (f (w f (xwz. (2.23 as the first orer an the secon orer expansions of f (x α aroun f (x. For these terms we have pointwise in x estimates an α in terms of the issipation present on the right sie of (2.7. First, we have that x [f ](x, α f (z f (x z x z α x α z x ( x /2 α (D[f ](x /2 z x 2 x x α Cα /2 (D[f ](x /2 (2.24 for any α > 0, for some universal constant C > 0. The boun for all α 0 trivially hols upon replacing α /2 with α /2. Similarly, it follows that 2 [f ](x, α α α x x α x x α z x ( z x C (D[f ](x /2 α f (w f (x w x wz w x f (w f (x 2 /2 ( z w x 2 w x x α z x 3/2 z x w x 2 w /2 z Cα 3/2 (D[f ](x /2 (2.25

9 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 9 for any α > 0, for some universal constant C > 0. The boun for all α 0 trivially hols upon replacing α 3/2 with α 3/2. Lastly, we note that for p (, 2 in a similar way to (2.24 an (2.25 we may boun with a constant C = C(p > 0. [f ](x, α Cα /p (D p [f ](x /p ( [f ](x, α Cα (p+/p (D p [f ](x /p ( Nonlinear lower bouns In this section we use (.2 in orer to obtain lower bouns for D f [f ](x at any x. These lower bouns follow in a similar way to the bouns previously establishe in [CV2, CTV5] for Λ. The main results of this section are. LEMMA 3.. Assume that f is Lipschitz continuous with Lipschitz boun B, i.e. that (.2 hols. Then we have that D f [f ](x 24B( + B 2 f (x 3. (3. hols pointwise for x. Moreover, for p (, 2, we have that D p f [f ](x 96B( + B 2 f (x p+ (3.2 hols for all x. The above lower boun however will only suffice to prove a small B result. For large values of B we must obtain a better lower boun. If we aitionally use a moulus of continuity obeye by f, we obtain: LEMMA 3.2. In aition to the assumption in Lemma 3., assume that f obeys a moulus of continuity ρ. Then there exists a continuous function L B : [0, [0, that implicitly also epens on ρ, such that hols pointwise for x, an we have that at a rate that epens on how fast lim r 0 + ρ(r = 0. D f [f ](x L B ( f (x (3.3 L B (y lim y y 3 = (3.4 Next we provie two lower bouns whose proofs follow by similar arguments to those in Lemma 3. (see also [CV2, CTV5]. LEMMA 3.3. Assume that f is Lipschitz continuous with Lipschitz boun B, an let p (,. Then the following lower boun D f [f ](x 8 p f p L ( + B 2 f (x 2+p (3.5 p hols for all x. Moreover, for p (, 2 we have that D p f [f ](x 28 f p L ( + B 2 f (x 2p (3.6 p hols. LEMMA 3.4. Assume that f is a regular function with Lipschitz boun B. Then the following lower boun hols for any x. D f [f ](x 24 f L ( + B 2 f (x 3. (3.7

10 0 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL POOF OF LEMMA 3.. For this purpose, let r > 0 to be chosen later, an let χ be an even cutoff function, with χ = on [,, χ = 0 on [0, /2], an χ = 2 on (/2,. We then have D f [f ](x + B 2 D[f f (x 2 2f (xf (x α ( α ](x + B 2 α 2 χ α r ( + B 2 f (x 2 α α 2 2 f α f (x α ( α (x α 2 χ α. (3.8 r α r Integration by parts yiels D f [f ( 2 f (x 2 ](x ( + B 2 4 f (x r 4 f (x r ( 2 f (x 2 ( + B 2 24B f (x r r 2. α r/2 r/2 α r f (x α α 3 α f (x α α 2 α Letting r = 24B f (x in the above estimate, we arrive at the boun (3.. Let p (, 2. In orer to prove (3.2 we appeal to the ( inequality r s p r p p s r which hols for any r, s an all p >. It follows that D p f [f ](x + B 2 Dp [f f (x f (x α p ( α ](x + B 2 α 2 χ r f (x p ( + B 2 α 2 α p f (x p ( + B 2 Upon choosing α r the proof of the Lemma is complete. α α f (x α α 2 χ ( α r α f (x p + B 2 24 B f (x p r + B 2 r 2. (3.9 r = 48B f (x POOF OF LEMMA 3.2. Similarly to the above proof, we boun D f from below as D f [f ( 2 f (x 2 ](x ( + B 2 2 f α (f (x α f (x ( α (x r α 2 χ α r ( 2 f (x 2 ( + B 2 4 f δ α f (x (x r α r/2 α 3 α 4 f (x δ α f (x r r/2 α r α 2 α ( 2 f (x 2 ( + B 2 8 f ρ(α (x r r/2 α 3 α 8 f (x r ρ(α r r/2 α 2 α ( 2 f (x 2 ( + B 2 2 f ρ(α (x r α 3 α. r/2

11 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE In this case we choose r = r( f (x as the smallest r > 0 which solves f (x 6 = r r/2 ρ(α α. (3.0 α3 The existence of such an r is guarantee by the intermeiate value theorem, an by computing the limits at r 0+ an r of the right sie of (3.0. Inee, these limits are since ρ is not Lipschitz, an Note that with this choice we have lim r r 0 r/2 lim r r r/2 ρ(α α = + α3 ρ(α α = 0. α3 With the choice (3.0, the lower boun obtaine on D f [f ] becomes where the function L B (y is efine on (0, implicitly by L B (y = lim r(y = 0. (3. y D f [f ](x L B ( f (x, (3.2 y 2 ( + B 2, with r(y r(y r(y/2 ρ(α α 3 α = y 6. (3.3 A short computation shows that since r(y is Lipschitz continuous on (0,, which implies that L B is also Lipschitz continuous on this interval. The important aspect to notice is that since ρ(0 = 0, we have L B (y lim y y 3 =, (3.4 which means that the lower boun (3.2 is sharper than (3., when f (x. In orer to prove (3.4, note that L B (y y 3 = ( + B 2 yr(y an yr(y = r(y 2 ρ(α 6 r(y/2 α 3 α. Therefore, in view of (3., the limit in (3.4 inee iverges if we establish that lim r 0 + r2 r/2 ρ(α α = 0. (3.5 α3 Clearly, it is sufficient to verify that r ρ(α lim r 0 r2 α = 0. + r α3 Inee, since ρ is a moulus of continuity of f an we have that f L B, we obtain r 2 ρ(α α 2Br2 r α 3 α Br 0 as r 0. r α3 On the other han, it is easy to see from the monotonicity of ρ that r 2 r ρ(α α 3 α ρ( rr 2 α α 3 = ρ( r 0 2 as r 0 since ρ(0 = 0. r r

12 2 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL POOF OF LEMMA 3.3. In orer to prove (3.5 we procee as in (3.8, but we o not integrate by parts in the secon term. This yiels ( (p /p D f [f 2 f (x 2 2 ](x + B 2 r + B 2 f (x f L p α r/2 α 2p/(p 2 f (x 2 8 f (x f L p + B 2 r + B 2 r (p+/p for all p >. The esire inequality follows upon choosing r = 8p f p L p f (x p. Similarly, for p (, 2 in orer to prove (3.6, we consier (3.9, but we o not integrate by parts in the secon term. We obtain D p f [f ](x f (x p 4 f (x p f L p + B 2 r + B 2 r (p+/p. Choosing exactly as above conclues the proof of the Lemma. POOF OF LEMMA 3.4. We consier the inequality analogous to (3.8 with f replace by f. After integrating by parts once, the same argument use to prove Lemma 3. yiels ( Bouns for the nonlinear terms In this section we give pointwise in x bouns for the nonlinear terms T i (x, with i {,..., 5} appearing on the right sies of (2.7 an (2.0. We first fix a small constant ε (0, ] to be chosen later. The bouns we obtain epen on this ε, an ε will be chosen ifferently in the proofs of Theorem., Theorem.2, an Theorem.3 respectively. The main result of this section is: LEMMA 4.. Let B > 0 be such that (.2 hols an fix ε (0, ]. There exists a positive universal constant C > 0 such that the bouns T (x + T 2 (x + T 3 (x + T 4 (x CB ε 2 f (x 2 + CεB 2 D[f ](x f (x hol for all x where f (x 0. (4. T 5 (x CB Λf (x + CB ε f (x + Cε 2 B 2 D[f ](x f (x 2 (4.2 In aition, we nee a pointwise estimate for the nonlinear terms in terms of the issipative term D p [f ], when p (, 2. LEMMA 4.2. Let B > 0 be such that (.2 hols, let p (, 2, an fix ε (0, ]. There exists a positive universal constant C > 0 such that the bouns T (x + T 2 (x + T 3 (x + T 4 (x CB ε 2/(p f (x 2 + CεB 2 Dp [f ](x f (x p (4.3 hol for all x where f (x 0. T 5 (x CB Λf (x + CB ε p f (x + CεB 2 Dp [f ](x f (x p (4.4

13 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 3 POOF OF LEMMA 4.. Throughout this proof, we fix a cutoff raius (for α η = η(x = εb f (x. (4.5 Note that at the points x where f (x = 0, there is no estimate to be one for T i (x terms as they are multiplie with f (x in (2.7 an (2.0. Estimate for the T term. We ecompose T into an inner piece an an outer piece accoring to T (x = 4 P.V. (δ α f (x αf (x δ α f(x δ α f (x ((δ α f(x 2 + α 2 2 α [f ](x, α α f(x (f (x + [f ](x, α = 4 P.V. (( α f(x α α [f ](x, α α f(x δ α f (x + 4 α >η (( α f(x α α 2 =: T,in (x + T,out (x. Using the pointwise in x an α bouns (2.20 an (2.24, an the efinition of η in (4.5, we obtain T,in (x CB f (x (D[f ](x /2 α α /2 + CBD[f ](x α an Cε /2 B 3/2 f (x /2 (D[f ](x /2 + CεB 2 D[f ](x f (x CB f (x 2 + CεB 2 D[f ](x f (x T,out (x CB 2 (D[f ](x /2 α >η α α 3/2 Cε /2 B 3/2 f (x /2 (D[f ](x /2 Cε 2 B f (x 2 + CεB 2 D[f ](x f (x for some universal constant C > 0. Combining (4.6 an (4.7 we arrive at (4.6 (4.7 2 f (xt (x Cε 2 B f (x 3 + CεB 2 D[f ](x (4.8 for some universal C > 0. This boun is consistent with (4.. Estimate for the T 2 term. We ecompose T 2 as ( α f(x f (x(δ α f (x 2 T 2 = 2 P.V. (( α f(x α α 3 ( αf (x/2 + 2 [f ](x, α(δ α f (x 2 = 2 P.V. (( α f(x α α 3 2 [f ](x, α(δ α f (x(f (x + [f ](x, α = 2 P.V. (( α f(x α α 2 f (f (x + [f ](x, α 2 (x P.V. (( α f(x α ( αf (x/2 + 2 [f ](x, α(δ α f (x α >η (( α f(x α α 3 =: T 2,,in + T 2,2,in + T 2,out.

14 4 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL By appealing to (2.25 an (2.24 we may estimate the inner terms as T 2,,in CB(D[f ](x ( f /2 α (x α /2 + (D[f ](x /2 Cε /2 B 3/2 f (x /2 (D[f ](x /2 + CεB 2 D[f ](x f (x CB f (x 2 + CεB 2 D[f ](x f (x α (4.9 an T 2,2,in C f (x ( f (x 2 α + f (x (D[f ](x /2 α /2 α + D[f ](x CεB f (x 2 + Cε 3/2 B 3/2 f (x /2 (D[f ](x /2 + Cε 2 B 2 D[f ](x f (x CB f (x 2 + CεB 2 D[f ](x f (x α α (4.0 while the outer terms may be boune as T 2,out CB ( f 2 α (x α >η α 2 + (D[f ](x /2 α >η α α 3/2 Cε B f (x 2 + Cε /2 B 3/2 f (x /2 (D[f ](x /2 Cε 2 B f (x 2 + CεB 2 D[f ](x f (x (4. for some universal C > 0. Combining (4.9, (4.0, (4., using the ientity (2.3 an the Cauchy- Schwartz inequality we arrive at 2 f (xt 2 (x Cε 2 B f (x 3 + CεB 2 D[f ](x (4.2 for some universal C > 0. This boun is consistent with (4.. Estimate for the T 3 term. We boun T 3 as T 3 = 2 P.V. ( α f(x f (x α CB(D[f ](x /2 ( CB(D[f ](x /2 ( + CB(D[f ](x /2 ( α >η =: T 3,in + T 3,out. α f(x δ α f (x (( α f(x α α ( α f(x f (x 2 α 2 /2 (αf (x/2 + 2 [f ](x, α] 2 α 2 ( [f ](x, α 2 α 2 /2 /2

15 Using (2.25 we may further estimate GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 5 T 3,in CB(D[f ](x /2 ( CB(D[f ](x /2 ( f (x 2 CB(D[f ](x /2 ( (αf (x/2 + 2 [f ](x, α] 2 α 2 α + f (x (D[f ](x /2 /2 Bε f (x + B 3/2 ε 3/2 (D[f ](x /2 f (x /2 + B 2 ε 2 D[f ](x f (x 2 /2 α /2 α + D[f ](x α α CBε /2 B /2 f (x /2 (D[f ](x /2 + CεB 2 D[f ](x f (x CB f (x 2 + CεB 2 D[f ](x f (x. (4.3 /2 Similarly, we have that T 3,out CB(D[f ](x /2 ( α >η /2 α α 2 Cε /2 B 3/2 f (x /2 (D[f ](x /2 Cε 2 B f (x 2 + CεB 2 D[f ](x f (x. (4.4 Combining (4.3 (4.4, leas to the estimate 2 f (xt 3 (x Cε 2 B f (x 3 + CεB 2 D[f ](x (4.5 for some positive universal constant C > 0. This boun is consistent with (4.. Estimate for the T 4 term. We ecompose T 4 as ( α f(x f (x( α f(x 2 (δ α f (x 2 T 4 = 8 P.V. (( α f(x α α 3 = 4f ( α f(x 2 (f (x + [f ](x, α 2 (x P.V. (( α f(x α ( 2 [f ](x, α( α f(x 2 (δ α f (x(f (x + [f ](x, α 8 P.V. (( α f(x α α 2 ( [f ](x, α( α f(x 2 (δ α f (x 2 8 α >η (( α f(x α α 3 =: T 4,,in + T 4,2,in + T 4,out.

16 6 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL Using (2.24 we may estimate T 4,,in C f (x ( f (x 2 C f (x α + f (x (D[f ](x /2 α /2 α + D[f ](x ( εb f (x + ε 3/2 B 3/2 (D[f ](x /2 f (x /2 + ε 2 B 2 D[f ](x f (x 2 α α CεB f (x 2 + Cε 2 B 2 D[f ](x f (x. (4.6 By also appealing to (2.25 we obtain the boun T 4,2,in CB(D[f ](x ( f /2 (x α α /2 + (D[f ](x /2 Cε /2 B 3/2 f (x /2 (D[f ](x /2 + CεB 2 D[f ](x f (x. (4.7 On the other han, for the outer term we obtain T 4,out CB 3 α >η α α α 3 Cε 2 B f (x 2. (4.8 Combining (4.6 (4.8, an using the Cauchy-Schwartz inequality we arrive at 2 f (xt 4 (x Cε 2 B f (x 3 + CεB 2 D[f ](x (4.9 for some positive universal constant C > 0. This boun is consistent with (4.. Estimate for the T 5 term. ecall that T 5 is compute by taking the erivative of (2.3 as T 5 (x = x v(x = ( α P.V. x ( α f(x 2 + α α f(x δ α f (x = 2P.V. (( α f(x α 2 α. The main iea here is that in orer to ecompose T 5 into an inner an an outer term, we first nee to subtract an a f (x T 5,pv (x = 2 (f (x P.V. δ α f (x f (x α 2 α = 2 (f (x Λf (x. The contribution from this term is boune by T 5,pv (x CB Λf (x = CB Hf (x (4.20 where H is the Hilbert transform. The ifference is now ecompose further as T 5 T 5,pv = P.V. K f (x, α ( α f(x f (x ( αf (x α, α where K f (x, α = 2 ( αf(x 3 f (x + ( α f(x 2 (f (x 2 + f (x( α f(x 3 + 2f (x α f(x (( α f(x (f (x Next we use that K f (x, α 2, for any value of α f(x an f (x.

17 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 7 Decomposing further, it is possible to fin T 5 T 5,pv = P.V. K f (x, α ( αf (x/2 + 2 [f ](x, α (f (x + [f ](x, α α <η (( α f(x α α [f ](x, α (δ α f (x + K f (x, α (( α f(x α 2 α α >η =: T 5,in + T 5,out. For the outer term we irectly obtain T 5,out CB 2 α >η α α 2 CB ε f (x. (4.2 We recall that η = η(x = εb f (x. For the inner term, we appeal to (2.25 an (2.24 an obtain that T 5,in f (x 2 α + C f (x (D[f ](x /2 α /2 α + CD[f ](x α α α <η CεB f (x + Cε 3/2 B 3/2 (D[f ](x /2 f (x /2 + Cε2 B 2 D[f ](x f (x 2. (4.22 Summarizing, (4.2 an (4.22 an using the Cauchy-Schwartz inequality, we obtain the esire boun α <η T 5 (x T 5,pv (x C B ε f (x + Cε 2 B 2 D[f ](x f (x 2 which combine with (4.20 yiels the esire boun (4.2. POOF OF LEMMA 4.2. The proof closely follows that of Lemma 4., but uses (2.26 (2.27 instea of (2.24 (2.25. We let ε (0, ] to be etermine, an as in (4.5 let η = η(x = εb f (x. In this proof the constant C may change from line to line, an may epen on p, but not on ε, B, or x. Estimate for the T, T 2, T 3, an T 4 terms. We first claim that T (x + T 2 (x + T 3 (x + T 4 (x CB ε 2 f (x 2 + CB (p+2/p ε 2/p (Dp [f ](x 2/p f (x 2/p (4.23 for some constant C > 0, an all x. We verify this estimate by checking each term iniviually. For T, similarly to (4.6 an (4.7, by using (2.26 (2.27 we have that α <η T,in (x C ε /p B (p+/p f (x (p /p (D p [f ](x /p + CB (p+2/p ε 2/p (Dp [f ](x 2/p T,out (x CB(p+/p ε (p /p f (x (p /p (D p [f ](x /p so that by Cauchy-Schwartz we obtain that T is boune as in (4.23. For the T 2 term, as in (4.9, (4.0, an (4., but using (2.26 (2.27, we obtain f (x 2/p T 2,,in (x C ε /p B (p+/p f (x (p /p (D p [f ](x /p + CB (p+2/p ε 2/p (Dp [f ](x 2/p T 2,2,in (x C εb f (x 2 + CB (p+2/p ε (p+2/p (Dp [f ](x 2/p f (x 2/p T 2,out (x CB ε f (x 2 + CB(p+/p ε (p /p f (x (p /p (D p [f ](x /p so that by Cauchy-Schwartz it follows that T 2 obeys the boun (4.23. f (x 2/p

18 8 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL For T 3, we procee slightly ifferently from (4.6 an (4.7, but still use (2.26 (2.27 an obtain T 3 (x = 2 P.V. [f ](x, α α f(x δ α f (x α 2 2/p (( α f(x α 2/p α ( CB(D p [f ](x /p ( CB(D p [f ](x /p ( [f ](x, α p/(p (p /p α 2 (p /p ( 2 [f ](x, α + αf (x/2 p/(p α 2 + Bη (p /p CB(D p [f ](x /p ( f (x η /p + (D p [f ](x /p η 2/p + Bη (p /p CB(p+/p ε (p /p f (x (p /p (D p [f ](x /p + C ε 2/p B (p+2/p (Dp [f ](x 2/p f (x 2/p which is consistent with (4.23. Lastly, for the T 4 term, we use bouns similar to (4.6, (4.7, an (4.8, combine with (2.26 (2.27, to euce ( T 4,,in C f (x εb f (x + ε (p+2/p B (p+2/p (Dp [f ](x, α 2/p f (x (p+2/p T 4,2,in C ε /p B (p+/p f (x (p /p (D p [f ](x /p + C ε 2/p B (p+2/p (Dp [f ](x 2/p f (x 2/p T 4,out CB ε 2 f (x 2 which conclues the proof of (4.23. In orer to prove (4.3, we now use (4.23 in which we choose ε epening on two cases: if { B 2/p (D p [f ](x 2/p f (x 2(p+/p, B 2/p (D p [f ](x 2/p f (x 2(p+/p, then let { ε =, ε = f (x (BD p [f ](x /(p+. We thus obtain the esire estimate T (x + T 2 (x + T 3 (x + T 4 (x CB max { f (x 2, B 2/(p+ (D p [f ](x 2/(p+} CεB2 D p [f ](x f (x p + CB f (x 2 ε 2/(p (4.24 where in the last inequality we have appeale to the ε-young inequality an the essential fact that 2 < p +. Estimate for the T 5 term. The first step is to show that ( T 5 (x CB Λf (x + ε f (x + ε (p+2/p B 2/p (D p[f ](x 2/p f (x (p+2/p (4.25 hols when p (, 2. For this purpose, similarly to (4.20, (4.2, an (4.22, by using (2.26 (2.27 we arrive at T 5,pv CB Λf (x T 5,in C εb f (x + C ε (p+2/p B (p+2/p (Dp [f ](x 2/p T 5,out CB ε f (x. f (x (p+2/p

19 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 9 Combining the above estimates proves (4.25. We now use (4.25 in which we choose ε precisely as in (4.24 to obtain the esire estimate T 5 (x CB ( Λf (x + f (x + B /(p+ (D p [f ](x /(p+ ( CB Λf (x + ε p f (x + εbdp [f ](x f (x p In the last inequality we have appeale to the ε-young inequality an the fact that < p Proof of Theorem.2, part (i, Blowup Criterium for the Curvature We evaluate (2.7 at a point x = x(t where the secon erivative achieves its maximum, i.e. such that At this point x we have f ( x, t = f (t L = M (t. x f ( x, t 2 = 0 an L f [f ]( x, t 0. By aemacher s theorem whose use is justifie by the assumptions, we may then show that t M (t 2 = t f (t 2 L = t f ( x, t 2 ( t + v( x, t x + L f f ( x, t 2 = 2f ( x, t(t + T 2 + T 3 + T 4 D f [f ]( x, t (5. for almost every t [0, T ]. In orer to complete the proof, we estimate the right han sie of (5.. The issipative term D f [f ](x is non-negative, an is boune from below explicitly as in (2.4, (3., an (3.3. In orer to estimate the T,..., T 4 terms on the right sie of (2.6, we appeal to the upper boun (4. obtaine in Lemma 4.. We thus obtain t M 2 + 2( + B 2 D[f ]( x + 2 L B(M C 0B ε 2 M 3 + C 0 εb 2 D[f ]( x (5.2 for some universal constant C 0. We now choose the value of ε in Lemma 4. as { } ε = min 2C 0 B 2 ( + B 2, in orer to obtain where t M L B(M K B M 3 K B = C 0 B max{, 4C 2 0B 4 ( + B 2 2 } is a constant that epens solely on B. To close the argument, we recall cf. (3.4 that L B (M lim M M 3 = an that L B is continuous, which shows that there exists M = M (B, C 0 such that Therefore, we obtain 2 L B(M K B M 3 for any M M. M (t max{m (0, M } for all t [0, T ], which conclues the proof of the theorem.

20 20 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL 6. Proof of Theorem., Local Existence In view of the available L p maximum principle for f, cf. (2.4, the proof consists of coupling the evolution of the maximal slope B(t = f (t L with that of the L p norm of f M p (t = f (t L p. Our goal is to obtain an a priori estimate of the type t (B2 + M 2 p polynomial(b 2 + M 2 p from which the existence of solutions on a time interval that only epens on B(0 2 + M p (0 2 follows by a stanar approximation proceure. For the evolution of M p (t we split the proof in three cases: (i p = (ii p [2, (iii p (, 2. We however note that in all of these three cases, but the D Sobolev embeing, as soon as f L p, we have f C (p /p, an we have the boun [f ] C (p /p f L p for some positive constant C that may epen on p. In particular, it follows from (2.2 that the boun [f ](x, α = α f(x f (x α (p /p f L p (6. for all x, α, an all p (, ]. In view of this estimate, we immeiately notice that the uniqueness of solutions in W 2,p follows irectly from Theorem.4, whose proof is given below. 6.. Evolution of the maximal slope B(t. We first multiply equation (2.5 by f (x, t an arrive at the equation ( t + v x + L f f 2 + D f [f ] = T 6 (6.2 pointwise in (x, t, where we have enote T 6 (x, t = 4f ( [f ](x, α( α f(x, t δ α f (x, t (x, t ( α f(x, t 2 + (δ α f(x 2 + α 2 α. Let x = x(t be a point such that f ( x(t, t = B(t. Using equation (6.2 an aemacher s theorem we fin that t B2 (t = t f ( x, t 2 = T 6 (x, t ( D f [f ]( x, t + L f f ( x, t 2 (6.3 for almost every t. Here we use that at a point of global maximum x f 2 = 0. Note that also that at x we have L f f 2 0. It remains to boun T 6 ( x, t. For this purpose we ecompose T 6 as follows ( [f ](x, α( α f(x, t (δ α f (x, t 2 T 6 (x = 2 α ε ( α f(x, t 2 + (δ α f(x 2 + α 2 α ( [f ](x, α( α f(x, t δ α (f (x, t α ε ( α f(x, t 2 + (δ α f(x 2 + α 2 α + 4f ( [f ](x, α( α f(x, t δ α f (x, t (x, t (( α f(x, t α 2 α α ε = T 6,,in (x + T 6,2,in (x + T 6,out (x

21 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 2 where ε = ε(x > 0 is to be etermine, an we ignore the t-epenence of all factors. Using (6. an the fact that L f f ( x 2 0 we may estimate while the boun (2.20 yiels T 6,,in ( x ε (p /p f L pd f [f ]( x T 6,2,in ( x ε (p /p f L pl f f ( x 2, T 6,out ( x 8B3 ε. Letting ( p/(p ε = 2 f L p in the above three estimates, we arrive at T 6 ( x ( Df [f ]( x + L f f ( x 2 + CB 3 f p/(p L 2 p which combine with (6.3 gives t B2 CB 3 f p/(p L (6.4 p for some positive constant C that may epen on p (, ] Case (i, p =. We recall the evolution of M (t, cf. (5.2, in which we take { } ε = ε(t = min 2C 0 B(t 2 ( + B(t 2, to arrive at the a priori estimate t M 2 C 0 B max { 2C 0 B 2 ( + B 2, } M 3. (6.5 Combining (6.4 with (6.5 we obtain that t (B(t2 + M (t 2 CB(t 3 M (t + CB(t( + B(t 2 2 M (t 3 C( + B(t 2 + M (t 2 4 (6.6 for some positive constant C. Integrating (6.6, we obtain that there exists T = T ( f 0 L, f 0 L > 0 on which the solution may be shown to exist an have finite W 2, norm Case (ii, 2 p <. We consier the evolution of f p in ivergence form, cf. (2.6, apply the upper boun given by Lemma 4. for the terms T,..., T 5 on the right sie of (2.6, an the lower boun given in Lemma 3.3, estimate (3.5, an the boun given by (2.4 for the issipative term D f [f ] on the left sie of (2.6, to arrive at ( t + L f f (x, t p + x (v(x, t f (x, t p + f (x, t p 2 2( + B(t 2 D[f f (x, t 2p ](x, t + C ( + B(t 2 f (t p L p C B(t f (x, t p Hf (x, t + C B(t ε(t 2 f (x, t p+ + C εb(t 2 f (x, t p 2 D[f ](x, t pointwise in x an t. Choosing { } ε(t = min 4C B(t 2 ( + B(t 2, (6.7

22 22 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL we conclue from (6.7 that ( t + L f f (x, t p + x (v(x, t f (x, t p + f (x, t p 2 4( + B(t 2 D[f f (x, t 2p ](x, t + C ( + B(t 2 f (t p L p C B(t f (x, t p Hf (x, t + CB(t 5 ( + B(t 2 f (x, t p+ (6.8 where C = C(C > 0 is a constant. At this stage we integrate (6.8 for x. First we note that L f [ f p f (x p f (x α p ](xx = P.V. (f(x f(x α 2 αx = 0. (6.9 + α2 This fact may be seen by changing variables x x α. We thus obtain an a priori estimate for the evolution of M p (t = f (t L p as t M p(t p M 2p (t 2p + C ( + B(t 2 M p (t p CB(t( + B(t2 3 M p+ (t p+ (6.0 by also using that the Hilbert transform is boune on L p. Furthermore, since for p > we have p+ < 2p, we may interpolate which combine with (6.0 an the ε-young inequality, yiels t M p(t p M 2p (t 2p + C ( + B(t 2 M p (t p M p+ (t M p (t (p /(p+ M 2p (t 2/(p+ (6. CB(t( + B(t 2 3 M p (t p M 2p (t 2 In conclusion, we obtain M 2p (t 2p 2C ( + B(t 2 M p (t p + CB(tp/(p ( + B(t 2 (3p+/(p M p (t p2 /(p (6.2 t M p(t 2 CB(t p/(p ( + B(t 2 (3p+/(p ( M p (t 2 (3p 2/(2p 2 which combine with (6.4 gives ( B(t 2 + M p (t 2 C( + B(t 2 + M p (t 2 5p/(p (6.3 t for some positive constant C. Integrating (6.3, we obtain that there exists T = T ( f 0 L, f 0 L p > 0 on which the solution may be shown to exist an have finite W 2,p W, norm Case (iii, < p < 2. We procee similarly to the case p 2, but instea of applying to (2.6, we use (2.9. For the issipative term on the left sie of (2.6, we use the minimum between the lower bouns provie by (3.5 an (3.6 in Lemma 3.3 an (2.4. For the nonlinear terms on the right sie of (2.9 we use the minimum between the upper bouns provie by Lemmas 4. an 4.2. The resulting pointwise in (x, t inequality is ( t + L f f p + x (v f p + C 2 ( + B 2 min { D[f C 2 εb 2 } ] min f 2 p, Dp [f ] { D[f } ] f 2 p, Dp [f ] + for some constant C 2 > 0. Choosing { } ε(t = min 2C2 2B(t2 ( + B(t 2, f 2p C 2 ( + B 2 f p L p + C 2B ε 2/(p f p+ + C 2 B Hf f p (6.4

23 we conclue from (6.4 that GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 23 ( t + L f f p + x (v f p + { D[f } 2C 2 ( + B 2 min ] f 2 p, Dp [f ] + f 2p C 2 ( + B 2 f p L p C( + B 2 (p+7/(2p 2 ( f p+ + Hf f p. (6.5 Upon integrating (6.5 for x, using the ientity (6.9, the bouneess of H on L p, an the interpolation boun (6., we thus arrive at t M p(t p M 2p (t 2p + C 2 ( + B(t 2 M p (t p C( + B(t2 (p+7/(2p 2 M p+ (t p+ C( + B(t 2 (p+7/(2p 2 M p (t p M 2p (t 2. (6.6 Similarly to (6.2 (6.3, since p + > 2, an by using the ε-young inequality, we conclue from (6.6 that t M p(t 2 C( + B(t 2 (p2 +9p 2/(2(p 2 (M p (t 2 (3p 2/(2p 2. (6.7 Combining with (6.4 we finally arrive at t (B(t2 + M p (t 2 C( + B(t 2 + M p (t 2 2p(p+/(p 2. Integrating (6.3, we obtain that there exists T = T ( f 0 L, f 0 Lp > 0 on which the solution may be shown to exist an have finite W 2,p W, norm. 7. Proof of Theorem.3, Global Existence for Small Datum The proof follows closely the estimates in Section 6. The major ifference is that assumption (.4 an the maximum principle for f establishe in [CG09, Section 5] show that f (t L B C (7. for all t > 0. Thus, we o not nee to consier the evolution of B(t, as we have B(t C for t [0, T, where T > 0 is the maximal existence time in W 2,p. For simplicity, we split the proof in three cases base on the value of p (, ]: (i p = (ii p [2, (iii p (, Case (i, p =. We use the estimate (5.2, but here we apply lower boun (3. instea of (3.2, an we set ε =. We arrive at t M 2 + 2( + B 2 D[f ]( x + M 3 48B( + B 2 C 0BM 3 + C 0 B 2 D[f ]( x where x = x(t is a point at which M = f ( x, t. For B small enough, so that hols, we thus obtain 2C 0 B 2 ( + B 2 an 00C 0 B 2 ( + B 2 t M + 50B( + B 2 M 2 0.

24 24 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL Integrating the above ODE we obtain that for all t 0, which proves (.5. M (t M (0 + M (0 00B t 7.2. Case (ii, 2 p <. We use the first line of estimate (6.8, but instea of using (3.5 to boun the issipative term D[f ] from below, we appeal to (3.. We arrive at ( t + L f f (x, t p + x (v(x, t f (x, t p + f (x, t p+ 96B( + B 2 + f (x, t p 2 D[f ](x, t 4( + B 2 C B f (x, t p Hf (x, t + CB 5 ( + B 2 f (x, t p+. Integrating the above over x, similarly to (6.0 we obtain t M p(t p + M p+(t p+ 96B( + B 2 + 4( + B 2 f (x, t p 2 D[f ](x, tx CB( + B 2 3 M p+ (t p+ (7.2 for some C > 0. If B is chosen small enough so that 200CB 2 ( + B 2 4 we thus obtain t M p(t p + M p+(t p+ 200B( + B 2 + 4( + B 2 f (x, t p 2 D[f ](x, tx 0. Integrating the above in time an noting that f (x, t p 2 D[f ](x, tx = ( f (x p 2 + f (x α p 2 (f (x f (x α 2 2 α 2 αx 4 ( f (x p/2 f (x α p/2 2 p 2 αx conclues the proof of (.6. = 4 p 2 f p/2 Ḣ/ Case (iii, < p < 2. We use estimate (6.5 in which we boun from below the issipative terms from below by appealing to Lemma 3., an arrive at ( t + L f f p + x (v f p + α 2 f p+ 200BC 2 ( + B 2 + C( + B 2 (p+7/(2p 2 ( f p+ + Hf f p. f 2p C 2 ( + B 2 f p L p Integrating the above over x, an using that H is boune on L p in this range of p, we thus arrive at t M p(t p + M p+(t p+ 200BC 2 ( + B 2 + M 2p (t 2p C 2 ( + B 2 f p C( + B 2 (p+7/(2p 2 M p+ (t p+. L p Lastly, choosing B small enough so that we arrive at 400C 2 CB( + B 2 (3p+5/(2p 2, t M p(t p + M p+(t p+ 400BC 2 ( + B 2 0 which upon integration in time conclues the proof of (.7 an thus of the theorem.

25 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE Proof of Theorem.2, part (ii, Blowup Criterium for Smooth Solutions We shall stuy the evolution of the f H k(t norms for k 3. We show that they can be controlle by sup [0,T ] f (t L an sup [0,T ] f (t L. Then Theorem.2 conclues the proof. In fact, the H k norm of a solution with k > 3 can be controlle alreay by H 3 -norm as shown in [CCG + 3a], Section 5.2. Therefore we may assume k = 3. We start by ealing with the evolution of f (t 2 L. We use inequality 2 (7.2 with ε small enough an p = 2, to obtain an therefore We use equation (2. to split where t f 2 L 2 + f 2 Ḣ /2 2C 0 ( + B 2 C(B f 3 L 3 C(B f L f 2 L 2, I = ( f (t 2 L f L exp C(B 2 f (x t f (xf t (xx = I + I 2 + I 3 + I 4, ( (f (xα δ α f (x 0 f (s L s. (8. (δ α f(x 2 + α 2 αx, ( I 2 = 3 f (x (f (xα δ α f (x x (δ α f(x 2 + α 2 αx, ( I 3 = 3 f (x (f (xα δ α f (x x 2 (δ α f(x 2 + α 2 αx, ( I 4 = f (x (f (xα δ α f(x x 3 (δ α f(x 2 + α 2 αx. In I it is possible to ecompose further an obtain I = f (xf α (xp.v. (δ α f(x 2 + α 2 αx D f [f ](xx := I, + I,2. 2 We boun I, as I, = = f (x 2 P.V. f (x 2 P.V. + αδ α f(xδ α f (x ((δ α f(x 2 + α 2 2 α ( α f(x α (( α f(x f (x ((f (x α f (xαx f (x 2 f (xλf (x ((f (x x := I,, + I,,2. In I,, one fins extra cancelation in such a way that splitting in the regions α < r an α > r an optimizing in r, it is possible to obtain as before I,, C(B f L f 2 L 2. For I,,2, the Gagliaro-Nirenberg interpolation inequality g L 4 C g /2 g /2 L 2 Ḣ I,,2 f L 2 f 2 L 4 C f L 2 f L 2 f Ḣ/2 f 2 C(B f 2 L f 2 Ḣ 2 L + /2 2 32( + B 2 /2 yiels

26 26 PETE CONSTANTIN, FANCISCO GANCEDO, OMAN SHVYDKOY, AND VLAD VICOL by using the ε-young inequality. This yiels I, = I,, + I,,2 C(B( f L + f 2 L 2 f 2 L 2 + Using (2.4, (2.3 together with (3.7 in I,2 we arrive at f 2 Ḣ /2 32( + B 2. I,2 4( + B 2 f 2 Ḣ /2 2 7 ( + B 2 3 f f 3 L. L 3 Aing the last two estimates it is possible to obtain I = I, + I,2 C(B( f L + f 2 L 2 f 2 L 2 We are one with I. For I 2 we rewrite as 7 f 2 Ḣ /2 32( + B 2 f 3 L ( + B 2 3 f. (8.2 L I 2 =6 f α f (x f (x α f(x (x α ( α f(x αf (xαx =6 f [f ](x, α α f(x (x α < f α ( L α f(x αf (xαx + 6 f [f ](x, α α f(x (x α > f α ( L α f(x αf (xαx :=I 2,in + I 2,out. Inequality (2.24 allows us to get I 2,in 6 f L C f /2 L f (x D[f](x /2 f (x D[f](x /2 x C(B f L f 2 L 2 + f 2 Ḣ /2 32( + B 2, α < f L α x α /2 an I 2,out 2B f (x D[f](x /2 α > f L C(B f /2 L f (x D[f](x /2 x α x α 3/2 C(B f L f 2 L 2 + f 2 Ḣ /2 32( + B 2. These last to inequalities give the appropriate boun for I 2 such that aing (8.2 we obtain I + I 2 C(B( f L + f 2 L 2 f 2 L 2 5 f 2 Ḣ /2 32( + B 2 f 3 L ( + B 2 3 f. (8.3 L

27 GLOBAL EGULAITY FO 2D MUSKAT EQUATIONS WITH FINITE SLOPE 27 It is possible to ecompose further in I 3 as follows I 3 =c 3, f (x (f (xα δ α f (δ α f (x 2 (x ((δ α f(x 2 + α 2 2 αx + c 3,2 f (x (f (xα δ α f (x (δ αf(x 2 (δ α f (x 2 ((δ α f(x 2 + α 2 3 αx + c 3,3 f (x (f (xα δ α f (x δ αf(xδ α f (x ((δ α f(x 2 + α 2 2 αx The ientity :=I 3, + I 3,2 + I 3,3. allows us to get I 3, C f 2 L f (xα δ α f (x = α 2 rf (x + (r αr, ( CB 2 0 α <B f L α >B f L C(B f L f 2 L 2. An analogous approach for I 3,2 gives 0 ( f (x 2 + f (x + (r α 2 xαr ( f (x 2 + f (x + (r α 2 x α α 2 r I 3,2 C(B f L f 2 L 2. For the I 3,3 term, we ecompose further: I 3,3 =c 3,3 f f (x α f (x α f(x (x α <ζ α ( α f(x αf (xαx + c 3,3 f (x α 2 (f (x α f α f(x (x (( α f(x δ αf (xαx :=I 3,3,in + I 3,3,out. For the outer term, inequality yiels Ientity (8.4 together with α >ζ allow us to obtain for the inner term I 3,3,out c 3,3 We take to fin 0 0 f 2 L 4 C f L f L 2 (8.5 I 3,3,out C ζ f 2 L 4 f L 2 CB ζ f 2 L 2. α <ζ 2c 3,3 ζ f 3 L 3. α f (x = 0 f (x + (s αs, (8.6 f (x f (x + (r α f (x + (s α xαsr ζ = 2 8 ( + B 2 3 c 3,3 f L I 3 = I 3, + I 3,2 + I 3,3 C(B f L f 2 f 3 L L ( + B 2 3 f. L