The Muskat problem with surface tension and equal viscosities in subcritical $L_p$-Sobolev spaces

In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $W^s_p(\mathbb{R})$, where ${p\in(1,2]}$ and ${s\in(1+1/p,2)}$. This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $W^{\overline{s}-2}_p(\mathbb{R})$, where ${\overline{s}\in(1+1/p,s)}$. Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.


Introduction
In this paper we study the evolution equation which is defined for t > 0 and x ∈ R. The function f is assumed to be known at time t = 0, that is f (0, •) = f 0 .
(1.1b) The evolution problem (1.1) is the contour integral formulation of the Muskat problem with surface tension and with/without gravity effects, see [24,25] for an equivalence proof of (1.1) to the classical formulation of the Muskat problem [29].The problem (1.1) describes the twodimensional motion of two fluids with equal viscosities µ − = µ + = µ and general densities ρ − and ρ + in a vertical/horizontal homogeneous porous medium which is identified with R 2 .The fluids occupy the entire plane, they are separated by the sharp interface {y = f (t, x) + V t}, and they move with constant velocity (0, V ), where V ∈ R. The fluid denoted by + is located above this moving interface.We use g ∈ [0, ∞) to denote the Earth's gravity, k > 0 is the permeability of the homogeneous porous medium, and σ > 0 is the surface tension coefficient at the free boundary.Moreover, to shorten the notation we have set x, y ∈ R. Finally, κ(f (t)) is the curvature of {y = f (t, x) + tV } and PV denotes the principal value.
The Muskat problem with surface tension has received much interest in the recent years.Besides the fundamental well-posedness issue also other important important features like the stability of stationary solutions [13,14,16,20,26,28,32,34], parabolic smoothing properties [24][25][26], the zero surface tension limit [8,31], and the degenerate limit when the thickness of the fluid layers (or a certain nondimensional parameter) vanishes [15,22] have been investigated in this context.We also refer to [19,21] for results on the Hele-Shaw problem with surface tension effects, which is the one-phase version of the Muskat problem, and to [35][36][37] for results on the related Verigin problem with surface tension.
Concerning the well-posedness of the Muskat problem with surface tension effects, this property has been investigated in bounded (layered) geometries in [14,16,17,32,34] where abstract parabolic theories have been employed in the analysis, the approach in [23] relies on Schauder's fixed-point theorem, and in [10] the authors use Schauder's fixed-point theorem in a setting which allows for a sharp corner point of the initial geometry.
The results on the Muskat problem with surface tension in the unbounded geometry considered in this paper (and possibly in the general case of fluids with different viscosities) are more recent, cf.[8,[24][25][26]30,31,38].While in [8,38] the initial data are chosen from H s (T), with s ≥ 6, the regularity of the initial data has been decreased in [24][25][26] to H 2+ε (R), with ε ∈ (0, 1) arbitrarily small.Finally, the very recent references [30,31] consider the problem with initial data in H 1+ d 2 +ε (R d ), with d ≥ 1 and ε > 0 arbitrarily small, covering all subcritical L 2 -based Sobolev spaces in all dimensions.
It is the aim of this paper to study the Muskat problem (1.1) in the subcritical L p -based Sobolev spaces W s p (R) with p ∈ (1, 2] and s ∈ (1 + 1/p, 2).This issue is new in the context of (1.1) (see [2,12] for results in the case when σ = 0).To motivate why W 1+1/p p (R) is a critical space for (1.1) we first emphasize that the surface tension is the dominant factor for the dynamics as it contains the highest spatial derivatives of f .Besides, if we set g = 0, then it is not difficult to show that if f is a solution to (1.1a), then, given λ > 0, the function f λ (t, x) := λ −1 f (λ 3 t, λx) also solves (1.1a).This scaling identifies W 1+1/p p (R) as a critical space for (1.1).The main result Theorem 1.1 establishes the well-posedness of (1.1) in W s p (R).This is achieved by showing that (1.1) can be recast as a quasilinear parabolic evolution equation, so that abstract results available for this type of problems, cf.[3,4,6,28], can be applied in our context.A particular feature of the Muskat problem (1.1) is the fact that the equations have to be interpreted in distributional sense as they are realized in the Sobolev space W s−2 p (R), where s is chosen such that 1 + 1/p < s < s < 2. Additionally to well-posedness, Theorem 1.1 provides two parabolic smoothing properties, showing in particular that (1.1a) holds pointwise, and a criterion for the global existence of solutions.
with T + = T + (f 0 ) ∈ (0, ∞] denoting the maximal time of existence.Moreover, the following properties hold true: (i) The solution depends continuously on the initial data; We emphasize that some of the arguments use in an essential way the fact that p ∈ (1, 2].More precisely, we employ several times the Sobolev embedding W s p (R) ֒→ W t p ′ (R), where p ′ is the adjoint exponent of p, that is p −1 + p ′ −1 = 1 (this notation is used in the entire paper).This provides the restriction p ∈ (1,2]. Additionally, we expect that the assertion (ii) of Theorem 1.1 can be improved to realanalyticity instead of smoothness.However, this would require to establish real-analytic dependence of the right-hand side of (1.1a) on f in the functional analytic framework considered in Section 3, which is much more involved than showing the smooth dependence (see [25,Proposition 5.1] for a related proof of real-analyticity).
p (R) that consists of functions for which the seminorm Here {τ ξ } ξ∈R is the group of right translations and The norm on W s p (R) is defined by For s < 0, W s p (R) is defined as the dual space of W −s p ′ (R).The following properties can be found e.g. in [39].
Outline.In Section 2 we introduce some multilinear singular integral operators and study their properties.These operators are then used in Section 3 to formulate (1.1) as a quasilinear evolution problem, cf.(3.1)-(3.2).Subsequently, we show in Theorem 3.5 that (3.1) is of parabolic type and we complete the section with the proof of Theorem 1.1.In Appendix A and Appendix B we prove some technical results used in the analysis.

Some singular integral operators
In this section we investigate a family of multilinear singular integral operators which play a key role in the analysis of the Muskat problem (and also of the Stokes problem [27]).Given n, m ∈ N and Lipschitz continuous functions a 1 , . . ., a m , b 1 , . . ., b n : R → R we set For brevity we write 2) The relevance of these operators in the context of (1.1) is enlightened by the fact that (1.1a) can be recast, at least at formal level, in a compact form as where B(f ) is defined by and f ′ := df /dx.A key observation that we exploit in our analysis is the quasilinear structure of the curvature operator.Indeed, it holds that where (2.5) We first recall the following result.
We point out that, given Lipschitz continuous functions a 1 , . . ., a m , a 1 , . . ., a m , b 1 , . . ., b n , we have The formula (2.6) was used to establish the Lipschitz continuity property (denoted by C1− ) in Lemma 2.1 and is also of importance for our later analysis.The strategy is as follows.In Lemma 2.4 we show that, given The last term on the right side of the previous identity vanishes if (n − 1) 2 + m 2 = 0. Otherwise, we use integration by parts and arrive at where, given x ∈ R and y = 0, we set We estimate the L p ′ -norm of the four terms on the right of (2.8) separately.Let q ∈ (p ′ , ∞) be defined as the solution to , and additionally we have W Hölder's inequality together with Lemma 2.1 (with p = 1/r) then yields (2.9) Minkowski's integral inequality, Hölder's inequality, and the property b (2.10) To estimate the integral term we choose again (R), 2 ≤ j ≤ n, Minkowski's integral inequality, Hölder's inequality, and a change of variables lead to and, similarly, (2.12) The desired claim follows now from (2.8)-(2.12).
Lemma 2.3 below is used to prove Lemma 2.4 (see also [2, Lemma 2.7] for a related result).
the desired estimate is immediate.
We are now in a position to prove that which proves (2.13) for r = 0. Let now r ∈ (0, 1 − 1/p).It remains to estimate the quantity Taking advantage of (2.6), we write Hence, and Lemma 2.1 (with p = p ′ ) yields Furthermore, using (2.7), Lemma 2.3 (with p = p ′ , t = s + 1 − 2/p, and and the embedding and, by similar arguments, The relations (2.14)-(2.17)lead to the desired estimate.The local Lipschitz continuity property follows from (2.6) and (2.13).

A functional analytic framework for the Muskat problem
In this section we take advantage of the mapping properties established in Section 2 and formulate the Muskat problem (1.1) as a quasilinear evolution problem in a suitable functional analytic setting, see (3.1)- (3.3).Afterwards, we show that the problem is of parabolic type.This enables us to employ theory for such evolution equations as presented in [6,28] to establish our main result in Theorem 1.1.The quasilinear structure of (1.1) is due to the quasilinearity of the curvature operator, the latter being established in Lemma 3.1.
Proof.The arguments are similar to those presented in [27, Appendix C] and are therefore omitted.
The Muskat problem (1.1) can thus be formulated as the evolution problem where with B introduced in (2.4).Arguing as in [27, Appendix C], we may infer from Lemma 2.5 that, given n, m ∈ N, p ∈ (1, 2], and s ∈ (1 + 1/p, 2), we have This property and Lemma 3.1 combined yield for all p ∈ (1, 2] and s ∈ (1 + 1/p, 2).Let p ∈ (1, 2], s ∈ (1 + 1/p, 2), and f ∈ W s p (R) be fixed in the remaining of this section.The analysis below is devoted to showing that the linear operator Φ(f ), viewed as an unbounded operator in W s−2 p (R) and with definition domain W s+1 p (R), is the generator of an analytic semigroup in L(W 2−s p (R)), which writes in the notation used in [7] as This property is established in Theorem 3.5 below and it identifies the quasilinear evolution problem (3.1) as being of parabolic type.To start, we note that π −1 B 0,0 = H, where H is the Hilbert transform, and therefore where (d 4 /dx 4 ) 3/4 denotes the Fourier multiplier with symbol m(ξ) := |ξ| 3 and (−d 2 /dx 2 ) 1/2 is the Fourier multiplier with symbol m(ξ) := |ξ|.We shall locally approximate the operator Φ(τ f ), with τ ∈ [0, 1], by certain Fourier multipliers A j,τ .Therefore we choose for each ε ∈ (0, 1) a so-called finite ε-localization family, that is a set

The real number x ε
N plays no role in the analysis below.To each finite ε-localization family we associate a second family {χ ε j : • supp χ ε j is an interval of length 3ε and with the same midpoint as supp π ε j , |j| ≤ N − 1.To each finite ε-localization family we associate a norm on W r p (R), r ∈ R, which is equivalent to the standard norm.
Proof.The claim follows from the fact that π ε j ∈ C ∞ (R) is a pointwise multiplier for W r p (R).
The next result is the main step in the proof of (3.4).
while, using some elementary arguments, we get Below we take advantage of (3.8) when considering the leading order term Step 1: The case |j| ≤ N − 1.Given |j| ≤ N − 1, we infer from Lemma A.2 and (3.8) that provided that ε is sufficiently small.Besides, we have where For ε sufficiently small to guarantee that , it follows from (1.7) (with r = s − 1), Lemma 2.5, and Finally, if ε is sufficiently small, we may argue as in the derivation of (3.9) to get (3.12) Gathering (3.10)-(3.12),we conclude that We now combine (3.7), (3.9), and (3.13) and obtain that As a final step we show that, if ε sufficiently small, then To start, we note that (3.17) Using (1.5) together with the identity χ ε j π ε j = π ε j , for ε sufficiently small we get Step 2: The case j = N .Similarly to (3.9), we obtain from Lemma A.5 and (3.8) that for all h ∈ W s+1 p (R) and τ ∈ [0, 1], provided that ε is sufficiently small.Moreover, where Because f ′ ∈ W s−1 p (R) vanishes at infinity, for ε sufficiently small to ensure that , it follows from (1.7) (with r = s − 1), Lemma 2.5, and (3.8) (if χ ε N f ′ is not identically zero, otherwise the estimate is trivial) that for all h ∈ W s+1 p (R) and τ ∈ [0, 1].It remains to show that for ε sufficiently small Arguing as in the first step (see (3.17)), we find in view of (3.16) that Using (1.5) and the fact that f ′ vanishes at infinity, for ε sufficiently small, we obtain This proves (3.23).The claim (3.6) for j = N follows now directly from (3.22) and (3.23).
We now consider a class of Fourier multipliers related to the multipliers from Theorem 3.3. .
We next present the proof of the main result.The arguments rely to a large extent on the theory of quasilinear parabolic problems presented in [3,4,6] (see also [28]).Besides, in order to obtain the parabolic smoothing properties, we additionally employ a parameter trick which was successfully applied also to other problems, cf., e.g., [9,11,18,33].
and the assumptions of [28,Theorem 1.1] are satisfied in the context of the Muskat problem (3.1).Applying [28, Theorem 1.1], we may conclude that (3.1) has for each Moreover, if the solution belongs to the set η∈(0,1) then it is also unique, cf.
Therewith the existence and uniqueness claim is proven.Moreover, the assertion (i) follows from the abstract theory (see the proof of [28,Theorem 1.1]).The parabolic smoothing property established at (ii) can be shown by arguing as in the more restrictive case considered in [25,Theorem 1.3].
Finally, in order to prove (iii), we assume that Using again Lemma 3.6 and arguing as above, we conclude that f : [0, T + ) → W s p (R) is uniformly continuous.Let now Choosing , and setting F θ := (F 0 , F 1 ) θ,p , θ ∈ {α ′ , β}, we have that Thus, we may apply again [28, Theorem 1.1] (iv) (α) to (3.1) and conclude that f can be extended to an interval [0, T + ) with T + > T + and such that Moreover, by (ii) (with (s, s) = (s, s)) we also have f ∈ C 1 ((0, T + ), W 3 p (R)), and this contradicts the maximality of f .This proves the claim (iii) and the argument is complete.
We finish the section by presenting a result used in the proof of Theorem 1.1.Lemma 3.6.Given M > 0, there exists a constant C = C(M ) such that In this proof the constants denoted by C depend only on M .Given f ∈ W s+1 p (R) with f W s p ≤ M , we have (3.30) Moreover, we infer from Lemma 2.1 that and it remains to prove that To this end we note that the L 2 -adjoint B * (f ) of B(f ) is identified by the relation We next show that Additionally, we may argue as in the proof of [25,Lemma 3.5] to infer in view of Lemma 2.1 that Invoking Lemma 2.4 (with r = 2 − s ∈ (0, 1 − 1/p)), we get that and since f This proves (3.34).Combining (3.33) and (3.34), a standard density argument leads us to If ε is sufficiently small, Lemma A.4 (with s = s − ρ) implies that The estimate (A.4) follows from (A.5) and (A.6).
In Lemma A.3 we gather some classical properties of mollifiers.
Lemma A.4 below provides the key estimate in the proof of Lemma A.2.
and f ∈ W s p (R) be given.For sufficiently small ε ∈ (0, 1) there exists a positive constant and we prove subsequently that, if ε sufficiently small, then In order to define the terms in (A.10), let {η δ } δ>0 be a mollifier as in Lemma A.3.We set is continuous, hence We thus conclude that We are now in a position to prove (1.7).

Lemma 2 . 1 .
Let p ∈ (1, ∞), n, m ∈ N, and let a 1 , . . ., a m , b 1 , . . ., b n : R → R be Lipschitz continuous.Then, there exists a constant C Lemma 2.2 below provides the key argument in the proof of Lemma 2.4.The desired mapping property B 0 n,m (f ) ∈ L(W s−2 p (R)) stated in Lemma 2.5, follows then from Lemma 2.4 via a duality argument.Lemma 2.5 and the fact that f

W 1
.1) together with (B.2) lead to the desired claim.