Mixing solutions for the Muskat problem with variable speed

We provide a quick proof of the existence of mixing weak solutions for the Muskat problem with variable mixing speed. Our proof is considerably shorter and extends previous results in Castro et al. (Mixing solutions for the Muskat problem, 2016, arXiv:1605.04822) and Förster and Székelyhidi (Comm Math Phys 363(3):1051–1080, 2018).


Introduction
The mathematical model for the evolution of two incompressible fluids moving in a porous medium, such as oil and water in sand, was introduced by Morris Muskat in his treatise [16], and is based on Darcy's law (see also [20,25]). In this paper we focus on the case of constant permeability under the action of gravity so that, after non-dimensionalizing, the equations describing the evolution of density ρ and velocity u are given by (see [17,7] and references therein) u`∇p "´p0, ρq , ρpx, 0q " ρ 0 pxq . (4) We assume that at the initial time the two fluids, with densities ρ`and ρ´, are separated by an interface which can be written as the graph of a function over the horizontal axis. That is, ρ 0 pxq " # ρ`x 2 ą z 0 px 1 q, ρ´x 2 ă z 0 px 1 q. (5) Thus, the interface separating the two fluids at the initial time is given by Γ 0 :" tps, z 0 psqq|s P Ru. Assuming that ρpx, tq remains in the form (5) for positive times, the system reduces to a non-local evolution problem for the interface Γ. If the sheet can be presented as a graph as above, one can show (see for example [7]) that the equation for zps, tq is given by B t zps, tq " ρ´´ρ2 π ż 8 8 pB s zps, tq´B s zpξ, tqqps´ξq ps´ξq 2`p zps, tq´zpξ, tqq 2 dξ.
Linearising (6) around the flat interface z " 0 reduces to B t f " ρ`´ρ2 HpB s f q, where H denotes the classical Hilbert transform. Thus one distinguishes the following cases: The case ρ`ą ρ´is called the unstable regime and amounts to the situation where the heavier fluid is on top. The case ρ`ă ρ´is called the stable regime. In the stable case, this equation is locally well-posed in Date: May 19, 2020. 1 H 3 pRq, see [7,5], whereas in the unstable case, we have an ill-posed problem, see [20,7], and there are no general existence results for (6) known. Thus, the description of (1)-(4) as a free boundary problem seems not suitable for the unstable regime. Indeed, as shown in experiments [25], in this regime the sharp interface seems to break down and the two fluids start to mix on a mesoscopic scale. In a number of applications [16,25], however, it is precisely this mixing process in the unstable regime which turns out to be highly relevant, calling for an amenable mathematical framework.
Mixing solutions and admissible subsolutions. A notion of solution, which allows for a meaningful existence theory and at the same time able to represent the physical features of the problem such as mixing, was introduced in [23]; it is based on the concept of subsolution, which appears naturally when considering stability of the nonlinear system (1)-(4) under weak convergence [10]. This point of view was pioneered by L. Tartar in the 1970s-80s in his study of compensated compactness [24], and experienced renewed interest in the past 10 years in connection with the theory of convex integration, applied to weak solutions in fluid mechanics [8,9,6,21]. In order to state the definition, we recall that after applying a simple affine change of variables we may assume |ρ˘| " 1. In particular, for the rest of the paper we will be concerned with the unstable case, so that ρ`"`1, ρ´"´1.
Furthermore, there exists a sequence of such admissible weak solutions pρ k , u k q such that ρ káρ as k Ñ 8.
In other wordsρ represents a sort of "coarse-grained, average" density. Recently this coarse-graining property of Theorem 1.2 was sharpened in [4] along the lines of [9] to the statement that, essentially,ρ denotes the average density not just on space-time balls B but also space-balls for every time t ą 0. Theorem 1.2 is based on a general and very robust Baire-category type argument (c.f. [10] as well as [23,Appendix]), and basically highlights the key observation that a central object in the study of unstable hydrodynamic interfaces is a suitably defined subsolution. In recent years this approach has been successfully applied in various contexts: for the incompressible Euler system in the presence of a Kelvin-Helmholtz instability [22,2,15], the density-driven Rayleigh-Taylor instability [13], the Muskat problem with fluids of different mobilities [14], as well as in the context of the compressible Euler system [11].
Evolution of the mixing region -the pseudointerface. An interesting phenomenon concerning the evolution of the coarse-grained interface was discovered by A. Castro, D. Cordoba and D. Faraco in [3]: for general (sufficiently smooth) initial curves Γ 0 the mixing (sub)solutions exhibit a two-scale dynamics. On a fast scale the sharp interface diffuses to a mixing zone Ω mix at some speed c ą 0, which has a stabilizing effect on the overall dynamics. On a slower scale the mixing zone itself begins to twist and evolve according to the now regularized evolution of the mid-section of Ω mix , called a pseudo-interface.
The authors in [3] showed that appearance of a mixing zone with speed c is compatible with the requirements of Definition 1.1 provided c ă 2 (for the flat initial condition c ă 2 was also the upper bound reached in [23], in agreement with the relaxation approach in [17]), and by a suitable ansatz exhibited the regularized evolution of the pseudo-interface as a nonlinear and nonlocal evolution equation of the form (see (1.11)-(1.12) in [3]) In a technical tour de force they were able to show well-posedness of (9) for initial data z 0 P H 5 pRq. Roughly speaking, the key point is that the linearization of (6), in Fourier space written as B tf " |ξ|f , is modified by the appearance of the mixing zone to which leads tof " p1`ct|ξ|q 1{cf 0 . The analysis of this equation was performed for constant c " 1 in [3], and recently extended to variable c " cpsq in [1] (in which case (10) has to be interpreted as a pseudodifferential equation) under certain restrictive conditions. In particular, the analysis in [1] applies under the condition that the range of mixing speed c 1 ď cpsq ď c 2 ă 2 is small: 0 ă c 1 ď c 2 ď c 1 1´c 1 ; furthermore, high regularity is required: It was observed in [12] that the ansatz of [3] is too restrictive since after a short initial time the macroscopic evolution of the pseudo-interface quite likely becomes non-universal. Thus, the authors in [12] replaced the equation (9) by a simple expansion in time upto second order, (11) zps, tq " z 0 psq`tz 1 psq`1 2 t 2 z 2 psq, and showed that a suitable choice of z 1 and z 2 leads to an evolution which is compatible with Definition 1.1 for any constant speed c P p0, 2q. More precisely, in the expansion (11) z 1 " B t z| t"0 :" u ν | t"0 is chosen as the normal velocity induced on the interface by (6) at time t " 0, whereas z 2 involves a non-local operator of the same type applied to z 0 plus a local curvature term: where κ 0 " κ 0 psq is the curvature of the initial interface Γ 0 . This expansion reveals an important difference to the approach in [3]: the regularity of the pseudointerface does not deteriorate with small c Ñ 0, in sharp contrast to the evolution in (10). From a physical point of view this is natural to expect, if one takes into account the scale separation in the two dynamics: once a mixing zone appears, the pseudointerface is a matter of arbitrary choice, the only relevant object for the system (7)-(8) being the set Ω mix . Thus, a coupling between the two dynamics should appear at most in terms of higher order fluctuations; indeed, a closer look [3,1] reveals that the deterioration of regularity observed in (10) in fact applies to f " B 4 s z. Motivated by this heuristic, in this short note we extend and simplify the analysis of [23] by (1) allowing for variable mixing speed c " cpsq within the whole range 0 ă inf R c ď sup R c ă 2, with no degeneration of regularity; (2) allowing for asymptotically vanishing mixing speed cpsq Ñ 0 as |s| Ñ 8 in case the initial interface z 0 is asymptotically horizontal. Moreover, our analysis shows that the expansion (11) above, obtained in [23], remains valid upto second order even in this generality, thus giving further evidence towards universality of the macroscopic evolution.
The main result. In this section we state the precise form of our main result.
Furthermore, we define the associated Hölder (semi-)norms as follows. We set and for any k P N }f }k ,α :" sup We denote by C k,α pRq :" tf P C k,α pRq : }f }k ,α ă 8u. Next, we describe the geometry of the coarse-grained evolution. Given a pseudointerface z : Rˆr0, T s Ñ R and mixing speed c : R Ñ p0, 8q define Ω˘ptq and Ω mix ptq as and set Ω˘" ď tą0 Ω˘ptq, Ω mix " ď tą0 Ω mix ptq. Theorem 1.3. Let z 0 psq " βs`z 0 psq with z 0 P C 3,α pRq for some 0 ă α ă 1 and β P R. Let c " cpsq ą 0 with sup s cpsq ă 2 and B s c P C α pRq. If inf s cpsq " 0, assume in addition that β " 0 and there exists c min ą 0 such that cpsq ě c min p1`|s| 2α{3 q´1. Then there exists T ą 0 such that there exists a pseudo-interface z P C 2 pr0, T s; C 1,α pRqq with z| t"0 " z 0 for which the mixing zone defined in (13) admits admissible subsolutions on r0, T s. In particular there exist infinitely many admissible weak solutions to (1)-(4) on r0, T s with mixing zone given by (13).
Observe that under the conditions in the theorem the function c has limits at infinity s Ñ˘8.
The paper is organised as follows. In Section 2 we show that an admissible subsolution exists provided certain smallness conditions are satisfied on the temporal expansion of the pseudo-interface -see Proposition 2.1. This section closely follows the construction in [23], in particular the construction of symmetric piecewise constant densities in [23,Section 5].
Then in Section 3 we obtain a regular expansion in time t for the normal component of the velocity across interfaces for arbitrary mixing speeds. Our key result in this section is Proposition 3.8, see also Remark 3.10 for a simplified statement. It is worth pointing out that validity of the expansion requires minimal smoothness assumptions on the pseudo-interface and, at variance with the approach in [1], does not degenerate as c Ñ 0 or c Ñ 2. Finally, in Section 4 we complete the proof of Theorem 1.3.
We remark in passing that if β ‰ 0, the statement of the theorem continues to hold provided the lower bound on cpsq is strengthened to cpsq ě c min p1`|s| α{2 q´1.

Subsolutions with variable mixing speed
We start by fixing N P N and setting (14) c Define the density ρpx, tq to be the piecewise constant function with i " 1, . . . , N´1. This definition of ρ already determines the velocity u by the kinematic part of (7), namely the Biot-Savart law (see Section 3 below) div u " 0, curl u "´B x 1 ρ. (16) Observe that ρ is piecewise constant, with jump discontinuities across 2N interfaces , z p˘iq ps, tq " zps, tq˘c i psqt.
It is well known [7] that, provided the interfaces are sufficiently regular, the solution u to (16) is then globally bounded, smooth in R 2 z Ť i Γ piq with well-defined traces on Γ˘, and the normal component (18) u piq ν ps, tq :" ups, z piq ps, tq, tq¨ˆ´B s z piq ps, tq 1i s continuous across the interfaces Γ piq for i "˘1, . . . ,˘N . Indeed, we will see in the next section that this is the case provided (19) zps, tq " βs`zps, tq, withzp¨, tq P C 1,α pRq, B s c P C α 1 pRq and some 0 ă α, α 1 ă 1 and β P R.
Our main result in this section is as follows: Proposition 2.1. Let zps, tq and cpsq be as in (19) withz P C 1 pr0, T s; C 1,α pRqq, B s c P C α 1 pRq and 0 ă cpsq ď c max @s for some c max ă 2N´1 N . Let ρ be defined by (15) and u the corresponding velocity field u according to (16), with normal traces as in (18). Assume that lim tÑ0 sup s 1 cpsqˇˇˇB t zps, tq´u p˘iq ν ps, tqˇˇ" 0 for all i , and furthermore, there exists M ą 0 such that (22) |B s zps, tq|, |B s cpsq| ď M c 1{2 psq @ s P R, t P r0, T s.
Then there exists T 1 P p0, T s and a vectorfield m : R 2ˆp 0, T 1 q Ñ R 2 such that pρ, u, mq is an admissible subsolution on r0, T 1 q with mixing zone Ω mix given by (13).
Next, for i ‰ 0 we make the choice that g piq is a function of x 1 , t only. Then, using the fact that g p˘N q " 0, we may use (26) to inductively define g p˘pN´1qq , g p˘pN´2qq , . . . , g p˘1q as In particular we obtain where op1q denotes terms going to zero uniformly in s as t Ñ 0 and we have used (20) in the last line. Since we also set B x 2 g piq " 0, and 0 ă cpsq ď c max ă 2N´1 N , we can deduce that (25) holds for sufficiently small t ą 0. Next, we turn our attention to g p0q on Ω p0q . For s P R, t P p0, T q and λ P r´1, 1s define (28)ĝps, λ, tq :" g p0q ps, zps, tq`λc 1 psqt, tq.
This concludes the proof.

The velocity u
In this section we analyse more closely the normal component of the velocity, given in (18), where the velocity u is the solution of the system (16) with piecewise constant density ρ given in (15). Following the computations in [7] and [12] we see that for any t ą 0 (30) u piq ν ps, tq " where the sum is over j "˘1, . . . ,˘N , the kernels Φ ij pξ, s, tq are defined as (31) Φ ij pξ, s, tq " ξ 2 ξ 2`p z piq ps, tq´z pjq ps´ξ, tqq 2 , and (32) z piq ps, tq " zps, tq`c i psqt.
with the convention c´ipsq "´c i psq, where c i psq is defined in (14) for i " 1, . . . , N . The principal value integral here refers to P V ş R " lim RÑ8 ş Ŕ R . Next, we recall the operator T Φ from [23], a weighted version of the Hilbert transform, defined for a weight function Φ " Φpξ, sq as Then (30) can be written as where we set Observe that in I ij for i ‰ j it again suffices to consider the principal value integral as above, with regularization as |ξ| Ñ 8. Nevertheless, for i " j also a principal value regularization at |ξ| Ñ 0 is necessary -see below in Lemmas 3.6-3.7. We next recall the following bound on T Φ on Hölder-spaces from [12], where for the weight we use the following norms: first of all we assume that Φ 8 psq :" lim |ξ|Ñ8 Φpξ, sq exists, Φp¨, sq P C 1 pRzt0uq, and set where we use the convention that }Φpξ,¨q} denotes a norm in the second argument only and }Φ} denotes a norm joint in both variables. In particular the Hölder-continuity of B k s Φ in both variables ξ, s is required in the norm Φ~k ,α . Accordingly, we define the spaces W 0 " tΦ P L 8 pR 2 q : Φ 8 and B ξ Φ exist, with~Φ~0 ă 8u, Then, the following version of the classical estimate on the Hilbert transform T 1 " H∇ on Hölder-spaces holds [12, Theorem 3.1]: Theorem 3.1. For any α ą 0, f P C 1,α pRq and Φ P W 0 we have Moreover, for any k P N, f P C k`1,α pRq and Φ P W k,α (38) }T Φ pf q}k ,α ď C~Φ~k ,α }f }k`1 ,α .
where the constant depends only on k and α.
In the following we analyse boundedness and continuity properties of the type of operators (33) arising in (34). This will ultimately enable us to derive an expansion in time for t Ñ 0 of the normal velocity components u piq ν in (30).
Then Φ P C l pr0, T s; W k,α q.
Proof. We start by introducing the following notation: for z " zps, tq define (39) Z " Zpξ, s, tq " zps, tq´zps´ξ, tq ξ " and furthermore, let Since K P C 8 pRq with derivatives of any order uniformly bounded on R, and since Φ " K˝Z, it follows easily from the chain rule that, for any j ď l, Concerning the far-field terms, note that Φ 8 " 1 1`β 2 , hencē Φ " β`Z p1`β 2 qp1`Z 2 q pzps´ξ, tq´zps, tqq " K β pZqpzps´ξ, tq´zps, tqq, where K β pxq " β`x p1`β 2 qp1`x 2 q is again a function with derivatives of all order uniformly bounded on R. Therefore the chain rule as above, together with the product rule, easily imply that, for any j ď l, B j tΦ P C k,α pR 2 q with sup t }B j tΦ p¨,¨, tq} C k,α pR 2 q ă 8.
so that, once again for any j ď l, ξB ξ pB j Using the notation introduced above in (39)-(40) we can write where Z pjq pξ, s, tq " z pjq ps, tq´z pjq ps´ξ, tq ξ , c ij psq " c i psq´c j psq.
Observe that c ii " 0 so that Lemma 3.2 applies to Φ ii , but the second term requires more care. In the next lemmata we address boundedness and continuity with respect to the functions z, c.
For the Lipschitz bound (43) we proceed entirely analogously, using the representation Next, we address continuity of the mapping c Þ Ñ KpZ`c ξ q at the singularity c " 0.
Proof. Let us fix s P R. Observe that if cpsq " 0, (45) is obvious; therefore we may assume in the following that cpsq ‰ 0, without loss of generality cpsq ą 0. We may then change variables in the integral defining T Φ f to obtain where Φ c pξ, sq :" Φpcpsqξ, sq. Note that where we denoted Z c pξ, sq " Zpcpsqξ, sq. Using that both K and K 2 are uniformly bounded, it follows that (48) |Φ c | ď C min`1, ξ´2˘.
Finally, we turn our attention to I ij defined in (35).
The conclusion follows from Lebesgue's dominated convergence theorem.
Lemma 3.7. There exists a constant C ą 1 such that the following holds. Let zpsq " βs`zpsq withz P C 1,α pRq, c P CpRq with cpsq ‰ 0, and let Then |I c psq| ď Cp1`}z} 1,α`} B s z} 2 0 q, and moreover Proof. Let us fix s P R and assume without loss of generality that cpsq ą 0. We perform the change of variables ξ Þ Ñ ξ cpsq to obtain where we denoted, as in the proof of Lemma 3.5, Z c pξ, sq " Zpcpsqξ, sq. Now we observe the following: using the uniform Lipschitz bound on K,ˇˇK´Z c`1 ξ¯´K pZ c qˇˇď 1 |ξ| , so that the first integrand is dominated by |ξ|´2 on |ξ| ą 1; furthermore, we have the lower bound implying that the third integrand is uniformly bounded. Finally, for the second integral we proceed as in the proof of Lemma 3.6: Collecting the estimates above we obtain the uniform bound on I c . In order to prove (50), we write KpZ c`1 ξ q´KpB s z`1 ξ q dξ ξ .
With Lemmas 3.2-3.7 we are now in a position to obtain an expansion in time as t Ñ 0 for the normal velocities u piq ν for sufficiently regular z, c.
where z 1 psq " B t zps, 0q, Ψ 0 pξ, sq " B t Φpξ, s, 0q and Proof. Using the representation (34) we write The estimate (51) then follows from applying Lemma 3.4, Lemma 3.3 and Lemma 3.7. Next, we calculate where Z 0 pξ, sq " z 0 psq´z 0 ps´ξq ξ , Z piq 1 pξ, sq " B t | t"0 Z piq pξ, s, tq and c ij " c i´cj . Recalling the convention that c´i "´c i , we see that On the first two terms we can argue as in Lemma 3.3 and Lemma 3.4, whereas Lemma 3.5 applies to the last term. Finally, Lemma 3.7 implies that |I ij´Iji | ď Ct 1`α . This concludes the proof of (53). Remark 3.9. In the statement of Proposition 3.8 we left the expression forc in the general form (54), which is valid for any choice of 0 ă c 1 psq ă c 2 psq ă¨¨¨ă c N psq with c´ipsq "´c i psq and B s c i P C α pRq. The explicit choice of c i psq in (14) leads to the following simplified expression: 2N`1 3N cpsq.
Remark 3.10. It is instructive to compare the expansions in Proposition 3.8 with the expansion of the sharp interface evolution in (6). In particular, recall that the right hand side of (6) is the normal component of the velocity at the interface. Let us denote this by v ν , so that v ν " T Φ z, so that also (22) holds. It remains to verify (21). To this end our aim is to choose ψ k in such a way that (59) ż I k B t zps, tq´1 2N ÿ i u piq ν ps, tq ds " 0 @ t P r0, T s, k " 1, 2.
To complete the proof it therefore remains to choose ψ k to ensure (59). Let where dependence on ψ " pψ 1 , ψ 2 q appears in the implicit dependence of u piq ν on z defined in (56) via (30)-(32). Then, recalling the choice of f k , (59) is equivalent to the ODE system ψ 1 k ptq " h k pt, ψptqq, k " 1, 2 with initial condition ψ k p0q " 0. Using estimate (43) in Lemma 3.3 we verify that x Þ Ñ hpt, xq is uniformly Lipschitz continuous. Therefore the Cauchy-Lipschitz theorem is applicable and yields a unique solution ψ : r0, T s Ñ R 2 .