Abstract
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 [22] and is based on Darcy’s law (see also [25, 30]). In this paper, we focus on the case of constant permeability under the action of gravity so that, after nondimensionalizing, the equations describing the evolution of density \(\rho \) and velocity u are given by (see [9, 24] and references therein)
We assume that at the initial time the two fluids, with densities \(\rho ^+\) and \(\rho ^\), are separated by an interface which can be written as the graph of a function over the horizontal axis is,
Thus, the interface separating the two fluids at the initial time is given by \(\Gamma _0 := \{(s,z_0(s)) s \in \mathbb {R}\}\). Assuming that \(\rho (x,t)\) remains in the form (5) for positive times, the system reduces to a nonlocal evolution problem for the interface \(\Gamma \). If the sheet can be presented as a graph as above, one can show (see for example [9]) that the equation for z(s, t) is given by
Linearising (6) around the flat interface \(z=0\) reduces to \(\partial _tf=\frac{\rho ^+\rho ^}{2}{\mathcal {H}}(\partial _sf)\), where \({\mathcal {H}}\) denotes the classical Hilbert transform. Thus one distinguishes the following cases: The case \(\rho ^+ > \rho ^\) is called the unstable regime and amounts to the situation where the heavier fluid is on top. The case \(\rho ^+< \rho ^\) is called the stable regime. In the stable case, this equation is locally wellposed in \(H^3(\mathbb {R})\), see [7, 9], as well as [1, 2, 19, 23] for recent developments. In the unstable case, however, which is our focus in this article, we have an illposed problem (see [9, 25]), 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 [30], 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 [22, 30], 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 [28]; it is based on the concept of subsolution, which appears naturally when considering stability of the nonlinear system (1)–(4) under weak convergence [13]. This point of view was pioneered by L. Tartar in the 1970s1980s in his study of compensated compactness [29], 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, 11, 12, 26]. In order to state the definition, we recall that after applying a simple affine change of variables we may assume \(\rho ^\pm =1\). In particular, for the rest of the paper we will be concerned with the unstable case, so that
With this normalization subsolutions are defined as follows (c.f. [28, Definition 4.1]).
Definition 1.1
Let \(T > 0\). We call a triple \((\rho ,u,m) \in L^{\infty }(\mathbb {R}^2 \times [0,T))\) an admissible subsolution of (1)–(4) if there exist open domains \(\Omega ^{\pm }, \Omega _\mathrm{mix}\) with \(\overline{\Omega ^+} \cup \overline{\Omega ^} \cup \Omega _\mathrm{mix} = \mathbb {R}^2\times [0,T)\) such that

(i)
The system
$$\begin{aligned} \begin{aligned} \partial _t \rho + \text {div }m&= 0\\ \text {div }u&= 0\\ \text {curl }u&= \partial _{x_1} \rho \\ \rho \vert _{t=0}&= \rho _0 \end{aligned} \end{aligned}$$(7)holds in the sense of distributions in \(\mathbb {R}^2 \times [0,T)\);

(ii)
The pointwise inequality
$$\begin{aligned} \left m\rho u + \frac{1}{2} (0,1\rho ^2) \right \le \frac{1}{2} \left( 1\rho ^2 \right) , \end{aligned}$$(8)holds almost everywhere;

(iii)
\(\rho (x,t) = 1\) in \(\Omega ^{+}\cup \Omega ^\);

(iv)
In \(\Omega _\mathrm{mix}\), the triple \((\rho ,u,m)\) is continuous and (8) holds with a strict inequality.
Observe that whenever \(\rho =\pm 1\) for an admissible subsolution, then by (8) the system (7) reduces to (1)–(3). Conversely, in the set \(\Omega _\mathrm{mix}\) the subsolution \(\rho \) represents the coarsegrained density of a microscopically mixed state. More precisely, we have the following theorem from [5, 28]:
Theorem 1.2
Suppose there exists an admissible subsolution \(({\bar{\rho }},{\bar{u}},{\bar{m}})\) to (1)–(4). Then there exist infinitely many admissible weak solutions \((\rho ,u)\) with the following additional mixing property: For any \(r>0\), \(0<t_0<T\) and \(x_0\in \mathbb {R}^2\) such that \(B:=B_r(x_0,t_0)\subset \Omega _\mathrm{mix}\), both sets \(\{(x,t)\in B:\,\rho (x,t)=\pm 1\}\) have strictly positive Lebesgue measure.
Furthermore, there exists a sequence of such admissible weak solutions \((\rho _k,u_k)\) such that \(\rho _k\overset{*}{\rightharpoonup }{\bar{\rho }}\) as \(k\rightarrow \infty \).
In other words \({\bar{\rho }}\) represents a sort of “coarsegrained, average” density. Recently, this coarsegraining property of Theorem 1.2 was sharpened in [6] along the lines of [12] to the statement that, essentially, \({{\bar{\rho }}}\) denotes the average density not just on spacetime balls B but also spaceballs for every time \(t>0\).
Theorem 1.2 is based on a general and very robust Bairecategory type argument (c.f. [13] as well as [28, 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 [4, 21, 27], the densitydriven Rayleigh–Taylor instability [16, 17], the Muskat problem with fluids of different mobilities [20], as well as in the context of the compressible Euler system [10]. This type of research is driven by the theoretical expectation, underpinned by recent numerical work [14, 18], that whereas the microscopic evolution in such problems is illposed, certain statistical quantities of interest obey a welldefined, “universal” macroscopic evolution, thus exhibiting universal features among the very large number of weak solutions.
Evolution of the mixing region: the pseudointerface
An interesting phenomenon concerning the evolution of the coarsegrained interface was discovered by A. Castro, D. Cordoba and D. Faraco in [5]: for general (sufficiently smooth) initial curves \(\Gamma _0\), the mixing (sub)solutions exhibit a twoscale dynamics. On a fast scale, the sharp interface diffuses to a mixing zone \(\Omega _\mathrm{mix}\) at some speed \(c>0\), 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 midsection of \(\Omega _\mathrm{mix}\), called a pseudointerface.
The authors in [5] showed that appearance of a mixing zone with speed c is compatible with the requirements of Definition 1.1 provided \(c<2\) (for the flat initial condition \(c<2\) was also the upper bound reached in [28], in agreement with the relaxation approach in [24]) and by a suitable ansatz exhibited the regularized evolution of the pseudointerface as a nonlinear and nonlocal evolution equation of the form (see (1.11)–(1.12) in [5])
In a technical tour de force, they were able to show wellposedness of (9) for initial data \(z_0\in H^5(\mathbb {R})\). Roughly speaking, the key point is that the linearization of (6), in Fourier space written as \(\partial _t{\hat{f}}=\xi {\hat{f}}\), is modified by the appearance of the mixing zone to
which leads to \({\hat{f}}=(1+ct\xi )^{1/c}{{\hat{f}}}_0\). The analysis of this equation was performed for constant \(c=1\) in [5], and recently extended to variable \(c=c(s)\) in [3] (in which case (10) has to be interpreted as a pseudodifferential equation) under certain restrictive conditions. In particular, the analysis in [3] applies under the condition that the range of mixing speed \(c_1\le c(s)\le c_2<2\) is small: \(0<c_1\le c_2\le \frac{c_1}{1c_1}\); furthermore, high regularity is required: \(c\in W^{k,\infty }(\mathbb {R})\) with \(k>c^{1}\) (c.f. [3, Definition 4]).
It was observed in [15] that the ansatz of [5] is too restrictive since after a short initial time the macroscopic evolution of the pseudointerface quite likely becomes nonuniversal. Thus, the authors in [15] replaced the equation (9) by a simple expansion in time up to second order,
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 \(c\in (0,2)\). More precisely, in the expansion (11) \(z_1=\partial _tz_{t=0}:=u_{\nu }_{t=0}\) is chosen as the normal velocity induced on the interface by (6) at time \(t=0\), whereas \(z_2\) involves a nonlocal operator of the same type applied to \(z_0\) plus a local curvature term:
where \(\kappa _0=\kappa _0(s)\) is the curvature of the initial interface \(\Gamma _0\). This expansion reveals an important difference to the approach in [5]: the regularity of the pseudointerface does not deteriorate with small \(c\rightarrow 0\), 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 \(\Omega _\mathrm{mix}\). Thus, a coupling between the two dynamics should appear at most in terms of higher order fluctuations; indeed, a closer look [3, 5] reveals that the deterioration of regularity observed in (10) in fact applies to \(f=\partial _s^4z\).
Motivated by this heuristic, in this short note we extend and simplify the analysis of [15] by

(1)
allowing for variable mixing speed \(c=c(s)\) within the whole range \(0<\inf _{\mathbb {R}}c\le \sup _{\mathbb {R}}c<2\), with no degeneration of regularity;

(2)
allowing for asymptotically vanishing mixing speed \(c(s)\rightarrow 0\) as \(s\rightarrow \infty \) in case the initial interface \(z_0\) is asymptotically horizontal.
Moreover, our analysis shows that the expansion (11) above, obtained in [15], remains valid up to second order even in this generality, thus giving further evidence towards universality, in the sense explained above, of the macroscopic evolution.
The main result
In this section, we state the precise form of our main result.
Our assumption on the initial datum is that the initial interface is asymptotically flat with some given slope \(\beta \in \mathbb {R}\), i.e. \(\rho _0\) is given by (5) with
for some \({\tilde{z}}_0\) with sufficiently fast decay at infinity, using—as in [15]—the following weighted Hölder norms: for any \(0<\alpha <1\) set
Furthermore, we define the associated Hölder (semi)norms as follows. We set
and for any \(k\in \mathbb {N}\)
We denote by \(C_*^{k,\alpha }(\mathbb {R}):=\{f\in C^{k,\alpha }(\mathbb {R}):\,\Vert f\Vert _{k,\alpha }^*<\infty \}\).
Next, we describe the geometry of the coarsegrained evolution. Given a pseudointerface \(z:\mathbb {R}\times [0,T] \rightarrow \mathbb {R}\) and mixing speed \(c:\mathbb {R}\rightarrow (0,\infty )\) define \(\Omega ^{\pm }(t)\) and \(\Omega _\mathrm{mix}(t)\) as
and set
Theorem 1.3
Let \(z_0(s) = \beta s + {\overline{z}}_0(s)\) with \({\overline{z}}_0 \in C^{3,\alpha }_*(\mathbb {R})\) for some \(0<\alpha <1\) and \(\beta \in \mathbb {R}\). Let \(c=c(s)>0\) with \(\sup _sc(s)<2\) and \(\partial _sc\in C^{\alpha }_*(\mathbb {R})\). If \(\inf _sc(s)=0\), assume in addition that \(\beta =0\) and there exists \(c_\mathrm{min}>0\) such that
Then there exists \(T>0\) such that there exists a pseudointerface \(z\in C^2([0,T];C^{1,\alpha }(\mathbb {R}))\) with \(z_{t=0}=z_0\) for which the mixing zone defined in (13) admits admissible subsolutions on [0, T]. In particular, there exist infinitely many admissible weak solutions to (1)–(4) on [0, T] with mixing zone given by (13).
Observe that under the conditions in the theorem the function c has limits at infinity \(s\rightarrow \pm \infty \).
The paper is organised as follows. In Sect. 2, we show that an admissible subsolution exists provided certain smallness conditions are satisfied on the temporal expansion of the pseudointerface—see Proposition 2.1. This section closely follows the construction in [15], in particular the construction of symmetric piecewise constant densities in [15, Section 5].
Then in Sect. 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 pseudointerface and, at variance with the approach in [3], does not degenerate as \(c\rightarrow 0\) or \(c\rightarrow 2\). Finally, in Sect. 4 we complete the proof of Theorem 1.3.
We remark in passing that if \(\beta \ne 0\), the statement of the theorem continues to hold provided the lower bound on c(s) is strengthened to
The proof of this requires minor modifications in Proposition 2.1, in particular replacing the term \(c^{3/2}\) in (21) by \(c^2\). As such modifications unnecessarily complicate the presentation without added value, we chose not to include the details here.
Subsolutions with variable mixing speed
We start by fixing \(N\in \mathbb {N}\) and setting
Define the density \(\rho (x,t)\) to be the piecewise constant function
with \(i=1,\dots ,N1\). This definition of \(\rho \) already determines the velocity u by the kinematic part of (7), namely the Biot–Savart law (see Sect. 3 below)
Observe that \(\rho \) is piecewise constant, with jump discontinuities across 2N interfaces
It is well known [9] that, provided the interfaces are sufficiently regular, the solution u to (16) is then globally bounded, smooth in \(\mathbb {R}^2\setminus \bigcup _{i}\Gamma ^{(i)}\) with welldefined traces on \(\Gamma ^{\pm }\), and the normal component
is continuous across the interfaces \(\Gamma ^{(i)}\) for \(i=\pm 1,\dots ,\pm N\).
Indeed, we will see in the next section that this is the case provided
with \({{\tilde{z}}}(\cdot ,t) \in C^{1,\alpha }_*(\mathbb {R})\), \(\partial _sc\in C^{\alpha '}_*(\mathbb {R})\) and some \(0<\alpha ,\alpha '<1\) and \(\beta \in \mathbb {R}\).
Our main result in this section is as follows:
Proposition 2.1
Let z(s, t) and c(s) be as in (19) with \({{\tilde{z}}}\in C^1([0,T];C^{1,\alpha }_*(\mathbb {R}))\), \(\partial _sc\in C^{\alpha '}_*(\mathbb {R})\) and \(0<c(s)\le c_\mathrm{max}\) \(\forall s\) for some \(c_\mathrm{max}<2\). Let \(N\in \mathbb {N}\) be the smallest integer such that \(c_\mathrm{max}<\frac{2N1}{N}\) and set \(\rho \) to be defined by (15) and u the corresponding velocity field u according to (16), with normal traces as in (18). Assume that
and furthermore, there exists \(M>0\) such that
Then there exists \(T'\in (0,T]\), depending on \(c_\mathrm{max}<2\), \(\tilde{z},c\) as well as on the rate of convergences in (20)–(21), and a vectorfield \(m:\mathbb {R}^2\times (0,T')\rightarrow \mathbb {R}^2\) such that \((\rho ,u,m)\) is an admissible subsolution on \([0,T')\) with mixing zone \(\Omega _\mathrm{mix}\) given by (13).
Remark 2.2
Formally, in the proof below the time of existence \(T'\) also depends on N, but this dependency is already taken into account with \(c_\mathrm{max}\).
Proof
The proof follows closely the proof of Theorem 5.1 in [15], adapted here to the variable speed setting. Set
for some \(\gamma =\gamma (x,t)\), with \(\gamma \equiv 0\) in \(\Omega ^{\pm }\). Then (8) amounts to the condition
whereas (7) is equivalent to \(\text {div }\gamma =0\) in \(\Omega _\mathrm{mix}\setminus \bigcup _{i=1}^N(\Gamma ^{(i)}\cup \Gamma ^{(i)})\) together with \(2(N1)\) jump conditions
for \(i=\pm 1,\dots ,\pm N\), where \([\cdot ]_{\Gamma ^{(i)}}\) denotes the jump on \(\Gamma ^{(i)}\). Observe that
with connected open sets \(\Omega ^{(i)}\) defined as
for \(i=1,\dots ,N1\) and
The divergencefree condition is then taken care by setting
where \(g^{(N)}=g^{{(N)}}=0\) and \(g^{(i)}\in C^1(\overline{\Omega ^{(i)}})\) for \(i=(N1)\dots (N1)\) are to be determined. Then, (8) amounts to the conditions
and (23) reduces to conditions on the tangential derivatives on each interface: for any \(i=1,\dots ,N\) we require on \(\Gamma ^{(\pm i)}\)
with
and
Next, for \(i\ne 0\) we make the choice that \(g^{(i)}\) is a function of \(x_1,t\) only. Then, using the fact that \(g^{(\pm N)}=0\), we may use (26) to inductively define \(g^{(\pm (N1))}\), \(g^{(\pm (N2))},\dots ,g^{(\pm 1)}\) as
In particular, we obtain
where o(1) denotes terms going to zero uniformly in s as \(t\rightarrow 0\) and we have used (20) in the last line. Since we also set \(\partial _{x_2}g^{(i)}=0\), and \(0<c(s)\le c_\mathrm{max}< \tfrac{2N1}{N}\), we can deduce that (25) holds for sufficiently small \(t>0\).
Next, we turn our attention to \(g^{(0)}\) on \(\Omega ^{(0)}\). For \(s\in \mathbb {R}\), \(t\in (0,T)\) and \(\lambda \in [1,1]\) define
In order to satisfy (26), we set
and, more generally, for \(\lambda \in [1,1]\)
Then
where we have used (20)–(21). By differentiating (28) with respect to s and \(\lambda \) and using (20)–(21), we deduce
where we have used (22) in the last line. Consequently
Since \(\tfrac{N}{2N1}c_\mathrm{max}<1\), we deduce that
This concludes the proof. \(\square \)
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 \(\rho \) given in (15). Following the computations in [9] and [15] we see that for any \(t>0\)
where the sum is over \(j=\pm 1,\dots ,\pm N\), the kernels \(\Phi _{ij}(\xi ,s,t)\) are defined as
and
with the convention \(c_{i}(s)=c_i(s)\), where \(c_i(s)\) is defined in (14) for \(i=1,\dots ,N\). The principal value integral here refers to \(PV\int _{\mathbb {R}} = \underset{R\rightarrow \infty }{lim}\int _{R}^R\). Next, we recall the operator \(T_\Phi \) from [15], a weighted version of the Hilbert transform, defined for a weight function \(\Phi =\Phi (\xi ,s)\) as
Then (30) can be written as
where we set
Observe that in \(I_{ij}\) for \(i\ne j\) it again suffices to consider the principal value integral as above, with regularization as \(\xi \rightarrow \infty \). Nevertheless, for \(i=j\) also a principal value regularization at \(\xi \rightarrow 0\) is necessary—see below in Lemmas 3.6–3.7.
We next recall the following bound on \(T_{\Phi }\) on Hölderspaces from [15], where for the weight we use the following norms: first of all we assume that \(\Phi ^{\infty }(s):=\lim _{\xi \rightarrow \infty }\Phi (\xi ,s)\) exists, \(\Phi (\cdot ,s)\in C^1(\mathbb {R}\setminus \{0\})\), and set
We introduce the norms
where we use the convention that \(\Vert \Phi (\xi ,\cdot )\Vert \) denotes a norm in the second argument only and \(\Vert \Phi \Vert \) denotes a norm joint in both variables. In particular, the Höldercontinuity of \(\partial _s^k\Phi \) in both variables \(\xi ,s\) is required in the norm \(\!\!\Phi \!\!\!_{k,\alpha }\). Accordingly, we define the spaces
Then, the following version of the classical estimate on the Hilbert transform \(T_1={\mathcal {H}}\nabla \) on Hölderspaces holds [15, Theorem 3.1]:
Theorem 3.1
For any \(\alpha >0\), \(f\in C^{1,\alpha }_*(\mathbb {R})\) and \(\Phi \in {\mathcal {W}}^0\) we have
Moreover, for any \(k\in \mathbb {N}\), \(f\in C_*^{k+1,\alpha }(\mathbb {R})\) and \(\Phi \in {\mathcal {W}}^{k,\alpha }\)
where the constant depends only on k and \(\alpha \).
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 \(t\rightarrow 0\) of the normal velocity components \(u^{(i)}_\nu \) in (30).
Lemma 3.2
Let \(z(s,t)=\beta s+{{\tilde{z}}}(s,t)\) with \({{\tilde{z}}}\in C^{l}([0,T];C^{k+1,\alpha }(\mathbb {R}))\) for some \(l,k\in \mathbb {N}\), \(0<\alpha <1\), \(\beta \in \mathbb {R}\) and \(T<\infty \), and let
Then \(\Phi \in C^{l}([0,T];{\mathcal {W}}^{k,\alpha })\).
Proof
We start by introducing the following notation: for \(z=z(s,t)\) define
and furthermore, let
Since \(K\in C^{\infty }(\mathbb {R})\) with derivatives of any order uniformly bounded on \(\mathbb {R}\), and since \(\Phi =K\circ Z\), it follows easily from the chain rule that, for any \(j\le l\), \(\partial _t^j\Phi \in C^{k,\alpha }(\mathbb {R}^2)\) with
Concerning the farfield terms, note that \(\Phi ^{\infty }=\frac{1}{1+\beta ^2}\), hence
where \(K_{\beta }(x)=\frac{\beta +x}{(1+\beta ^2)(1+x^2)}\) is again a function with derivatives of all order uniformly bounded on \(\mathbb {R}\). Therefore the chain rule as above, together with the product rule, easily imply that, for any \(j\le l\), \(\partial _t^j{{\bar{\Phi }}}\in C^{k,\alpha }(\mathbb {R}^2)\) with
Similarly, \(\xi \partial _\xi \Phi =K'(Z)\xi \partial _\xi Z\), where
so that, once again for any \(j\le l\), \(\xi \partial _\xi (\partial _t^j\Phi )=\partial _t^j(\xi \partial _\xi \Phi )\in C^{k,\alpha }(\mathbb {R}^2)\) with
We deduce that \(\partial _t^j\Phi \in {\mathcal {W}}^{k,\alpha }\) with
as required. \(\square \)
Using the notation introduced above in (39)–(40), we can write
where
Observe that \(c_{ii}=0\) so that Lemma 3.2 applies to \(\Phi _{ii}\), but the second term requires more care. In the next lemmata, we address boundedness and continuity with respect to the functions z, c.
Lemma 3.3
There exists a constant \(C>1\) such that the following holds. Let \(z(s)=\beta s+{{\tilde{z}}}(s)\) with \({{\tilde{z}}}\in C^1(\mathbb {R})\), \(c\in C(\mathbb {R})\), and let
where \(K(Z)=(1+Z^2)^{1}\) as in (40) and \(Z(\xi ,s)=\frac{z(s)z(s\xi )}{\xi }\). Then \(\Phi \in {\mathcal {W}}^0\), with
Furthermore, if \(\Phi _1,\Phi _2\) are defined as above with \(\tilde{z}_1,{{\tilde{z}}}_2\in C^1(\mathbb {R})\), then
Proof
Using that \(K\le 1\), we deduce \(\sup _{\xi ,s}\Phi \le 1\). Moreover, since also \(K'\le 1\), we have
Similarly, for \(\xi \ge 1\) we also have
The estimate (42) follows.
For the Lipschitz bound (43), we proceed entirely analogously, using the representation
and the bound \(Z_1Z_2\le \Vert \partial _s{{\tilde{z}}}_1\partial _s\tilde{z}_2\Vert _0\). \(\square \)
Next, we address continuity of the mapping \(c\mapsto K(Z+\frac{c}{\xi })\) at the singularity \(c=0\).
Lemma 3.4
There exists a constant \(C>1\) such that the following holds. Let \(z(s)=\beta s+{{\tilde{z}}}(s)\) with \({{\tilde{z}}}\in C(\mathbb {R})\), \(c\in C(\mathbb {R})\), and let
where \(K(Z)=(1+Z^2)^{1}\) as in (40) and \(Z(\xi ,s)=\frac{z(s)z(s\xi )}{\xi }\). Then, for any f with \(\partial _sf\in C^{\alpha }_*(\mathbb {R})\) we have
Proof
Using the fact that \(K,K'\le 1\), we have
Next, recalling the definition of \(T_{\Phi }f\) from (33) we have
\(\square \)
Lemma 3.5
There exists a constant \(C>1\) such that the following holds. Let \(z(s)=\beta s+{{\tilde{z}}}(s)\) with \({{\tilde{z}}}\in C^{1,\alpha }(\mathbb {R})\), \(c\in C(\mathbb {R})\), and let
where \(K(Z)=(1+Z^2)^{1}\) as in (40) and \(Z(\xi ,s)=\frac{z(s)z(s\xi )}{\xi }\). Then, for any \(f\in C^{2,\alpha }_*(\mathbb {R})\) and any \(s\in \mathbb {R}\) we have
where
Proof
Let us fix \(s\in \mathbb {R}\). Observe that if \(c(s)=0\), (45) is obvious; therefore, we may assume in the following that \(c(s)\ne 0\), without loss of generality \(c(s)>0\). We may then change variables in the integral defining \(T_{\Phi }f\) to obtain
where \(\Phi _c(\xi ,s):=\Phi (c(s)\xi ,s)\). Note that
where we denoted \(Z_c(\xi ,s)=Z(c(s)\xi ,s)\). Using that both K and \(K''\) are uniformly bounded, it follows that
Furthermore, letting
we also obtain, using that both K and \(K''\) are uniformly Lipschitz, that
Since \(\xi \mapsto \Phi _0(\xi ,s)\) is a rational function of \(\xi \), its precise integral in \(\xi \) may be calculated by elementary methods (as done in [15]), leading to
Thus \(\frac{1}{2\pi }\int _\mathbb {R}\Phi _0(\xi ,s)\,\mathrm{d}\xi =\sigma (\partial _sz(s))\). We then write (47) as
Using the bounds (48)–(49) on the two integrals, we then deduce (45). \(\square \)
Finally, we turn our attention to \(I_{ij}\) defined in (35).
Lemma 3.6
There exists a constant \(C>1\) such that the following holds. Let \(z(s,t)=\beta s+{{\tilde{z}}}(s,t)\) with \({{\tilde{z}}}\in C([0,T];C^{1,\alpha }(\mathbb {R}))\), and let
Then \(I\in C([0,T];C(\mathbb {R}))\) with
Proof
Using that \(PV\int _{\xi <1}\frac{\xi }{\xi }=PV\int _{\xi >1}\frac{\mathrm{d}\xi }{\xi }=0\) and using the notation introduced in (39)–(40), we have
Observe that the integrands in these two integrals are uniformly integrable. Indeed, \(K:\mathbb {R}\rightarrow \mathbb {R}\) is uniformly Lipschitz continuous, so that for the first integral we may use \(Z\partial _sz\le [\partial _sz]_{\alpha }\xi ^\alpha \) and for the second integral \(Z\beta \le 2\Vert {{\tilde{z}}}\Vert _{0}\xi ^{1}\). Thus
The conclusion follows from Lebesgue’s dominated convergence theorem. \(\square \)
Lemma 3.7
There exists a constant \(C>1\) such that the following holds. Let \(z(s)=\beta s+{{\tilde{z}}}(s)\) with \({{\tilde{z}}}\in C^{1,\alpha }(\mathbb {R})\), \(c\in C(\mathbb {R})\) with \(c(s)\ne 0\), and let
Then
and moreover
Proof
Let us fix \(s\in \mathbb {R}\) and assume without loss of generality that \(c(s)>0\). We perform the change of variables \(\xi \mapsto \frac{\xi }{c(s)}\) to obtain
where we denoted, as in the proof of Lemma 3.5, \(Z_c(\xi ,s)=Z(c(s)\xi ,s)\). Now we observe the following: using the uniform Lipschitz bound on K,
so that the first integrand is dominated by \(\xi ^{2}\) on \(\xi >1\); 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
For the first term, we use uniform boundedness of \(K''\) to obtain the bound
which suffices to bound the integral. For the second and third terms, we obtain analogously
which again allows to bound the integrals. The estimate (50) follows. \(\square \)
With Lemmas 3.2–3.7, we are now in a position to obtain an expansion in time as \(t\rightarrow 0\) for the normal velocities \(u^{(i)}_\nu \) for sufficiently regular z, c.
Proposition 3.8
Let \(z(s,t)=\beta s+{{\tilde{z}}}(s,t)\) with \({{\tilde{z}}}\in C^1([0,T];C^{1,\alpha }_*(\mathbb {R}))\), \(c>0\) with \(\partial _sc\in C^{\alpha }_*(\mathbb {R})\), and define \(c_i, \Phi _{ij}, z^{(i)}\) as in (14), (31) and (32). Then there exists a constant \(C_{z,c}\) depending on \(\Vert \tilde{z}\Vert _{1,\alpha }^*\), \(\Vert \partial _t{{\tilde{z}}}\Vert _{1,\alpha }^*\), \(\Vert \partial _sc\Vert _{\alpha }^*\), and N, such that
where \(z_0(s)=z(s,0)\), \(\Phi _0(\xi ,s)=\Phi (\xi ,s,0)\) and
Furthermore
where \(z_1(s)=\partial _tz(s,0)\), \(\Psi _0(\xi ,s)=\partial _t\Phi (\xi ,s,0)\) and
Proof
Using the representation (34) and the identity \(\sum _jc_j=0\) (recall that here and below \(\sum _j\) means summing over \(1\le j\le N\), and that \(c_{j}=c_j\)), we write
For the first term in the summation, we can argue as in the proof of Lemma 3.4: indeed, \(\Phi _{ij}\tfrac{1}{2}\Phi _0=K(Z^{(j)}+\tfrac{tc_{ij}}{\xi })K(Z)\) and \(Z^{(j)}+\tfrac{tc_{ij}}{\xi }Z\le 2t\tfrac{\Vert c\Vert _0}{\xi }\). For the second term in the summation, we apply Lemma 3.3 and Theorem 3.1, and for the third term Lemma 3.7. This way we arrive at estimate (51).
Next, we calculate
Now, let us write
where \(Z_0(\xi ,s)=\frac{z_0(s)z_0(s\xi )}{\xi }\), \(Z_1^{(i)}(\xi ,s)=\partial _t_{t=0}Z^{(i)}(\xi ,s,t)\) and \(c_{ij}=c_ic_j\). Recalling the convention that \(c_{i}=c_i\), we see that
On the first two terms, we can argue as in Lemmas 3.3 and 3.4, whereas Lemma 3.5 applies to the last term.
Furthermore, we have
and for the second term on the RHS, we argue again as above, using the strategy of Lemma 3.4. Finally, Lemma 3.7 implies that
Collecting all the terms, we then conclude the proof of (53). \(\square \)
Remark 3.9
In the statement of Proposition 3.8, we left the expression for \({\bar{c}}\) in the general form (54), which is valid for any choice of \(0<c_1(s)<c_2(s)<\dots <c_N(s)\) with \(c_{i}(s)=c_i(s)\) and \(\partial _sc_i\in C^\alpha _*(\mathbb {R})\). The explicit choice of \(c_i(s)\) in (14) leads to the following simplified expression:
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 \(v_{\nu }\), so that
with \(\Phi \) defined in (52). In view of Lemma 3.2\(v_{\nu }\in C^2([0,T];C^{\alpha }_*(\mathbb {R}))\) and therefore (51),(53) amount to the expansions
Construction of the curve z
In this section, we construct a function \(z=z(s,t)\) satisfying the conditions of Proposition 2.1.
Proposition 4.1
Let \(z_0(s)=\beta s+{\tilde{z}}_0(s)\) with \({\tilde{z}}_0\in C^{3,\alpha }_*(\mathbb {R})\) for some \(0<\alpha <1\) and \(\beta \in \mathbb {R}\). Let \(c=c(s)>0\) with \(\partial _sc\in C^{\alpha }_*(\mathbb {R})\). If \(\inf _sc(s)=0\), assume in addition that \(\beta =0\) and there exists \(c_\mathrm{min}>0\) such that
For any \(T>0\) there exists \({{\tilde{z}}}\in C^2([0,T];C^{1,\alpha }_*(\mathbb {R}))\) such that the conditions (20)–(22) of Proposition 2.1 are satisfied.
Proof
First of all, let us fix \(f_1,f_2\in C_c^\infty (\mathbb {R})\) such that \(\int _{I_k}f_l(s)\,\mathrm{d}s=\delta _{kl}\), where \(I_1=(\infty ,0)\) and \(I_2=(0,\infty )\). We define
where
with \(\Phi _0\), \(\Psi _0\) and \({\bar{c}}\) as defined in Proposition 3.8, and \(\psi _k\in C^2([0,T])\) are functions of time still to be fixed, such that \(\psi _k(0)=\psi _k'(0)=\psi _k''(0)=0\) for \(k=1,2\).
Let us check that z satisfies the conditions of Proposition 3.8. Since \({\tilde{z}}_0\in C^{3,\alpha }_*(\mathbb {R})\), Lemma 3.2 (applied with \(z(s,t)=z_0(s)\)) implies that \(\Phi _0\in {\mathcal {W}}^{2,\alpha }\). Theorem 3.1 then implies that \(z_1\in C^{2,\alpha }_*(\mathbb {R})\). Consequently, from the expression for \(\Psi _0\) in Proposition 3.8 and using Lemma 3.2 again we deduce that \(\Phi _1\in {\mathcal {W}}^{1,\alpha }\). Hence \(z_2\in C^{1,\alpha }_*(\mathbb {R})\), using once more Theorem 3.1. Thus we have shown that \(z\in C^2([0,T];C^{1,\alpha }_*(\mathbb {R}))\). It follows now from Proposition 3.8 that
In the case \(\inf _s c(s)>0\) conditions (20)–(22) follow directly from (57)–(58). In particular, in this case we can take \(\psi _k\equiv 0\) for \(k=1,2\).
In what follows, let us then assume \(\inf _s c(s)=0\), so that also \(\beta =0\) and (55) holds. Observe first of all that (55) together with (57) directly implies (20). Furthermore,
so that also (22) holds. It remains to verify (21). To this end, our aim is to choose \(\psi _k\) in such a way that
Indeed, assume for the moment that (59) holds. Then (58) implies for \(s>0\)
and similarly for \(s<0\). Thus in this case also (21) is verified.
To complete the proof, it therefore remains to choose \(\psi _k\) to ensure (59). Let
where dependence on \(\psi =(\psi _1,\psi _2)\) appears in the implicit dependence of \(u_\nu ^{(i)}\) on z defined in (56) via (30)–(32). Then, recalling the choice of \(f_k\), (59) is equivalent to the ODE system
with initial condition \(\psi _k(0)=0\). Using estimate (43) in Lemma 3.3, we verify that \(x\mapsto h(t,x)\) is uniformly Lipschitz continuous. Therefore, the Cauchy–Lipschitz theorem is applicable and yields a unique solution \(\psi :[0,T]\rightarrow \mathbb {R}^2\).
Then, by recalling our choice for \(z_1\), \(z_2\) and the expansion (53), we see that \(h(t,0)=o(t)\). By differentiating the ODE, we also obtain \(\tfrac{\mathrm{d}}{\mathrm{d}t}\psi '(t)=\partial _th+\partial _xh\psi '\). Since \(\psi (0)=0\) and \(h(0,0)=0\), we obtain \(\psi '(0)=0\) and hence \(\psi (t)=o(t)\). Furthermore, since \(\partial _t h(0,0)=0\) also, we deduce \(\psi ''(0)=0\), and furthermore, from the equation we deduce \(\psi '(t)=o(t)\). This completes the proof.
\(\square \)
References
 1.
Alazard, T. and Lazar, O. Paralinearization of the Muskat equation and application to the Cauchy problem. Arch. Ration. Mech. Anal. (2020), 237 (2), 545–583.
 2.
Alazard, T. and Nguyen, Q.H. On the Cauchy problem for the Muskat equation. II: Critical initial data. arXiv:2009.08442 (2020)
 3.
Arnaiz, V., Castro, Á., and Faraco, D. Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime. arXiv:2001.06361 (2020)
 4.
Bardos, C., Székelyhidi Jr., L., and Wiedemann, E. Nonuniqueness for the Euler equations: The effect of the boundary. Dedicated to the memory of Professor Mark Vishik. Uspekhi Mat. Nauk 69:2(416) (2014), 3–22.
 5.
Castro, A., Córdoba, D., and Faraco, D. Mixing solutions for the Muskat problem. arXiv:1605.04822 (2016).
 6.
Castro, A., Faraco, D., and Mengual, F. Degraded mixing solutions for the Muskat problem arXiv:1805.12050 (2018)
 7.
Constantin, P., F.Gancedo, Vicol, V., and Shvydkoy, R. Global regularity for 2D Muskat equation with finite slope. Annales de l’Institut Henri Poincare (C) Non Linear Analysis (2015) 34 (4), 1041–1074.
 8.
Córdoba, D., Faraco, D., and F.Gancedo. Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch.Ration.Mech.Anal.,200,3 (2011), 725–746.
 9.
Córdoba, D., and F.Gancedo. Contour dynamics of incompressible 3D fluids in a porous medium with different densities. Comm. Math. Phys. 273,2 (2007), 445–471.
 10.
Chiodaroli, E., De Lellis, C., and Kreml, O. Global illposedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math. (2014).
 11.
De Lellis, C., and Székelyhidi Jr, L. The Euler equations as a differential inclusion. Ann of Math (2) (2009) 170 (3), 1417–1436.
 12.
De Lellis, C., and Székelyhidi Jr., L. On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. (2010) 195(1), 225–260.
 13.
De Lellis, C., and Székelyhidi Jr., L. The hprinciple and the equations of fluid dynamics. Bull. Amer. Math. Soc. (N.S.) (2012), 49(3), 347–375.
 14.
Fjordholm, U. S., Käppeli, R., Mishra, S. and Tadmor, E. Construction of approximate entropy measurevalued solutions for hyperbolic systems of conservation laws Found. Comput. Math. (2017), 17 (3), 763–827.
 15.
Förster, C. and Székelyhidi Jr., L. Piecewise constant subsolutions for the Muskat problem. Comm. Math. Phys. (2018), 363 (3), 1051–1080.
 16.
Gebhard, B., Kolumbán, J. J., and Székelyhidi Jr., L. A new approach to the RayleighTaylor instability. arXiv:2002.08843 (2020)
 17.
Gebhard, B. and Kolumbán, J. J. Relaxation of the Boussinesq system and applications to the RayleighTaylor instability. arXiv:2008.08853 (2020)
 18.
Lanthaler, S. and Mishra, S. Computation of measurevalued solutions for the incompressible Euler equations. Math. Models Methods Appl. Sci. (2015), 25 (11), 2043–2088.
 19.
Matioc, BV. The Muskat problem in two dimensions: equivalence of formulations, wellposedness, and regularity results. Anal. PDE (2019), 12 (2), 281–332.
 20.
Mengual, F. Hprinciple for the 2D incompressible porous media equation with viscosity jump. arXiv:2004.03307 (2020)
 21.
Mengual, F. and Székelyhidi Jr., L. Dissipative Euler flows for vortex sheet initial data without distinguished sign. arXiv:2005.08333 (2020)
 22.
Muskat, M. The flow of homogeneous fluids through porous media McGrawHill, New York (1937).
 23.
Nguyen, Q. H. and Pausader, B. A paradifferential approach for wellposedness of the Muskat problem. Arch. Ration. Mech. Anal. (2020), 237 (1), 35–100
 24.
Otto, F.. Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm. Pure Appl. Math. (1999), (52) 7, 873–915.
 25.
Saffman, P. G., and Taylor, G. I. The penetration of a fluid into a porous medium or HeleShaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 1242 (1958), 312–329.
 26.
Shvydkoy, R., Convex integration for a class of active scalar equations. J. Amer. Math. Soc. (2011), 24 (4), 1159–1174.
 27.
Székelyhidi Jr., L. Weak solutions to the incompressible Euler equations with vortex sheet initial data. C.R.Acad.Sci. Paris Ser. I Math., (2011), 349, 1063–1066.
 28.
Székelyhidi Jr., L. Relaxation of the incompressible Porous Media Equation. Ann.Sci.Éc.Norm.Supér.,45,3 (2012), 491–509.
 29.
Tartar, L. The compensated compactness method applied to systems of conservation laws. in Systems of nonlinear partial differential equations. Dordrecht (1977), 263–285.
 30.
Wooding, R. A. and MorelSeytoux, H. J. Multiphase fluid flow through porous media. Ann. Review Fluid Mech. (1976), 8, 233–274.
Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 724298). This work was initiated during the visit of the authors to the Hausdorff Research Institute for Mathematics (HIM), University of Bonn, in March 2019. This visit was supported by the HIM. Both the support and the hospitality of HIM are gratefully acknowledged.
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Dedicated to the 60th birthday of Matthias Hieber.
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Noisette, F., Székelyhidi, L. Mixing solutions for the Muskat problem with variable speed. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028020006551
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