Abstract
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.
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1 Introduction
In this paper we study the evolution equation
which is defined for \(t>0\) and \(x\in {\mathbb R}\). The function f is assumed to be known at time \(t=0\), that is
The evolution problem (1a) 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 (1a) to the classical formulation of the Muskat problem [29]. The problem (1a) describes the two-dimensional motion of two fluids with equal viscosities \(\mu _-=\mu _+=\mu\) and general densities \(\rho _-\) and \(\rho _+\) in a vertical/horizontal homogeneous porous medium which is identified with \({\mathbb R}^2\). The fluids occupy the entire plane, they are separated by the sharp interface \({\{y=f(t,x)+Vt\}}\), and they move with constant velocity (0, V), where \({V\in {\mathbb R}}\). The fluid denoted by \(+\) is located above this moving interface. We use \({g\in [0,\infty )}\) to denote the Earth’s gravity, \(k>0\) is the permeability of the homogeneous porous medium, and \(\sigma >0\) is the surface tension coefficient at the free boundary. Moreover, to shorten the notation we have set \(\Delta _\rho :=g(\rho _--\rho _+)\in {\mathbb R}\) and
Finally, \(\kappa (f(t))\) is the curvature of \(\{y=f(t,x)+tV\}\) and \(\mathop \mathrm{PV}\nolimits\) 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 features like the stability of stationary solutions [13, 14, 16, 20, 26, 28, 33, 36], 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, 37] 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 [34, 35, 38] 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, 33, 36] 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, 39]. While in [8, 39] the initial data are chosen from \({H^s(\mathbb {T})}\), with \(s\ge 6\), the regularity of the initial data has been decreased in [24,25,26] to \(H^{2+\varepsilon }({\mathbb R})\), with \(\varepsilon \in (0,1)\) arbitrarily small. Finally, the very recent references [30, 31] consider the problem with initial data in \(H^{1+\frac{d}{2}+\varepsilon }({\mathbb R}^d)\), with \(d\ge 1\) and \(\varepsilon >0\) 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 (1a) in the subcritical \(L_p\)-based Sobolev spaces \(W^s_p({\mathbb R})\) with \(p\in (1,2]\) and \(s\in (1+1/p,2)\). This issue is new in the context of (1a) (see [2, 12] for results in the case when \(\sigma =0\)). To motivate why \(W^{1+1/p}_p({\mathbb R})\) is a critical space for (1a) 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 (1a), then, given \(\lambda >0\), the function
also solves (1a). This scaling identifies \(W^{1+1/p}_p({\mathbb R})\) as a critical space for (1a). The main result Theorem 1.1 establishes the well-posedness of (1a) in \(W^s_p({\mathbb R})\). This is achieved by showing that (1a) 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 (1a) is the fact that the equations have to be interpreted in distributional sense as they are realized in the Sobolev space \(W^{\overline{s}-2}_p({\mathbb R})\), where \(\overline{s}\) is chosen such that \(1+1/p<\overline{s}<s<2\). Additionally to well-posedness, Theorem 1.1 provides two parabolic smoothing properties, showing in particular that (1a) holds pointwise, and a criterion for the global existence of solutions.
Theorem 1.1
Let \(p\in (1,2]\) and \(1+1/p<\overline{s}<s<2\). Then, the Muskat problem (1a) possesses for each \(f_0\in W^s_p({\mathbb R})\) a unique maximal solution \(f:=f(\,\cdot \,; f_0)\) such that
with \(T^+=T^+(f_0)\in (0,\infty ]\) denoting the maximal time of existence. Moreover, the following properties hold true:
-
(i)
The solution depends continuously on the initial data;
-
(ii)
Given \(k\in {\mathbb N}\), we have \(f\in \mathrm{C}^\infty ((0,T^+ )\times {\mathbb R},{\mathbb R})\cap \mathrm{C}^\infty ((0,T^+), W^k_p({\mathbb R}));\)
-
(iii)
The solution is global if
$$\begin{aligned} \sup _{[0,T^+)\cap [0,T]}\Vert f(t)\Vert _{ W^s_p({\mathbb R})}<\infty \qquad \text {for all}\quad T>0. \end{aligned}$$
We emphasize that some of the arguments use in an essential way the fact that \(p\in (1,2]\). More precisely, we employ several times the Sobolev embedding \({W^s_p({\mathbb R})\hookrightarrow W^{t}_{p'}({\mathbb R})},\) 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 \(p\in (1,2]\).
Additionally, we expect that the assertion (ii) of Theorem 1.1 can be improved to real-analyticity instead of smoothness. However, this would require to establish real-analytic dependence of the right-hand side of (1a) on f in the functional analytic framework considered in Sect. 3, which is much more involved than showing the smooth dependence (see [25, Proposition 5.1] for a related proof of real-analyticity).
1.1 Notation
Given \(k\in {\mathbb N}\) and \(p\in (1,\infty ),\) we let \(W^k_p({\mathbb R})\) denote the standard \(L_p\)-based Sobolev space with norm
Given \(0<s\not \in {\mathbb N}\) with \(s=[s]+\{s\}\), where \({[s]\in {\mathbb N}}\) and \(\{s\}\in (0,1)\), the Sobolev space \(W^s_p({\mathbb R})\) is the subspace of \(W^{[s]}_p({\mathbb R})\) that consists of functions for which the seminorm
is finite. Here \(\{\tau _\xi \}_{\xi \in {\mathbb R}}\) is the group of right translations and \(\Vert \cdot \Vert _q:=\Vert \cdot \Vert _{L_q({\mathbb R})}\), \(q\in [1,\infty ].\) The norm on \(W^s_p({\mathbb R})\) is defined by
For \(s<0\), \(W^s_p({\mathbb R})\) is defined as the dual space of \(W^{-s}_{p'}({\mathbb R})\).
The following properties can be found e.g. in [40].
-
(i)
\(\mathrm{C}^\infty _0({\mathbb R})\) lies dense in \(W^s_p({\mathbb R})\) for all \(s\in {\mathbb R}\). Moreover, \(W^s_p({\mathbb R})\hookrightarrow \mathrm{C}^{s-1/p}({\mathbb R})\) holds provided that \(0<s-1/p\not \in {\mathbb N}\).
-
(ii)
\(W^s_p({\mathbb R})\) is an algebra for \(s>1/p\).
-
(iii)
If \(\rho >\max \{s, -s\},\) then \(f\in \mathrm{C}^\rho ({\mathbb R})\) is a pointwise multiplier for \(W^s_p({\mathbb R})\), that is
$$\begin{aligned} \Vert fg\Vert _{W^s_p}\le C\Vert f\Vert _{\mathrm{C}^\rho }\Vert g\Vert _{W^s_p} \qquad \text {for all}\; g\in W^s_p({\mathbb R}), \end{aligned}$$(2)with C independent of f and g.
-
(iv)
Given \(\theta \in (0,1)\) and \(p\in (1,\infty )\), let \((\cdot ,\cdot )_{\theta ,p}\) denote the real interpolation functor of exponent \(\theta\) and parameter \(p\in (1,\infty ).\) Given \(s_0,\, s_1\in {\mathbb R}\) with \((1-\theta )s_0+\theta s_1\not \in {\mathbb Z}\), it holds that
$$\begin{aligned} \left( W^{s_0}_{p}({\mathbb R}), W^{s_1}_{p}({\mathbb R})\right) _{\theta ,p}=W^{(1-\theta )s_0+\theta s_1}_{p}({\mathbb R}). \end{aligned}$$(3) -
(v)
\(W^s_p({\mathbb R})\hookrightarrow W^{t}_q({\mathbb R}))\) if \(1<p\le q<\infty\) and \(s-1/p\ge t-1/q.\)
Besides, we need also the following properties.
-
(a)
Given \(r\in [0,1)\) and \(p\in (1,\infty )\), there exists a constant \(C>0\) such that
$$\begin{aligned} \Vert gh\Vert _{W^{r}_p}\le 2\Vert g\Vert _\infty \Vert h\Vert _{W^{r}_p}+C\Vert g\Vert _{W^{r+1}_p}\Vert h\Vert _p,\quad g\in W^{r+1}_p({\mathbb R}), h\in W^{r}_p({\mathbb R}). \end{aligned}$$(4) -
(b)
Given \(r\in (1/p,1)\) and \(p\in (1,\infty )\), there exists a constant \(C>0\) such that
$$\begin{aligned} \Vert gh\Vert _{W^{r}_p}\le 2\left( \Vert g\Vert _\infty \Vert h\Vert _{W^{r}_p}+\Vert h\Vert _\infty \Vert g\Vert _{W^{r}_p}\right) ,\quad g,\, h\in W^{r}_p({\mathbb R}). \end{aligned}$$(5) -
(c)
Given \(p\in (1,2]\), \(r\in (1/p,1)\), and \(\rho \in (0,\min \{r-1/p,1-r\}),\) there exists a constant \(C>0\) such that
$$\begin{aligned} \Vert gh\Vert _{W^{1-r}_{p'}}\le 2\Vert g\Vert _\infty \Vert h\Vert _{W^{1-r}_{p'}}+C\Vert g\Vert _{W^{r}_p}\Vert h\Vert _{W^{1-r-\rho }_{p'}} \end{aligned}$$(6)for all \(g\in W^{r}_p({\mathbb R})\) and \(h\in W^{1-r}_{p'}({\mathbb R}),\) and
$$\begin{aligned} \Vert \varphi h\Vert _{W^{r-1}_p}\le 5\Vert \varphi \Vert _\infty \Vert h\Vert _{W^{r-1}_p}+C\frac{ \Vert \varphi \Vert _{W^{r}_{p}}^{1+2r/\rho }}{\Vert \varphi \Vert _{\infty }^{2r/\rho }}\Vert h\Vert _{W^{r-1-\rho }_p} \end{aligned}$$(7)for all \(h\in W^{r-1}_p({\mathbb R})\) and \(0\ne \varphi \in W^r_p({\mathbb R})\).
The estimates (4) and (5) are straightforward consequences of the properties (i)-(v) listed above. The inequalities (6) and (7) are established in Appendix 2. Let us point out that (6) implies in particular that the multiplications
are both continuous if \(p\in (1,2]\) and \(r\in (1/p,1)\). The continuity of (8)\(_1\) is a straightforward consequence of (6), while the continuity of (8)\(_2\) follows by a standard duality argument.
1.2 Outline
In Sect. 2 we introduce some multilinear singular integral operators and study their properties. These operators are then used in Sect. 3 to formulate (1a) as a quasilinear evolution problem, cf. (27)–(28). Subsequently, we show in Theorem 3.5 that (27) is of parabolic type and we complete the section with the proof of Theorem 1.1. In Appendixes 1 and 2 we prove some technical results used in the analysis.
2 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 \({n,\,m\in {\mathbb N}}\) and Lipschitz continuous functions \({a_1,\ldots , a_{m},\, b_1, \ldots , b_n:\mathbb {R}\rightarrow \mathbb {R}}\) we set
For brevity we write
The relevance of these operators in the context of (1) is enlightened by the fact that (1a) can be recast, at least at formal level, in a compact form as
where \(\mathbb {B}(f)\) 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
Given \(f\in W^s_p({\mathbb R})\) and \(h\in W^{s+1}_p({\mathbb R})\), with \(p\in (1,\infty )\) and \(s\in (1+1/p,2),\) Lemma 3.1 ensures that in (11) the term
belongs to \(W^{s-2}_p({\mathbb R})\). Therefore, it is natural to ask weather \(\mathbb {B}(f)\in \mathcal {L}(W^{s-2}_p({\mathbb R}))\). The proof of this boundedness property, which enables us to view (11) as an evolution equation in \({W^{s-2}_p({\mathbb R})}\), see Sect. 3, is the main goal of this section. As already mentioned, some of the arguments require \(p\in (1,2]\).
We first recall the following result.
Lemma 2.1
Let \(p\in (1,\infty )\), \(n,\,m \in {\mathbb N}\), and let \(a_1,\ldots , a_{m},\, b_1, \ldots , b_n:{\mathbb R}\rightarrow {\mathbb R}\) be Lipschitz continuous. Then, there exists a constant \(C=C(n,\, m,\,\max _{i=1,\ldots , m}\Vert a_i'\Vert _{\infty } )\) such that
Moreover,Footnote 1\(B_{n,m}\in \mathrm{C}^{1-}((W^1_\infty ({\mathbb R}))^{m},\mathcal {L}^n_\mathrm{sym}(W^1_\infty ({\mathbb R}),\mathcal {L}( L_p({\mathbb R}))).\)
Proof
See [2, Lemma 2].
We point out that, given Lipschitz continuous functions \(a_1,\ldots , a_m, \,\widetilde{a}_1,\ldots , \widetilde{a}_m,\, b_1, \ldots , b_n\), we have
The formula (14) was used to establish the Lipschitz continuity property (denoted by \(\mathrm{C}^{1-}\)) 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 \(f\in W^s_p({\mathbb R})\), with \(p\in (1,2]\) and \(s\in (1+1/p,2)\), we have \(B_{n,m}^0(f)\in \mathcal {L}(W^{r}_{p'}({\mathbb R}))\) for all \(r\in [0,1-1/p)\) (in particular also for \(r=2-s\)). Lemma 2.2 below provides the key argument in the proof of Lemma 2.4. The desired mapping property \(B_{n,m}^0(f)\in \mathcal {L}(W^{s-2}_p({\mathbb R}))\) stated in Lemma 2.5, follows then from Lemma 2.4 via a duality argument. Lemma 2.5 and the fact that \(f'\in W^{s-1}_p({\mathbb R})\) is a pointwise multiplier for \(W^{s-2}_p({\mathbb R})\), see (8), provide the desired property \({\mathbb {B}(f)\in \mathcal {L}(W^{s-2}_p({\mathbb R}))}\).
Lemma 2.2
Given \(p\in (1,2]\), \(s\in (1 +1/p,2)\), \(r\in (0, 1-1/p)\), \(n,\, m\in {\mathbb N}\), \(n\ge 1\), and \({a_1,\ldots , a_m \in W^s_p({\mathbb R})}\), there exists a positive constant \( C \), which depends only on \( n,\, m,\, s,\,r,\,p,\,\max _{1\le i\le m}\Vert a_i\Vert _{W^s_p}\) , such that
for all \(b_1,\ldots , b_n\in W^s_p({\mathbb R})\) and \(\overline{\omega }\in W^{r}_{p'}({\mathbb R}).\)
Proof
Without loss of generality we may assume that \(\overline{\omega }\in \mathrm{C}^\infty _0({\mathbb R}).\) Using the relation
algebraic manipulations lead us to
for \(x\in {\mathbb R}\), where
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 \(x\in {\mathbb R}\) and \(y\ne 0\), we set
We estimate the \(L_{p'}\)-norm of the four terms on the right of (16) separately. Let \(q\in ( p',\infty )\) be defined as the solution to
Term 1. We note that \({\overline{\omega }\in W^{r}_{p'}({\mathbb R})\hookrightarrow L_q({\mathbb R})}\), \({b_1'\in W^{s-2/p}_{p'}({\mathbb R})\hookrightarrow W^{s-r-2/p}_{p'}({\mathbb R})}\), and additionally we have \(W^{s-r-2/p}_{p'}({\mathbb R})\hookrightarrow L_{1/r}({\mathbb R})\). Hölder’s inequality together with Lemma 2.1 (with \(p=1/r\)) then yields
Term 2. Let \(s_0\in (1+1/p, s]\) be chosen such that \(s_0-r-1/p<1\). Taking advantage of Minkowski’s integral inequality, of Hölder’s inequality, and of the embedding property \(b_1\in W^{s+1-r-2/p}_{p'}({\mathbb R})\hookrightarrow \mathrm{C}^{s_0-r-1/p}({\mathbb R})\) we get
where
We arrive at
Terms 3 & 4. Given \(2\le j\le n\), Hölder inequality and Minkowski’s integral inequality yield
To estimate the integral term we choose \(s_0\in (1+1/p, s]\) with \( s_0-r-1/p<1. \) Since \( b_1\in W^{s+1-r-2/p}_{p'}({\mathbb{R}})\hookrightarrow W^1_{1/r}({\mathbb{R}})\cap {\mathrm{C}}^{s_0-r-1/p}({\mathbb{R}})\), we get
for \(2\le j\le n.\) Since \(b_j\in W^{s_0}_p({\mathbb R})\hookrightarrow W^{s_0+r-\frac{1}{p} }_{1/r}({\mathbb R})\), \(2\le j\le n\), Minkowski’s integral inequality, Hölder’s inequality, and a change of variables lead to
Consequently, given \(2\le j\le n\), we have
and, similarly,
The desired claim follows now from (16) to (20).
Lemma 2.3 below is used to prove Lemma 2.4 (see also [2, Lemma 7] for a related result).
Lemma 2.3
Let \(p\in (1,\infty )\), \(n\in {\mathbb N}\), and \(n<t'<t<n+1\). Then, there exists \(C>0\) such that
Proof
We have
and, given \(k\in \{0,\ldots ,n\}\),
Moreover, since \({\mathbb R}^2=\overline{\{|\xi |<|h|\}}\cup \{|\xi |>|h|\}\) and
with
the desired estimate is immediate.
We are now in a position to prove that \(B_{n,m}^0(f)\in \mathcal {L}(W^{r}_{p'}({\mathbb R}))\) for all \(r\in [0,1-1/p)\).
Lemma 2.4
Given \(p\in (1,2]\), \(s\in (1 +1/p,2)\), \(n,\, m\in {\mathbb N}\), \({a_1,\ldots , a_m \in W^s_p({\mathbb R})}\), and \({r\in [0,1-1/p)}\), there exists a constant \(C=C(n,\, m,\,s,\,r,\,p,\, \max _{1\le i\le m}\Vert a_i\Vert _{W^s_p})\) such that
for all \(b_1,\ldots , b_n\in W^s_p({\mathbb R})\) and \(\overline{\omega }\in W^{r}_{p'}({\mathbb R}).\)
Moreover, \(B_{n,m}\in \mathrm{C}^{1-}((W^s_p({\mathbb R}))^m,\mathcal {L}^{n}_\mathrm{sym}( W_p^{s}({\mathbb R}) , \mathcal {L}(W^{r}_{p'}({\mathbb R})))).\)
Proof
Let \(B_{n,m}:=B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,\cdot ].\) Recalling Lemma 2.1 (with \(p=p'\)), we get
which proves (21) for \(r=0\). Let now \({r\in (0,1-1/p)}\). It remains to estimate the quantity
Taking advantage of (14), we write
with
Hence,
and Lemma 2.1 (with \(p=p'\)) yields
Furthermore, using (15), Lemma 2.3 (with \(p=p'\), \(t=s+1-2/p\), and \(t'=s+1-r-2/p\)), and the embedding \(W^s_p({\mathbb R})\hookrightarrow W^{s+1-2/p}_{p'}({\mathbb R})\), we deduce that
and, by similar arguments,
The relations (22)–(25) lead to the desired estimate. The local Lipschitz continuity property follows from (14) and (21).
Together with Lemma 2.4 (with \(r=2-s\in (0,1-1/p)\)) we obtain the following result.
Lemma 2.5
Given \(p\in (1,2]\), \(s\in (1 +1/p,2)\), \(n,\, m\in {\mathbb N},\) and \({a_1,\ldots , a_m \in W^s_p({\mathbb R})}\), there exists a constant \(C=C(n,\, m,\,s,\,p,\,\max _{1\le i\le m}\Vert a_i\Vert _{W^s_p})\) such that
for all \(b_1,\ldots , b_n\in W^s_p({\mathbb R})\) and \(\overline{\omega }\in L_{p}({\mathbb R}).\)
Moreover, \(B_{n,m}\in \mathrm{C}^{1-}((W^s_p({\mathbb R}))^m,\mathcal {L}^{n}_\mathrm{sym}(W_p^{s}({\mathbb R}),\mathcal {L}( W^{s-2}_{p}({\mathbb R})))).\)
Proof
We recall that \(W^{s-2}_p({\mathbb R})=(W^{2-s}_{p'}({\mathbb R}))'\). Let \({B_{n,m}:=B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,\cdot ]}\). It is not difficult to prove that the \(L_2\)-adjoint of \(B_{n,m}\) is the operator \(-B_{n,m}\). Therefore, given \(\overline{\omega },\,\varphi \in \mathrm{C}^\infty _0({\mathbb R})\), we obtain, in view of Lemma 2.4,
The estimate (26) follows via a standard density argument.The Lipschitz continuity property is a consequence of (26) and of (14).
3 A functional analytic framework for the Muskat problem
In this section we take advantage of the mapping properties established in Sect. 2 and formulate the Muskat problem (1a) as a quasilinear evolution problem in a suitable functional analytic setting, see (27)–(29). 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 (1a) is due to the quasilinearity of the curvature operator, the latter being established in Lemma 3.1.
Lemma 3.1
Given \(p\in (1,\infty )\) and \(s\in (1+1/p,2)\), the operator \({\kappa (\cdot )[\cdot ]}\) defined in (13) satisfies \(\kappa \in \mathrm{C}^\infty (W^s_p({\mathbb R}),\mathcal {L}(W^{s+1}_p({\mathbb R}), W^{s-1}_p({\mathbb R}))).\)
Proof
The arguments are similar to those presented in [27, Appendix C] and are therefore omitted.
The Muskat problem (1a) can thus be formulated as the evolution problem
where
with \(\mathbb {B}\) introduced in (12). Arguing as in [27, Appendix C], we may infer from Lemma 2.5 that, given \(n,\, m\in {\mathbb N}\), \(p\in (1,2]\), and \(s\in (1+1/p,2),\) we have
This property and Lemma 3.1 combined yield
for all \(p\in (1,2]\) and \(s\in (1+1/p,2).\)
Let \(p\in (1,2]\), \(s\in (1+1/p,2)\), and \(f\in W^s_p({\mathbb R})\) be fixed in the remaining of this section. The analysis below is devoted to showing that the linear operator \(\Phi (f)\), viewed as an unbounded operator in \(W^{s-2}_p({\mathbb R})\) and with definition domain \(W^{s+1}_p({\mathbb R})\), is the generator of an analytic semigroup in \(\mathcal {L}(W^{2-s}_p({\mathbb R}))\), which reads in the notation used in [7] as
This property is established in Theorem 3.5 below and it identifies the quasilinear evolution problem (27) as being of parabolic type. To start, we note that \(\pi ^{-1}B_{0,0 }= H\), where H is the Hilbert transform, and therefore
where \((d^4/dx^4)^{3/4}\) denotes the Fourier multiplier with symbol \(m(\xi ):=|\xi |^3\) and \((-d^2/dx^2)^{1/2}\) is the Fourier multiplier with symbol \(m(\xi ):=|\xi |\). We shall locally approximate the operator \(\Phi (\tau f)\), with \(\tau \in [0,1]\), by certain Fourier multipliers \({\mathbb A}_{j,\tau }\). Therefore we choose for each \({\varepsilon \in (0,1)}\) a so-called finite \(\varepsilon\)-localization family, that is a set
such that
The real number \(x_N^\varepsilon\) plays no role in the analysis below. To each finite \(\varepsilon\)-localization family we associate a second family \(\{\chi _j^\varepsilon \,:\, -N+1\le j\le N\}\subset \mathrm{C}^\infty (\mathbb {R},[0,1])\) such that
To each finite \(\varepsilon\)-localization family we associate a norm on \(W^r_p({\mathbb R}),\) \(r\in {\mathbb R}\), which is equivalent to the standard norm.
Lemma 3.2
Let \(\varepsilon \in (0,1)\) and let \(\{(\pi _j^\varepsilon ,x_j^\varepsilon )\,:\, -N+1\le j\le N\}\) be a finite \(\varepsilon\)-localization family. Given \(p\in (1,\infty )\) and \(r\in {\mathbb R}\), there exists \(c=c(\varepsilon ,r,p)\in (0,1)\) such that
Proof
The claim follows from the fact that \(\pi _j^\varepsilon \in \mathrm{C}^\infty ({\mathbb R})\) is a pointwise multiplier for \(W^r_p({\mathbb R})\).
The next result is the main step in the proof of (30).
Theorem 3.3
Let \(p\in (1,2]\), \(s\in (1+1/p,2)\), \(\rho \in (0,\min \{(s-1-1/p)/2),2-s\})\), and \(\nu >0\) be given. Then, there exist \(\varepsilon \in (0,1)\), a \(\varepsilon\)-localization family \(\{(\pi _j^\varepsilon ,x_j^\varepsilon )\,:\, -N+1\le j\le N\}\), a constant \(K=K(\varepsilon )\), and bounded operators
such that
for all \(-N+1\le j\le N\), \(\tau \in [0,1],\) and \(h\in W^{s+1}_p({\mathbb R})\). The operators \({\mathbb A}_{j,\tau }\) are defined by
and \(\alpha _\tau :=(k\sigma /(2\mu ))(1+\tau ^2f'^2)^{-3/2}\).
Proof
In this proof we denote by C constants that do not depend on \(\varepsilon\) and we write K for constants that depend on \(\varepsilon\).
Given \(-N+1\le j\le N\), \(\tau \in [0,1],\) and \(h\in W^{s+1}_p({\mathbb R})\), Lemma 2.1 yields
while, using some elementary arguments, we get
Below we take advantage of (34) when considering the leading order term \(\mathbb {B}(\tau f)[(\kappa (\tau f)[h])']\) of \(~\Phi (\tau f) [h]\). Let \(C_1:=C_0k\sigma /2\pi \mu .\)
Step 1: The case \(|j|\le N-1\). Given \(|j|\le N-1\), we infer from Lemma 4.2 and (34) that
provided that \(\varepsilon\) is sufficiently small. Besides, we have
where
For \(\varepsilon\) sufficiently small to guarantee that
it follows from (7) (with \(r=s-1\)), Lemma 2.5, and (34) (if \(\chi _j^\varepsilon (f'-f'(x_j^\varepsilon ))\) is not identically zero, otherwise the estimate is trivial) that
As \(\chi _j^\varepsilon (f'-f'(x_j^\varepsilon ))\in W^{s-1}_p({\mathbb R})\) is a pointwise multiplier for \(W^{s-2}_p({\mathbb R}),\) cf. (8), Lemma 4.1 yields
Finally, if \(\varepsilon\) is sufficiently small, we may argue as in the derivation of (35) to get
Gathering (36)–(38), we conclude that
We now combine (33), (35), and (39) and obtain that
As a final step we show that, if \(\varepsilon\) sufficiently small, then
for all \(h\in W^{s+1}_p({\mathbb R})\), \(\tau \in [0,1]\), and \(|j|\le N-1.\) To start, we note that
and Lemma 2.5 yields for \(-N+1\le j\le N\) that
Moreover, letting \(C_2:=\Vert B_{0,0}\Vert _{\mathcal {L}(W^{s-2}_p({\mathbb R}))}\), the algebra property of \(W^{s-1}_p({\mathbb R})\) leads us to
Using (5) together with the identity \(\chi _j^\varepsilon \pi _j^\varepsilon =\pi _j^\varepsilon\), for \(\varepsilon\) sufficiently small we get
Gathering (42)–(44), we conclude that (41) holds true. The desired estimate (32) follows now, for \(|j|\le N-1,\) by combining (40) and (41).
Step 2: The case \(j=N\). Similarly to (35), we obtain from Lemma 4.5 and (34) that
for all \(h\in W^{s+1}_p({\mathbb R})\) and \(\tau \in [0,1]\), provided that \(\varepsilon\) is sufficiently small. Moreover,
where
Since \(\chi _N^\varepsilon f'\in W^{s-1}_p({\mathbb R})\) is a pointwise multiplier for \(W^{s-2}_p({\mathbb R}),\) cf. (8), Lemma 4.1 yields
Because \(f'\in W^{s-1}_p({\mathbb R})\) vanishes at infinity, for \(\varepsilon\) sufficiently small to ensure that
it follows from (7) (with \(r=s-1\)), Lemma 2.5, and (34) (if \(\chi _N^\varepsilon f'\) is not identically zero, otherwise the estimate is trivial) that
Gathering (33) and (45)–(47), we have shown that if \(\varepsilon\) is sufficiently small, then
for all \(h\in W^{s+1}_p({\mathbb R})\) and \(\tau \in [0,1]\). It remains to show that for \(\varepsilon\) sufficiently small
for all \(h\in W^{s+1}_p({\mathbb R})\) and \(\tau \in [0,1]\). Arguing as in the first step (see (43)), we find in view of (42) that
Using (5) and the fact that \(f'\) vanishes at infinity, for \(\varepsilon\) sufficiently small, we obtain
This proves (49). The claim (32) for \(j= N\) follows now directly from (48) and (49).
We now consider a class of Fourier multipliers related to the multipliers from Theorem 3.3.
Lemma 3.4
Let \(\eta \in (0,1)\). Given \(\alpha \in [\eta ,1/\eta ]\), let
Then, there exits a constant \(\kappa _0=\kappa _0(\eta )\ge 1\) such that
Proof
We first consider the realizations
Since \(W^k_p({\mathbb R})=H^k_p({\mathbb R})\), \(k\in {\mathbb Z}\), Mikhlin’s multiplier theorem, cf. e.g. [1, Theorem 4.23], shows that the properties (50)–(51) (in the appropriate spaces) are valid for these realizations. Then, using the interpolation property (3), we obtain that (50)-(51) hold true.
The next result provides the generator property announced in (30).
Theorem 3.5
Given \(p\in (1,2]\), \(s\in (1+1/p,2)\), and \(f\in W^{s}_p({\mathbb R})\) we have
Proof
Fix \(\rho \in (0,\min \{(s-1-1/p)/2),2-s\})\). Since \(f'\in W^{s-1}_p({\mathbb R})\) is bounded, there exists a constant \(\eta >0\) with the property that the function \(\alpha _\tau\), \(\tau \in [0,1]\), from Theorem 3.3 satisfies \(\eta \le |\alpha _\tau |\le 1/\eta\) for all \(\tau \in [0,1]\). Hence, regardless of \(\varepsilon >0,\) the operators \(A_{j,\tau },\) with \(-N+1\le j\le N\) and \(\tau \in [0,1],\) defined in Theorem 3.3 satisfy (50)-(51) with a constant \(\kappa _0\ge 1.\)
We set \(\nu :=(2\kappa _0)^{-1}\). Theorem 3.3 provides an \(\varepsilon \in (0,1)\), a \(\varepsilon\)-localization family, a constant \(K=K(\varepsilon )>0\), and bounded operators \({\mathbb A}_{j,\tau }\in \mathcal {L}(W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R}))\), \({-N+1\le j\le N}\) and \(\tau \in [0,1],\) such that
for all \(-N+1\le j\le N\), \(\tau \in [0,1],\) and \(h\in W^{s+1}_p({\mathbb R})\). Besides, Lemma 3.4 yields
for all \(-N+1\le j\le N\), \(\tau \in [0,1],\) \(\mathop \mathrm{Re}\nolimits \lambda \ge 1\), and \(h\in W^{s+1}_p({\mathbb R})\). The latter inequalities imply
Summing up over j and using Lemma 3.2, the interpolation property (3), and Young’s inequality we find constants \(\kappa =\kappa (f)\ge 1\) and \(\omega =\omega ( f)>0\) such that
for all \(\tau \in [0,1],\) \(\mathop \mathrm{Re}\nolimits \lambda \ge \omega\), and \(h\in W^{s+1}_p({\mathbb R})\).
Recalling (31) and arguing as in Lemma 3.4 we may choose \(\omega\) sufficiently large to guarantee that \(\omega -\Phi (0) \in \mathrm{Isom}(W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R}))\). The method of continuity together with (52) consequently yields
The relations (52) (with \(\tau =1\)) and (53) imply the desired claim, cf. [7, Chapter I].
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, 32].
Proof of Theorem 1.1
Fix
We further set \(E_1:=W^{\overline{s}+1}_p({\mathbb R})\), \(E_0:=W^{\overline{s}-2}_p({\mathbb R})\), and \(E_\theta :=(E_0,E_1)_{\theta ,p}\), with \(\theta \in \{\alpha ,\, \beta \}\). Recalling the interpolation property (3), we have \(E_\alpha =W^{s}_p({\mathbb R})\) and \(E_\beta =W^{\overline{s}}_p({\mathbb R})\). In view of (29) and (30) (with \({s=\overline{s}}\)), we then get
and the assumptions of [28, Theorem 1.1] are satisfied in the context of the Muskat problem (27). Applying [28, Theorem 1.1], we may conclude that (27) has for each \(f_0\in W^s_p({\mathbb R})\) a unique maximal classical solution \(f= f(\,\cdot \, ; f_0)\) such that
Moreover, if the solution belongs to the set
then it is also unique, cf. [28, Remark 1.2 (ii)]. We now prove that each solution to (27) that satisfies (54) belongs to \(\mathrm{C}^{\eta }([0,T^+),W^{\overline{s}}_p(\mathbb {R}))\) with \(\eta =(s-\overline{s})/(3+s-\overline{s})\). Indeed, since \(f\in \mathrm{C}([0,T^+),W^s_p(\mathbb {R}))\), we infer from Lemma 3.6 that
for each \(T\in (0,T^+)\). Since for \(\theta :=3/(s-\overline{s}+3)\) we have \((W^{\overline{s}-3}_{p}({\mathbb R}), W^{s}_{p}({\mathbb R}))_{\theta ,p}=W^{\overline{s}}_{p}({\mathbb R}),\) cf. (3), (54) and the latter estimate now yield
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 \(f= f(\,\cdot \, ; f_0):[0,T^+)\rightarrow W^{s}_p({\mathbb R})\) is a maximal classical solution with \(T^+<\infty\) and that
Using again Lemma 3.6 and arguing as above, we conclude that \(f:[0,T^+)\rightarrow W^{\overline{s}}_p(\mathbb {R})\) is uniformly continuous. Let now
Choosing \(F_1:=W^{\widetilde{s}+1}_p({\mathbb R})\), \(F_0:=W^{\widetilde{s}-2}_p({\mathbb R})\), and setting \(F_\theta :=(F_0,F_1)_{\theta ,p}\), \(\theta \in \{\alpha ',\, \beta \}\), we have that \(F_\alpha =W^{\overline{s}}_p({\mathbb R})\) and \(F_\beta =W^{\widetilde{s}}_p({\mathbb R})\). Moreover (29) and (30) (with \({s=\widetilde{s}}\)), yield
Thus, we may apply again [28, Theorem 1.1] (iv) (\(\alpha\)) to (27) and conclude that f can be extended to an interval \([0,{\widetilde{T}}^+)\) with \({\widetilde{T}}^+>T^+\) and such that
Moreover, by (ii) (with \((s,\overline{s})=(\overline{s},\widetilde{s})\)) we also have \(f\in \mathrm{C}^1((0,{\widetilde{T}}^+), W^{3}_p(\mathbb {R})),\) 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
for all \(f\in W^{s+1}_p({\mathbb R})\) with \(\Vert f\Vert _{W^s_p}\le M\).
Proof
In this proof the constants denoted by C depend only on M. Given \(f\in W^{s+1}_p({\mathbb R})\) with \(\Vert f\Vert _{W^s_p}\le M\), we have
Moreover, we infer from Lemma 2.1 that
and it remains to prove that
To this end we note that the \(L_2\)-adjoint \(\mathbb {B}^*(f)\) of \(\mathbb {B}(f)\) is identified by the relation
Therefore, given \(\overline{\omega },\) \(\psi \in \mathrm{C}^\infty _0({\mathbb R})\), we have
We next show that
for all \(\psi \in W^{3-s}_{p'}({\mathbb R})\) and \(f\in W^{s+1}_p({\mathbb R})\) with \(\Vert f\Vert _{W^s_p}\le M\). Indeed, Lemma 2.1 (with \(p=p'\)), yields
Additionally, we may argue as in the proof of [25, Lemma 3.5] to infer in view of Lemma 2.1 that \({\mathbb {B}^*(f)[\psi ]\in W^1_{p'}({\mathbb R})}\) with
Invoking Lemma 2.4 (with \(r=2-s\in (0,1-1/p)\)), we get that
and since \(f'\in W^{s-1}_p({\mathbb R})\) is a pointwise multiplier for \(W^{2-s}_{p'}({\mathbb R})\), cf. (8), we may conclude from (61) and (62) that
This proves (60). Combining (59) and (60), a standard density argument leads us to
and (58) follows via (56). Recalling the definition (28) of \(\Phi\), the bound (55) is a straightforward consequence of (57) and (58).
Notes
Given \(n\in {\mathbb N}\) and Banach spaces X and Y, we let \(\mathcal {L}^n_\mathrm{sym}(X,Y)\) denote the space of n-linear, bounded symmetric maps \(A:\;X^n\longrightarrow Y\).
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Acknowledgements
The authors were partially supported by the RTG 2339 ”Interfaces, Complex Structures, and Singular Limits” of the German Science Foundation (DFG). The support is gratefully acknowledged.
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Appendices
Freezing the kernels of singular operators in Sobolev spaces with negative exponent
In this section we establish some technical results which enable us to locally approximate the Fréchet derivative \(\partial \Phi (\tau f)\), \(\tau \in [0,1]\) and \(f\in W^s_p({\mathbb R})\), by certain explicit Fourier multipliers, cf. Theorem 3.3. These results can be viewed as a generalization of the method of freezing the coefficients of elliptic differential operators. We extend this method to a particular class of singular integral operators, one of the main difficulties arising from the fact that the Sobolev space where the error is measured has negative exponent. As a first result we establish a commutator type estimate.
Lemma 4.1
Let \(p\in (1,2]\), \(s\in (1+1/p,2),\) \(\rho \in (0,(s-1-1/p)/2)\), \(n,\, m\in {\mathbb N},\) \({f\in W^s_p({\mathbb R})}\), and \(\varphi \in \mathrm{C}^1({\mathbb R})\) with uniformly continuous derivative \(\varphi '\) be given. Then, there exists a constant \(K=K(n, m, s,p,\rho , \Vert \varphi \Vert _{\mathrm{C}^1}, \Vert f\Vert _{W^s_p})\) such that
Proof
Let \(\overline{\rho }:=(3-s)\rho /(1-\rho )\). We point out that \(\overline{\rho }\in (\rho ,s-1-1/p)\) and that \(\varphi \in \mathrm{C}^1({\mathbb R})\) is a pointwise multiplier for \(W^{ s-2-\overline{\rho }}_p({\mathbb R})\). Lemma 2.5 (with \({s=s-\overline{\rho }}\)) then yields
Besides, in view of [2, Lemma 12], we also have
Noticing that \(W^{(1-\rho )( s-2-\overline{\rho })}_p({\mathbb R})=W^{ s-2-\overline{\rho }}_p({\mathbb R})\) and \(W^{(1-\rho )( s-2-\overline{\rho })+\rho }_p({\mathbb R})= W^{s-2}_p({\mathbb R})\), the interpolation property (3) together with (64) and (65) leads us to the desired claim.
The first important result of this section is the following lemma.
Lemma 4.2
Let \(n,\, m \in {\mathbb N}\), \(p\in (1,2],\) \(s\in (1+1/p,2)\), \(\rho \in (0,(s-1-1/p)/2)\), \({\nu \in (0,\infty )}\), and \(f\in W^s_p({\mathbb R})\) be given. For sufficiently small \(\varepsilon \in (0,1)\) there exists a constant K that depends on \(\varepsilon ,\) n, m, s, p, \(\rho ,\) and \(\Vert f\Vert _{W^s_p}\) such that
for all \(|j|\le N-1\) and \(\overline{\omega }\in W^{s-2}_p({\mathbb R})\).
Proof
Taking advantage of the relation \(\chi _j^\varepsilon \pi _j^\varepsilon =\pi _j^\varepsilon\), we have
where
Since \(\pi _j^\varepsilon\) is a pointwise multiplier for \(W^{s-2-\rho }_p({\mathbb R})\) and \(\chi _j^\varepsilon\) is a pointwise multiplier for \(W^{s-2}_p({\mathbb R})\), Lemma 4.1 leads to
It remains to estimate \(T_{b}.\) Using (14) and the relation \({f'(x_j^\varepsilon )=\delta _{[x,y]} (f'(x_j^\varepsilon )\mathop \mathrm{id}\nolimits _{{\mathbb R}})/y},\) we get
If \(\varepsilon\) is sufficiently small, Lemma 4.4 (with \(\widetilde{s}=s-\rho\)) implies that
The estimate (66) follows from (67) and (68).
In Lemma 4.3 we gather some classical properties of mollifiers.
Lemma 4.3
Let \(\eta _\delta (x):=\delta ^{-1}\eta (x/\delta )\), \(x\in {\mathbb R}\) and \(\delta >0\), where \(\eta \in \mathrm{C}^\infty _0({\mathbb R})\) is a nonnegative function with
Given \(\delta >0\) and \(f\in L_p({\mathbb R})\), with \(p\in [1,\infty )\), let \(f_\delta :=f*\eta _\delta\). The following properties hold true.
-
(i)
Given \(r\ge 0,\) it holds \(\sup _{\delta >0} \big \Vert [f\mapsto f_\delta ]\Vert _{\mathcal {L}(W^r_p({\mathbb R}))}= 1;\)
-
(ii)
Given \(r\ge 0\), there exists a constant \(C>0\) such that
$$\begin{aligned}&\Vert f_\delta \Vert _{W^{r}_p}\le C\delta ^{r'-r}\Vert f\Vert _{W^{r'}_p}\qquad \text {for all}\quad {f\in W^{r'}_p({\mathbb R})}, r'\in [0,r], \delta \in (0,1]. \end{aligned}$$(69) -
(iii)
There exists a constant \(C>0\) such that
$$\begin{aligned}&\Vert f_\delta -f\Vert _{W^{r'}_p}\le C\delta ^{r-r'}\Vert f\Vert _{W^r_p}\qquad \text {for all} {f\in W^r_p({\mathbb R})}, 0\le r'\le r\le 1, \delta >0. \end{aligned}$$(70)
Lemma 4.4 below provides the key estimate in the proof of Lemma 4.2.
Lemma 4.4
Let \(n,\, m \in {\mathbb N}\), \(n\ge 1\), \(p\in (1,2],\) \(1+1/p<\widetilde{s}<s<2\), \(\nu \in (0,\infty )\), and \({f\in W^s_p({\mathbb R})}\) be given. For sufficiently small \(\varepsilon \in (0,1)\) there exists a positive constant \(K=K(\varepsilon ,\, n,\, m,\, s,\, \widetilde{s}, \Vert f\Vert _{W^s_p})\) such that
for all \(|j|\le N-1\) and \(\overline{\omega }\in W^{s-2}_p({\mathbb R})\).
Proof
Given \(\varphi ,\, \overline{\omega }\in \mathrm{C}^\infty _0({\mathbb R})\), it holds in view of \(\chi _j^\varepsilon \pi _j^\varepsilon =\pi _j^\varepsilon\) and of the fact that the \(L_2\)-adjoint of \(B_{n,m}^0(f)\) is the operator \(-B_{n,m}^0(f)\), that
Let \(T_{j,\varepsilon }[\varphi ]:=\chi _j^\varepsilon B_{n,m}(f,\ldots ,f)[f,\ldots , f ,f-f'(x_j^\varepsilon )\mathop \mathrm{id}\nolimits _{{\mathbb R}},\chi _j^\varepsilon \varphi ]\). We decompose below
and we prove subsequently that, if \(\varepsilon\) sufficiently small, then
for all \(\varphi \in W^{2-s}_{p'}({\mathbb R}).\) Having established (73), we get
and the claim (71) follows via a standard density argument.
In order to define the terms in (72), let \(\{\eta _\delta \}_{\delta >0}\) be a mollifier as in Lemma 4.3. We set
where \(\delta =\delta (\varepsilon )\in (0,1]\) will be fixed later on and \(\varphi _\delta :=\varphi *\eta _\delta\).
Since \(2-\widetilde{s}\in [0,1-1/p)\), Lemma 2.4 (with \(r=2-\widetilde{s}\)) and the fact that \({\chi _j^\varepsilon \varphi _\delta \in W^{2-\widetilde{s}}_{p'}({\mathbb R})}\) yield \(T_{j,\varepsilon }^2[\varphi ]\in W^{2-\widetilde{s}}_{p'}({\mathbb R}),\) and together with (69) we obtain the estimate
It remains to estimate \(T_{j,\varepsilon }^1[\varphi ].\) Therefore, we first infer from Lemma 2.1 and (70) that
We now consider the seminorm \([T_{j,\varepsilon }^1[\varphi ]]_{W^{2-s}_{p'}}\). In view of (14) we write
where
with
Lemma 2.1 (with \(p=p'\)) together wit (70) yields
With regard to \(T_3(\xi )\) we fix \(\rho \in (s-\widetilde{s},\min \{2-\widetilde{s},2(s-\widetilde{s})\})\). Combining Lemma 2.2 (with \(s=\widetilde{s}\) and \(r=2-\widetilde{s}-\rho \in (0,1-1/p)\)) and (70) we get
In remains to estimate the function \(T_{2}(\xi ),\) \(\xi \in {\mathbb R}.\) To this end we choose \(a_{j,\varepsilon }\in {\mathbb R}\) such that \({\mathop \mathrm{supp}\nolimits \chi _j^\varepsilon \subset [a_{j,\varepsilon }-3\varepsilon /2,a_{j,\varepsilon }+3\varepsilon /2]}\) and denote by \(F_\varepsilon\) the Lipschitz continuous function defined by \(F_\varepsilon =f\) on \({[a_{j,\varepsilon }-2\varepsilon ,a_{j,\varepsilon }+2\varepsilon ]}\) and \(F_\varepsilon '=f'(x_j^\varepsilon )\) on \({\mathbb R}\setminus [a_{j,\varepsilon }-2\varepsilon ,a_{j,\varepsilon }+2\varepsilon ]\). Given \(\xi \in {\mathbb R}\) with \(|\xi |\ge \varepsilon /2,\) it follows from Lemma 2.1 and (70) that
If \(|\xi |<\varepsilon /2\), then \(\xi + \mathop \mathrm{supp}\nolimits \chi _j^\varepsilon \subset [a_{j,\varepsilon }-2\varepsilon ,a_{j,\varepsilon }+2\varepsilon ]\), and, since \(f'\in \mathrm{C}^{s-1-1/p}({\mathbb R})\), Lemma 2.1 and the properties defining \(F_\varepsilon\) lead to
provided that \(\varepsilon\) is sufficiently small.
Combining (76)–(79), we conclude that if \(\varepsilon\) is sufficiently small, then
Invoking (4) and Lemma 4.3 (i), the estimates (75) and (80) lead us to
We now chose \(\delta =\delta (\varepsilon )\in (0,1]\) sufficiently small to ensure that \(K\delta ^{\rho +\widetilde{s}-s}\le \nu /2.\) This choice together with (74) shows that the estimates in (73) hold true and the proof is complete.
The next lemma is the second main result of this section and describes how to freeze the kernels at infinity.
Lemma 4.5
Let \(n,\, m \in {\mathbb N}\), \(p\in (1,2],\) \(s\in (1+1/p,2)\), \(\rho \in (0,(s-1-1/p)/2)\), \({\nu \in (0,\infty )}\), and \(f\in W^s_p({\mathbb R})\) be given. For sufficiently small \(\varepsilon \in (0,1)\) there exists a constant K that depends on \(\varepsilon ,\) n, m, s, p, \(\rho ,\) and \(\Vert f\Vert _{W^s_p}\) such that
and
for all \(\overline{\omega }\in W^{s-2}_p({\mathbb R})\).
Proof
Similarly as in the proof of Lemma 4.2 we write
where
Since \(\pi _j^\varepsilon\) is a pointwise multiplier for \(W^{s-2-\rho }_p({\mathbb R})\) and \(\chi _j^\varepsilon\) is a pointwise multiplier for \(W^{s-2}_p({\mathbb R})\), we get in view of Lemma 4.1 that
With respect to \(T_b\) we infer from (14) that
If \(\varepsilon\) sufficiently small, it follows from Lemma 4.6 that
The claim (81) follows now from (83) and (84). Finally, the assertion (82) is obtained by arguing as in the proof of (81).
Lemma 4.6 below provides the crucial estimate in the proof of Lemma 4.5.
Lemma 4.6
Let \(n,\, m \in {\mathbb N}\), \(n\ge 1\), \(p\in (1,2],\) \(1+1/p<\widetilde{s}<s<2\), \(\nu \in (0,\infty )\), and \({f\in W^s_p({\mathbb R})}\) be given. For sufficiently small \(\varepsilon \in (0,1)\), there exists a positive constant \(K=K(\varepsilon ,\, n,\, m,\, s,\,\widetilde{s},\,\Vert f\Vert _{W^s_p})\) such that
for all \(\overline{\omega }\in W^{s-2}_p({\mathbb R})\).
Proof
Given \(\varphi ,\, \overline{\omega }\in \mathrm{C}^\infty _0({\mathbb R})\), it follows by arguing as in the proof of Lemma 4.4 that
We write \(\chi _N^\varepsilon B_{n,m}^0(f)[\chi _N^\varepsilon \varphi ]=T_{\varepsilon }^1[\varphi ]+T^2_{\varepsilon }[\varphi ]\) where, for sufficiently small \(\varepsilon\), we have
for all \(\varphi \in W^{2-s}_{p'}({\mathbb R}).\) The estimates in (86) together with the previous identity imply
and (85) follows.
Let again \(\{\eta _\delta \}_{\delta >0}\) be a mollifier and set
where \(\varphi _\delta :=\varphi *\eta _\delta\) and with \(\delta =\delta (\varepsilon )\in (0,1]\) which we fix later on. Taking advantage of Lemma 2.4 (with \({r=2-\widetilde{s}\in (0,1-1/p)}\)) and of the fact that \(\chi _N^\varepsilon \varphi _\delta \in W^{2-\widetilde{s}}_{p'}({\mathbb R})\), we conclude that \(T_{\varepsilon }^2[\varphi ]\in W^{2-\widetilde{s}}_{p'}({\mathbb R})\) and together with (69) we obtain the estimate
We now estimate \(T_{\varepsilon }^1[\varphi ].\) Combining Lemma 2.1 and (70), we have
and it remains to consider the seminorm \([T_{\varepsilon }^1[\varphi ]]_{W^{2-s}_{p'}}\). Using (14), we get
where
with \(B_{n+2,m+1}^j\) as defined in the proof of Lemma 4.4. Arguing as in Lemma 4.4, we obtain
and, for some fixed \(\rho \in (s-\widetilde{s},\min \{2-\widetilde{s},2(s-\widetilde{s})\})\),
In order to estimate \(T_{2}(\xi ),\) let \(F_\varepsilon\) denote the Lipschitz continuous function equal to f on the set \(\{|x|\ge 1/\varepsilon -2\varepsilon \}\) and which is linear in the interval \(\{|x|\le 1/\varepsilon -2\varepsilon \}\). If \(|\xi |\ge \varepsilon ,\) Lemma 2.1 and (70) yield
If \(|\xi |<\varepsilon\), we note that \(\xi + \mathop \mathrm{supp}\nolimits \chi _N^\varepsilon \subset \{|x|\ge 1/\varepsilon -2\varepsilon \}\). Lemma 2.1, the definition of \(F_\varepsilon\), and the observation that \(\Vert F_\varepsilon '\Vert _{\infty }\le 2\varepsilon \Vert f\Vert _\infty +\Vert f'\Vert _{L_\infty (\{|x|\ge 1/\varepsilon -2\varepsilon \})}\) (which holds if \(\varepsilon\) is sufficiently small), then lead to
provided that \(\varepsilon\) is sufficiently small. To obtain the last inequality we have taken advantage of the fact that \(f'\in W^{s-1}_p({\mathbb R})\) vanishes at infinity.
From (89) to (92), we conclude, for \(\varepsilon\) sufficiently small, that
Combining (4), Lemma 4.3 (i), (88), and (93) we get
Choosing now \(\delta =\delta (\varepsilon )\) sufficiently small to ensure that \(K\delta ^{\rho +\widetilde{s}-s}\le \nu /2,\) we obtain, together with (87), the desired estimates (86) and the proof is complete.
Estimates for some pointwise multipliers
In this appendix we present the proofs of the estimates (6)–(7). The estimate (7) is used in the proof of Theorem 3.3 and (6) is an important argument when establishing (7).
Proof of (6)
Since \(W^{r}_p({\mathbb R})\hookrightarrow \mathrm{C}({\mathbb R})\), we have
and it remains to estimate the term
According to [5, Theorem 4.1], the multiplication
is continuous, hence
Lemma 2.3 (with \(t'=2(r-1/p)\) and \(t=r+1-2/p\)) yields
We thus conclude that
and (94) together with (95) lead to the desired claim.
We are now in a position to prove (7).
Proof of (7)
Given \(h,\, \psi \in \mathrm{C}^\infty _0({\mathbb R})\), we have
where \(T_1[\psi ]\in W^{1-r}_{p'}({\mathbb R})\) and \(T_2[\psi ]\in W^{1- r+\rho }_{p'}({\mathbb R})\) are defined in (97) and satisfy
Having established (96), we get
and the estimate (7) follows. The functions in (96) are defined by
where \(\{\eta _\delta \}_{\delta >0}\) is a mollifier, \(\psi _\delta =\psi *\eta _\delta\), and \(\delta \in (0,1]\) is chosen below. Combining (6), Lemma 4.3 (i), and (70), we get
After eventually choosing C to be larger than the norm of the embedding \(W^r_p({\mathbb R})\hookrightarrow L_\infty ({\mathbb R}),\) we set \(\delta = (\Vert \varphi \Vert _\infty /(C\Vert \varphi \Vert _{W^r_p}))^{1/\rho }\in (0,1]\), and obtain that
With respect to \(T_2[\psi ]\), we note that since \(\varphi \in W^{r+1-2/p}_{p'}({\mathbb R})\) and \(\psi _\delta \in W^{r+1}_{p'}({\mathbb R})\), it holds that \(T_2[\psi ]\in W^{r+1-2/p}_{p'}({\mathbb R})\hookrightarrow W^{1-r+\rho }_{p'}({\mathbb R})\), cf. (2), and together with (2) and (69) we get
Hence, both estimates in (96) hold true and the proof is complete.
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Matioc, AV., Matioc, BV. The Muskat problem with surface tension and equal viscosities in subcritical \(L_p\)-Sobolev spaces. J Elliptic Parabol Equ 7, 635–670 (2021). https://doi.org/10.1007/s41808-021-00104-1
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DOI: https://doi.org/10.1007/s41808-021-00104-1