# Global solutions for the Muskat problem in the scaling invariant Besov space
$\dot{B}^{1}_{\infty,1}$

Huy Q. Nguyen  Department of Mathematics, Brown University, Providence, RI
02912<EMAIL_ADDRESS>
###### Abstract.

The one-phase and two-phase Muskat problems with arbitrary viscosity contrast
are studied in all dimensions. They are quasilinear parabolic equations for
the graph free boundary. We prove that small data in the scaling invariant
homogeneous Besov space $\dot{B}^{1}_{\infty,1}$ lead to unique global
solutions. The proof exploits a new structure of the Dirichlet-Neumann
operator which allows us to implement a robust fixed-point argument. As a
consequence of this method, the initial data is only assumed to be in
$\dot{B}^{1}_{\infty,1}$ and the solution map is Lipschitz continuous in the
same topology. For the general Muskat problem, the only known scaling
invariant result was obtained in the Wiener algebra (plus an $L^{2}$
assumption) which is strictly contained in $\dot{B}_{\infty,1}^{1}$.

###### Key words and phrases:

Muskat problem, viscosity contrast, global well-posedness, Besov spaces

MSC Classification: 35R35, 35Q35, 35A01, 35A02.

##  1\. Introduction

###  1.1. The Muskat problem

The two-phase Muskat problem concerns the dynamics of the free interface
between two immiscible fluids permeating a porous medium. The fluids have
different constant densities $\rho^{\pm}$ and, in general, different
viscosities $\mu^{\pm}$. Assume that the free interface is the graph of a
time-dependent function $\eta(x,t)$, i.e.

$\Sigma_{t}=\\{(x,\eta(x,t)):x\in\mathbb{R}^{d}\\},$

so that the fluids occupy the regions

$\Omega^{+}_{t}=\\{(x,y)\in\mathbb{R}^{d+1}:y>\eta(x,t)\\},\quad\Omega^{-}_{t}=\\{(x,y)\in\mathbb{R}^{d+1}:y<\eta(t,x)\\}.$

The fluid motion is modeled by Darcy’s law:

$\mu^{\pm}u^{\pm}+\nabla_{x,y}p^{\pm}=-(0,\rho^{\pm}),\quad\operatorname{div}_{x,y}u^{\pm}=0\quad\text{in}~{}\Omega^{\pm}_{t},$
(1.1)

Here $u^{\pm}$ are the velocity fields, $p^{\pm}$ are the pressure fields, and
we have normalized the gravity constant to $1$ for notational simplicity. The
fluids are assumed to be quiescent at great depths,

$\lim_{y\to\pm\infty}u^{\pm}(x,y)=0.$ (1.2)

The normal velocity is continuous across the interface, i.e.

$u^{+}\cdot n=u^{-}\cdot n\quad\text{on}~{}\Sigma_{t}\quad\text{on
}\Sigma_{t},$ (1.3)

where $n=\frac{1}{\sqrt{1+|\nabla\eta|^{2}}}(-\nabla\eta,1)$ is the upward
pointing unit normal to $\Sigma_{t}$. Then the interface moves with the
fluids:

$\partial_{t}\eta=\sqrt{1+|\nabla\eta|^{2}}u^{-}\cdot n|_{\Sigma_{t}}.$ (1.4)

Neglecting the effect of surface tension, the pressure is continuous at the
interface, i.e.

$p^{+}=p^{-}\quad\text{on}~{}\Sigma_{t}.$ (1.5)

We shall refer to the system (1.1)-(1.5) as the two-phase Muskat problem. When
the top phase corresponds to vacuum, i.e. $\mu^{+}=\rho^{+}=0$, the two-phase
Muskat problem reduces to the one-phase Muskat problem, in which case (1.5)
becomes

$p^{-}=0\quad\text{on}~{}\Sigma_{t}.$ (1.6)

###  1.2. Main result

A remarkable feature of the Muskat problem is that it can be recast as a
quasilinear nonlocal parabolic equation for the free boundary $\eta$. When the
initial data is sufficiently smooth, local well-posedness for large data and
global well-posedness for small data were established in [21, 11, 22, 16, 19,
5, 6, 17, 18, 12]. It turns out that the problem is invariant with respect to
the rescaling

$\eta(x,t)\to\lambda^{-1}\eta(\lambda x,\lambda t)\quad\forall\lambda>0.$

Therefore, it is natural to develop well-posedness theories in function spaces
that respect this scaling invariance. These include, among others, the
Lipschitz space $\dot{W}^{1,\infty}(\mathbb{R}^{d})$, the Wiener algebra
$\mathbb{A}^{1}(\mathbb{R}^{d})$, and the homogeneous Besov space
$\dot{B}^{1+\frac{d}{p}}_{p,q}(\mathbb{R}^{d})$ for all $p$, $q\in[1,\infty]$.
The latter includes the $L^{2}$-based Sobolev space
$\dot{H}^{1+\frac{d}{2}}(\mathbb{R}^{d})$ as a special case.

There have been significant developments on well-posedness theories in or
close to scaling invariant spaces. Local well-posedness for large data
arbitrarily close to scaling (“scaling$+\varepsilon$”) is quite well
understood. The case of no viscosity contrast ($\mu^{+}=\mu^{-}$) was treated
in [15] with $W^{2,p}(\mathbb{R})$ data for all $p\in(1,\infty]$, and in [32]
with $H^{\frac{3}{2}+\varepsilon}(\mathbb{R})$ data. The latter was then
improved to the homogeneous space
$\dot{H}^{1}(\mathbb{R})\cap\dot{H}^{\frac{3}{2}+}(\mathbb{R})$ [1], which is
natural since the PDE annihilates constants. For arbitrary viscosity contrast
and in all dimensions, local well-posedness was obtained in [33] for
$H^{1+\frac{d}{2}+\varepsilon}(\mathbb{R}^{d})$ data. See also [1] for the
one-phase case. The method developed in [33] could also handle the effect of
surface tension [34] and the zero surface tension limit [23] at the same
regularity level. The only existing scaling invariant (modulo low-frequency
assumptions) existence and uniqueness results are [3, 4] for the case of no
viscosity contrast. They were obtained for data in
$H^{\frac{3}{2}}(\mathbb{R})$ and
$W^{1,\infty}(\mathbb{R}^{2})\cap\dot{H}^{2}(\mathbb{R}^{2})$, respectively.

On the other hand, global well-posedness for small data is more developed
owing to the parabolic nature of the Muskat problem. The first result of this
type is [14] which considers the equal viscosities case with data in the
subcritical space $H^{3}(\mathbb{R})$ but small (with explicit bound) in the
Wiener norm

$\|\eta_{0}\|_{\mathbb{A}^{1}}=\int_{\mathbb{R}}|\xi||\widehat{f}(\xi)|d\xi.$
(1.7)

This Wiener framework was then sucessfully employed to study a variety of
problems such as the Navier-Stokes equations [31], the Landau and non-cutoff
Boltzmann equations [25], the Peskin problem [30]. The results in [14] were
extended to 3D in [13]. Smallness in $\mathbb{A}^{1}$ implies smallness of the
slope, i.e. of $\|\eta_{0}\|_{\dot{W}^{1,\infty}}$. Still for the case of no
viscosity contrast, if the initial Lipschitz norm is smaller than an explicit
number and the profile grows sublinearly at infinity, then there exists a
unique global classical solution [8, 9]. Recent developments can address
arbitrarily large slopes [20, 29, 4] and even infinite slope [3]. Precisely,
[20, 29, 4] consider Lipschitz data that are small in the critical Sobolev
space $\dot{H}^{1+\frac{d}{2}}(\mathbb{R}^{d})$, $d\in\\{1,2\\}$, whereas the
Lipschitz condition is removed in [3] for $d=1$. As far as the general case of
arbitrary viscosity contrast is concerned, to the best of our knowledge, the
only scaling invariant result is [27] for data in $L^{2}(\mathbb{R}^{d})$ and
small in $\mathbb{A}^{1}(\mathbb{R}^{d})$, $d\in\\{2,3\\}$. See also [28] for
the global stability of bubbles. The case of viscosity contrast, including the
one-phase problem, is more practical and mathematically challenging since in
addition to the evolution equation for the free boundary, one has to invert a
nonlocal operator for the vorticity amplitude at the free boundary. Unlike the
case of equal viscosities, there is no currently available $L^{\infty}$ or
Lipschitz maximum principle when the viscosity contrast is nonzero. See [27]
for the equations and further discussions. For the one-phase problem, by
taking advantage of the maximum principle for the slope, it is proved in [24]
that for any periodic Lipschitz initial data, there exists a global strong
solution which is also the unique viscosity solution.

The purpose of this paper is to provide an alternative scaling invariant
framework for the general Muskat problem: the Besov framework through the
Dirichlet-Neumann operator. To ensure the finiteness of the slope as in the
Wiener framework, we consider the scaling invariant Besov spaces
$\dot{B}^{\frac{d}{p}+1}_{p,1}(\mathbb{R}^{d})$, $p\in[1,\infty]$. The largest
space in this class is $\dot{B}^{1}_{\infty,1}(\mathbb{R}^{d})$ which strictly
contains $\mathbb{A}^{1}(\mathbb{R}^{d})$ and

$\|\cdot\|_{\dot{B}_{\infty,1}^{1}}\leq C\|\cdot\|_{\mathbb{A}^{1}}.$ (1.8)

For example, the function $f(x)=\cos(\sqrt{2}x)+\cos x$ belongs to
$\dot{B}^{1}_{\infty,1}(\mathbb{R})$ but not to $\mathbb{A}^{1}$. Note that
$f$ is neither periodic nor decaying at infinity.

We establish global well-posedness for small data in $\dot{B}_{\infty,1}^{1}$
by exploiting a semilinear-like structure of the problem for such data. Our
starting point is the reformulation [33] of the general Muskat problem in
terms of the Dirichlet-Neumann operators $G^{\pm}(\eta)$ associated to the
fluid domains $\Omega^{\pm}$. Precisely, for a given function $f$, if
$\phi^{\pm}$ solve

$\begin{cases}\Delta_{x,y}\phi^{\pm}=0\quad\text{in}~{}\Omega^{\pm},\\\
\phi^{\pm}=f\quad\text{on}~{}\Sigma,\\\ \nabla_{x,y}\phi\to 0\quad\text{as
}y\to\pm\infty,\end{cases}$ (1.9)

then

$G(\eta)^{\pm}f:=\sqrt{1+|\nabla\eta|^{2}}\frac{\partial\phi^{\pm}}{\partial
n}.$ (1.10)

######  Proposition 1.1 ([33]).

Let $d\geq 1$.
1\. $(u,p,\eta)$ solve the one-phase Muskat problem if and only if
$\eta:\mathbb{R}^{d}\to\mathbb{R}$ obeys the equation

$\partial_{t}\eta+\frac{\rho^{-}}{\mu^{-}}G^{-}(\eta)\eta=0.$ (1.11)

2\. $(u^{\pm},p^{\pm},\eta)$ solve the two-phase Muskat problem if and only if

$\partial_{t}\eta=-\frac{1}{\mu^{-}}G^{-}(\eta)f^{-},$ (1.12)

where $f^{\pm}:=p^{\pm}|_{\Sigma}+\rho^{\pm}\eta$ satisfy

$\begin{cases}f^{+}-f^{-}=(\rho^{+}-\rho^{-})\eta,\\\
\frac{1}{\mu^{+}}G^{+}(\eta)f^{+}-\frac{1}{\mu^{-}}G^{-}(\eta)f^{-}=0.\end{cases}$
(1.13)

We denote throughout this paper

$\kappa=\frac{\llbracket\rho\rrbracket}{\mu^{+}+\mu^{-}},\quad\llbracket\rho\rrbracket=\rho^{-}-\rho^{+}.$

It will be shown that $\kappa$ is the dissipation coefficient.

Our main result is the following global well-posedness in Hadamard’s sense for
small data in $\dot{B}_{\infty,1}^{1}(\mathbb{R}^{d})$.

######  Theorem 1.2.

Consider either the one-phase or the two-phase Muskat problem in the stable
case $\rho^{+}<\rho^{-}$. Let
$\eta_{0}\in\dot{B}^{1}_{\infty,1}(\mathbb{R}^{d})$, $d\geq 1$, be an initial
datum. Then there exist positive constants $\delta$ and $C$, both depending
only on $d$, such that the following holds: if
$\|\eta_{0}\|_{\dot{B}^{1}_{\infty,1}}\leq\delta$ then for any $T>0$, (1.11)
and (1.12)-(1.13) have a unique solution $\eta\in
C([0,T];\dot{B}^{1}_{\infty,1}(\mathbb{R}^{d}))$ satisfying

$\|\eta\|_{\widetilde{L}^{\infty}([0,T];\dot{B}^{1}_{\infty,1})}+\kappa\|\eta\|_{\widetilde{L}^{1}([0,T];\dot{B}^{2}_{\infty,1})}\leq
C\|\eta_{0}\|_{\dot{B}_{\infty,1}^{1}}.$ (1.14)

Here $\widetilde{L}^{q}([0,T];\dot{B}_{\infty,1}^{s})$ denotes the Chemin-
Lerner space. Moreover, if $\eta^{1}$ and $\eta^{2}$ are two solutions as
described above, then for all $T>0$,

$\|\eta^{1}-\eta^{2}\|_{\widetilde{L}^{\infty}([0,T];\dot{B}^{1}_{\infty,1})}+\kappa\|\eta^{1}-\eta^{2}\|_{\widetilde{L}^{1}([0,T];\dot{B}^{2}_{\infty,1})}\leq
C\|\eta^{1}(0)-\eta^{2}(0)\|_{\dot{B}_{\infty,1}^{1}}.$ (1.15)

Theorem 1.2 does not require any additional assumption on low frequencies of
initial data. This is natural since for any constant $c$, $\eta+c$ is a
solution if $\eta$ is a solution. This and the Lipschitz continuity of the
solution map in $\dot{B}_{\infty,1}^{1}$ follow as a direct consequence of the
fact that our solutions are constructed by means of a fixed-point argument.
Since we do not rely on explicit contour equations [19, 27], the proof of
Theorem 1.2 works regardless of the dimension and extends to the periodic
setting. We stress that the smallness $\delta$ is independent of all
parameters of the problem.

######  Remark 1.3.

It would be interesting to investigate the long time decay and instant
analyticity of the solutions constructed in Theorem 1.2. However, in order to
elucidate the main ideas, we do not pursue these issues in the current paper.

###  1.3. The method of proof

By virtue of Proposition 1.1, the analysis of the Muskat problem reduces to
that of the Dirichlet-Neumann operator. To fix ideas let us consider first the
one-phase equation (1.11) in which the nonlinearity $G^{-}(\eta)\eta$ is fully
nonlinear. In [33], a precise structure of $G^{-}(\eta)f$ is required to
handle large data. Precisely, we proved that for subcritical Sobolev
regularity, $G^{-}(\eta)f$ can be decomposed into a first order elliptic
(paradifferential) operator acting on $f$ plus a transport (paradifferential)
operator acting on $\eta$, and a lower order remainder. On the other hand, for
small data, it turns out that the following different but simpler structure
suffices

$G^{-}(\eta)f=|D_{x}|f+R^{-}(\eta)f,$ (1.16)

where $R^{-}(\eta)f$ contains only quadratic and higher nonlinearities. The
definition of $R^{-}(\eta)f$ involves $\eta$ and
$v:\mathbb{R}^{d}\times\mathbb{R}_{-}\to\mathbb{R}$, where the latter is the
harmonic potential $\phi$, defined by (1.9), in the flattened domain
$\\{(x,y):y<0\\}$. We shall prove that $v$ can be found as the unique fixed
point in Chemin-Lerner spaces of some operator $\mathcal{T}$ depending on
$\eta$ (see (3.16)), provided that $\eta$ is small in
$\dot{B}_{\infty,1}^{1}$. Moreover, $R^{-}(\eta)f$ satisfies the good Besov
bound

$\displaystyle\|R^{-}(\eta)f\|_{\dot{B}^{1}_{\infty,1}}$ $\displaystyle\leq
C\|\eta\|_{\dot{B}^{1}_{\infty,1}}\|f\|_{\dot{B}^{2}_{\infty,1}}+C\|\eta\|_{\dot{B}^{2}_{\infty,1}}\|f\|_{\dot{B}^{1}_{\infty,1}}.$
(1.17)

Interestingly, at the same $\dot{B}_{\infty,1}^{1}$ regularity, $R^{-}(\eta)f$
has the contraction property

$\displaystyle\|[R^{-}(\eta_{1})-R^{-}(\eta_{2})]f\|_{\dot{B}^{1}_{\infty,1}}$
(1.18) $\displaystyle\leq
C\|(\eta_{1},\eta_{2})\|_{\dot{B}^{2}_{\infty,1}}\|f\|_{\dot{B}^{1}_{\infty,1}}+C\|\eta_{\delta}\|_{\dot{B}^{1}_{\infty,1}}\|f\|_{\dot{B}^{2}_{\infty,1}}+C\|\eta_{\delta}\|_{\dot{B}^{2}_{\infty,1}}\|f\|_{\dot{B}^{1}_{\infty,1}},\quad\eta_{\delta}=\eta_{1}-\eta_{2}.$

Now in view of (1.16), the one-phase equation (1.11) can be written in the
Duhamel form

$\eta(t)=e^{-\kappa
t|D_{x}|}\eta_{0}-\kappa\int_{0}^{t}e^{-\kappa(t-\tau)|D_{x}|}(R^{-}(\eta)\eta)(\tau)d\tau.$
(1.19)

This is reminiscent of mild solutions for the semilinear Navier-Stokes
equations [26, 10] but our nonlinearity $R^{-}(\eta)\eta$ is implicit and not
purely quadratic. Nevertheless, by virtue of the boundedness (1.17) and the
contraction (1.18) of $R^{-}(\eta)f$, a unique fixed point $\eta$ of (1.19)
can be easily obtained in the space
$\widetilde{L}^{\infty}([0,T];\dot{B}_{\infty,1}^{1})\cap\widetilde{L}^{1}([0,T];\dot{B}_{\infty,1}^{2})$
for any $T>0$, provided that
$\|\eta_{0}\|_{\dot{B}_{\infty,1}^{1}}\leq\delta(d)\ll 1$. In other words, for
small data, the quasilinear Muskat problem can be treated as a semilinear
equation. The above fixed point argument allows us to bypass any additional
assumption on low frequencies of data (for instance, $\eta_{0}\in L^{2}$ in
[27]) and prove at once the Lipschitz continuity of the solution map in the
top topology of $\dot{B}_{\infty,1}^{1}$.

Regarding the two-phase problem, owing to the linearization (1.16) we find
that $f^{-}$, the solution of (1.13), can be again obtained as the unique
fixed point of some operator, provided that $\eta$ is small in
$\dot{B}_{\infty,1}^{1}$. Moreover, $f^{-}$ has the same Besov regularity as
$\eta$, so that the two-phase equation (1.12) can be analyzed exactly the same
as the one-phase equation (1.11).

A review of homogeneous Besov spaces is given in Section 2. Section 3 is
devoted to the linearization, boundedness and contraction properties of the
Dirichlet-Neumann operator in Besov spaces. The proof of Theorem 1.2 is given
in Section 4. After the preliminaries in Section 2, the remainder of the paper
is self-contained.

##  2\. A review of homogeneous Besov spaces

###  2.1. Homogeneous Besov spaces

######  Proposition 2.1 ([7, Proposition 2.10]).

Let $\mathcal{C}$ be the annulus
$\\{\xi\in\mathbb{R}^{d}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$. There exists
a radial function $\psi$ valued in $[0,1]$ and belonging to
$C_{c}^{\infty}(B(0,\frac{4}{3}))$ such that the following hold

$\displaystyle\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1\quad\forall\xi\in\mathbb{R}^{d}\setminus\\{0\\},$
(2.1)
$\displaystyle\operatorname{supp}\varphi(2^{-j}\cdot)\cap\operatorname{supp}\varphi(2^{-j^{\prime}}\cdot)=\emptyset\quad\text{for}~{}|j-j^{\prime}|\geq
2,$ (2.2)
$\displaystyle\frac{1}{2}\leq\sum_{j\in\mathbb{Z}}\varphi^{2}(2^{-j}\xi)\leq
1\quad\forall\xi\in\mathbb{R}^{d}\setminus\\{0\\}.$ (2.3)

######  Definition 2.2.

The Littlewood-Paley dyadic block $\dot{\Delta}_{j}$ is defined by the Fourier
multiplier $\dot{\Delta}_{j}u=\varphi(2^{-j}D_{x})u$. The low-frequency cut-
off operator $\dot{S}_{j}$ is defined by

$\dot{S}_{j}=\sum_{j^{\prime}\leq j-1}\dot{\Delta}_{j^{\prime}}.$ (2.4)

Let $\mathcal{P}(\mathbb{R}^{d})$ and $\mathcal{S}^{\prime}(\mathbb{R}^{d})$
respectively denote the set of polynomials on $\mathbb{R}^{d}$ and the space
of tempered distributions on $\mathbb{R}^{d}$. It is well-known that

$\text{I}=\sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}\quad\text{in
}\mathcal{S}^{\prime}(\mathbb{R}^{d})/\mathcal{P}(\mathbb{R}^{d}).$

######  Definition 2.3.

1) For $(p,r)\in[1,\infty]^{2}$ and $s\in\mathbb{R}$, the homogeneous Besov
space $\dot{B}^{s}_{p,r}(\mathbb{R}^{d})$ is the space of
$u\in\mathcal{S}^{\prime}(\mathbb{R}^{d})/\mathcal{P}(\mathbb{R}^{d})$ such
that the following norm is finite

$\|u\|_{\dot{B}^{s}_{p,r}(\mathbb{R}^{d})}=\|2^{sj}\|\dot{\Delta}_{j}u\|_{L^{p}(\mathbb{R}^{d})}\|_{\ell^{r}(\mathbb{Z})}.$

2) For $I\subset\mathbb{R}$, $(p,q,r)\in[1,\infty]^{3}$ and $s\in\mathbb{R}$,
the Chemin-Lerner norm is defined by

$\|u\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r}(\mathbb{R}^{d}))}=\|2^{sj}\|\dot{\Delta}_{j}u\|_{L^{q}(I;L^{p}(\mathbb{R}^{d}))}\|_{\ell^{r}(\mathbb{Z})}.$

We then define the Chemin-Lerner space
$\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r}(\mathbb{R}^{d}))$ to be the space of
tempered distributions
$u\in\mathcal{S}^{\prime}(\mathbb{R}^{d+1})/\mathcal{P}(\mathbb{R}^{d})$ and
$\|u\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r}(\mathbb{R}^{d}))}<\infty$.

######  Remark 2.4.

The inequality (1.8) holds since for $u\in\mathbb{A}^{1}$ we have

$\displaystyle\|u\|_{\dot{B}_{\infty,1}^{1}}$
$\displaystyle=\sum_{j\in\mathbb{Z}}2^{j}\|\dot{\Delta}_{j}u\|_{L^{\infty}(\mathbb{R}^{d})}\leq\sum_{j\in\mathbb{Z}}2^{j}\|\psi_{j}\widehat{u}\|_{L^{1}(\mathbb{R}^{d})}$
$\displaystyle\leq\int_{\mathbb{R}^{d}}\sum_{j\in\mathbb{Z}}2^{j}\psi_{j}(\xi)|\widehat{u}(\xi)|d\xi\leq\frac{4}{3}\int_{\mathbb{R}^{d}\setminus\\{0\\}}\sum_{j\in\mathbb{Z}}\psi_{j}(\xi)|\xi||\widehat{u}(\xi)|d\xi=\frac{4}{3}\|u\|_{\mathbb{A}^{1}}.$

For all $s\in\mathbb{R}$ and $(p,q)\in[1,\infty]^{2}$,
$\dot{B}^{s}_{p,q}(\mathbb{R}^{d})$ is a Banach space (see [35], page 240).
The following proposition can be proved analogously to Theorem 2.25 [7].

######  Proposition 2.5.

For all $s\in\mathbb{R}$ and $(p,q,r)\in[1,\infty]^{3}$,
$\widetilde{L}^{q}(I;B^{s}_{p,r}(\mathbb{R}^{d}))$ is a Banach space and
satisfies the Fatou property: for any sequence $(u_{n})$ bounded in
$\widetilde{L}^{q}(I;B^{s}_{p,r}(\mathbb{R}^{d}))$, there exist a subsequence
$(u_{n_{k}})$ and $u\in\widetilde{L}^{q}(I;B^{s}_{p,r}(\mathbb{R}^{d}))$ such
that

$\lim_{k\to\infty}u_{n_{k}}=u\quad\text{in
}\mathcal{S}^{\prime}(\mathbb{R}^{d+1})/\mathcal{P}(\mathbb{R}^{d})\quad\text{and}\quad\|u\|_{\widetilde{L}^{q}(I;B^{s}_{p,r})}\leq\liminf_{k\to\infty}\|u_{n_{k}}\|_{\widetilde{L}^{q}(I;B^{s}_{p,r})}.$
(2.5)

By Minkowski’s inequality, we have

$\|u\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r})}\leq\|u\|_{L^{q}(I;\dot{B}^{s}_{p,r})}~{}~{}\text{if}~{}~{}q\leq
r,\quad\|u\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r})}\geq\|u\|_{L^{q}(I;\dot{B}^{s}_{p,r})}~{}~{}\text{if}~{}~{}q\geq
r.$ (2.6)

We shall need to following lemma to prove the continuity in time of solutions.

######  Lemma 2.6.

Let $p\in[1,\infty]$ and $s,~{}s^{\prime}\in\mathbb{R}$ with $s\geq
s^{\prime}$. If
$u\in\widetilde{L}^{\infty}([a,b];\dot{B}^{s}_{p,1}(\mathbb{R}^{d}))$ and
$\partial_{z}u\in\widetilde{L}^{1}([a,b];\dot{B}^{s^{\prime}}_{p,1}(\mathbb{R}^{d}))$,
then $u\in C([a,b];\dot{B}^{s}_{p,1}(\mathbb{R}^{d}))$.

###### Proof.

We first note that if $v\in\widetilde{L}^{q}([a,b];\dot{B}^{s_{1}}_{p,r})$
then $\dot{S}_{j}v\in\widetilde{L}^{q}([a,b];\dot{B}^{s_{2}}_{p,r})$ for all
$s_{2}\geq s_{1}$ and all $j\in\mathbb{Z}$. Applying this we obtain

$\dot{S}_{j}u\in\widetilde{L}^{\infty}([a,b];\dot{B}^{s}_{p,1}),\quad\partial_{z}\dot{S}_{j}u=\dot{S}_{j}\partial_{z}u\in\widetilde{L}^{1}([a,b];\dot{B}^{s}_{p,1})=L^{1}([a,b];\dot{B}^{s}_{p,1}),$

whence $\dot{S}_{j}u\in C([a,b];\dot{B}^{s}_{p,1})$. It then suffices to prove
that the sequence of $\dot{B}^{s}_{p,1}$-valued functions $\dot{S}_{j}u$
converges to $u$ uniformly on $[0,T]$ as $j\to\infty$. Indeed, since
$u-\dot{S}_{j}u=\sum_{k\geq j}\dot{\Delta}_{j}u$ we have

$\|u-\dot{S}_{j}u\|_{L^{\infty}([a,b];\dot{B}^{s}_{p,1})}\leq\sum_{k\geq
j}\sum_{\ell\in\mathbb{Z}}2^{sk}\|\dot{\Delta}_{\ell}\dot{\Delta}_{k}u\|_{L^{\infty}([a,b];L^{p})}\leq
C\sum_{k\geq j-3}2^{sk}\|\dot{\Delta}_{k}u\|_{L^{\infty}([a,b];L^{p})}$

which converges to $0$ as $j\to\infty$ because it is the tail of the series
convergent to $\|u\|_{\widetilde{L}^{\infty}([a,b];\dot{B}^{s}_{p,1})}$. ∎

The following product rules and nonlinear estimates shall be used frequently.

######  Proposition 2.7 ([7, Corollary 2.54]).

For $I\subset\mathbb{R}$, $(p,r,q)\in[1,\infty]^{3}$ and $s>0$, we have

$\displaystyle\|u_{1}u_{2}\|_{\dot{B}^{s}_{p,r}}\leq
C\|u_{1}\|_{L^{\infty}}\|u_{2}\|_{\dot{B}^{s}_{p,r}}+C\|u_{2}\|_{L^{\infty}}\|u_{1}\|_{\dot{B}^{s}_{p,r}}$
(2.7)
$\displaystyle\|u_{1}u_{2}\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r})}\leq
C\|u_{1}\|_{L^{\infty}(I;L^{\infty})}\|u_{2}\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r})}+C\|u_{2}\|_{L^{\infty}(I;L^{\infty})}\|u_{1}\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r})}.$
(2.8)

We note that when $s=0$, (2.7) and (2.8) fail in general. In particular,
$\dot{B}_{\infty,1}^{0}$ is not an algebra.

######  Theorem 2.8 ([7, Theorem 2.61]).

Consider $F\in C^{\infty}(\mathbb{C}^{N})$ such that $F(0)=0$. Let $s>0$ and
$(p,r,q)\in[1,\infty]^{3}$. If

$U\in\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r}(\mathbb{R}^{d})^{N})\cap
L^{\infty}(I;L^{\infty}(\mathbb{R}^{d})^{N}),$

then $F(U)$ belongs to the same space and

$\|F(U)\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r})}\leq\mathcal{F}\bigl{(}\|U\|_{L^{\infty}(I;L^{\infty})}\bigr{)}\|U\|_{\widetilde{L}^{q}(I;\dot{B}^{s}_{p,r})}$
(2.9)

for some nondecreasing function $\mathcal{F}:\mathbb{R}^{+}\to\mathbb{R}^{+}$
independent of $U$.

###  2.2. Fractional heat kernel in homogeneous Besov spaces

######  Lemma 2.9 ([7, Lemma 2.4]).

There exist positive constants $c$ and $C$, both depending only on $d$, such
that for all $p\in[1,\infty]$, $t>0$ and $k\in\mathbb{Z}$ we have

$\|e^{-t|D_{x}|}\dot{\Delta}_{k}u\|_{L^{p}(\mathbb{R}^{d})}\leq
Ce^{-ct2^{k}}\|\dot{\Delta}_{k}u\|_{L^{p}(\mathbb{R}^{d})}.$

######  Proposition 2.10.

Let $s\in\mathbb{R}$, $\nu>0$, and $(p,r,q_{1},q_{2})\in[1,\infty]^{4}$ such
that $q_{2}\leq q_{1}$. Let $I=[a,b]$ where $a\in\mathbb{R}\cup\\{-\infty\\}$
and $b\in\mathbb{R}$. Then there exist positive constants $C_{1}$ and $C_{2}$,
both depending only on $d$, such that

$\displaystyle\|e^{-\nu
z|D_{x}|}u(x)\|_{\widetilde{L}^{q_{1}}(I;\dot{B}^{s+\frac{1}{q_{1}}}_{p,r})}\leq\frac{C_{1}}{\nu^{\frac{1}{q_{1}}}}\|u\|_{\dot{B}^{s}_{p,r}},$
(2.10)
$\displaystyle\left\|\int_{a}^{z}e^{-\nu(z-z^{\prime})|D_{x}|}f(x,z^{\prime})dz^{\prime}\right\|_{\widetilde{L}^{q_{1}}(I;\dot{B}^{s+\frac{1}{q_{1}}}_{p,r})}\leq\frac{C_{2}}{\nu^{1+\frac{1}{q_{1}}-\frac{1}{q_{2}}}}\|f\|_{\widetilde{L}^{q_{2}}(I;\dot{B}^{s-1+\frac{1}{q_{2}}}_{p,r})}.$
(2.11)

###### Proof.

We first prove (2.11). By virtue of Lemma 2.9,

$\|\dot{\Delta}_{j}e^{-\nu z|D_{x}|}u\|_{L^{p}_{x}}\leq C_{1}e^{-c\nu
z2^{j}}\|\dot{\Delta}_{j}u\|_{L^{p}_{x}},$

where $C_{1}=C_{1}(d)$. This implies

$\|\dot{\Delta}_{j}e^{-\nu
z|D_{x}|}u\|_{L^{q_{1}}(I;L^{p}_{x})}\leq\frac{C_{2}}{\nu^{\frac{1}{q_{1}}}}2^{-\frac{j}{q_{1}}}\|\dot{\Delta}_{j}u\|_{L^{p}_{x}},$

where $I=[a,b]$ and $C_{2}=C_{1}(cp)^{\frac{-1}{p}}\leq C_{3}=C_{3}(d)$.
Consequently,

$\|e^{-\nu
z|D_{x}|}u\|_{\widetilde{L}^{q_{1}}(I;\dot{B}^{s+\frac{1}{q_{1}}}_{p,r})}\leq\frac{C_{3}}{\nu^{\frac{1}{q_{1}}}}\|u\|_{\dot{B}^{s}_{p,r}}$

which proves (2.11).

As for (2.10), we apply Lemma 2.9 and Young’s inequality in $z$ to have

$\displaystyle\left\|\dot{\Delta}_{j}\int_{a}^{z}e^{-\nu(z-z^{\prime})|D_{x}|}f(x,z^{\prime})dz^{\prime}\right\|_{L^{q_{1}}_{z}(I;L^{p}_{x})}$
$\displaystyle=\left\|\|\int_{a}^{z}e^{-\nu(z-z^{\prime})|D_{x}|}\dot{\Delta}_{j}f(\cdot,z^{\prime})dz^{\prime}\|_{L^{p}_{x}}\right\|_{L^{q_{1}}_{z}(I)}$
$\displaystyle\leq C_{1}\|\int_{a}^{b}e^{-\nu
c(z-y)2^{j}}\|\dot{\Delta}_{j}f(\cdot,z^{\prime})\|_{L^{p}_{x}}dz^{\prime}\|_{L^{q_{1}}_{z}(I)}$
$\displaystyle\leq\frac{C_{4}}{\nu^{1+\frac{1}{q_{1}}-\frac{1}{q_{2}}}}2^{j(-1+\frac{1}{q_{2}}-\frac{1}{q_{1}})}\|\Delta_{j}f\|_{L^{q_{2}}_{z}(I;L^{p}_{x})},$

where $C_{4}=C_{4}(d)$. It follows that

$\left\|\int_{a}^{z}e^{-(z-y)|D_{x}|}f(x,y)dy\right\|_{L^{q_{1}}_{z}(I;\dot{B}^{s+\frac{1}{q_{1}}}_{p,r})}\leq\frac{C_{4}}{\nu^{1+\frac{1}{q_{1}}-\frac{1}{q_{2}}}}\|f\|_{L^{q_{2}}_{z}(I;\dot{B}^{s-1+\frac{1}{q_{2}}}_{p,r})}$

which completes the proof of (2.10). ∎

##  3\. The Dirichlet-Neumann operator in $\dot{B}^{r}_{\infty,1}$

###  3.1. Linearization and boundedness

We study the Dirichlet-Neumann operators associated to the fluid domains
$\Omega^{\pm}$. The time variable is forgotten throughout. In $\Omega^{-}$, we
consider the elliptic problem

$\begin{cases}\Delta_{x,y}\phi=0\quad\text{in}~{}\Omega^{-},\\\
\phi=f\quad\text{on}~{}\Sigma,\\\ \nabla_{x,y}\phi\to
0\quad\text{as}~{}y\to-\infty.\end{cases}$ (3.1)

The Dirichlet-Neuman operator associated to $\Omega^{-}$ is (formally) defined
by

$G^{-}(\eta)f=\sqrt{1+|\nabla\eta|^{2}}\frac{\partial\phi}{\partial n},$ (3.2)

where we recall that $n$ is the upward-pointing unit normal to $\Sigma$.
Similarly, if $\phi$ solves the elliptic problem (3.1) with $\Omega^{-}$
replaced by $\Omega^{+}$, then we also define $G^{+}(\eta)f$ by the right-hand
side of (3.2). Note that $n$ is inward-pointing for $\Omega^{+}$, so that
$G^{+}(\eta)$ is a nonpositive operator. We shall only state results for
$G^{-}(\eta)$ since corresponding results for $G^{+}(\eta)$ are completely
parallel.

Assume that $\phi$ is a smooth solution of (3.1). We straighten the free
boundary using the change of variables $\mathbb{R}^{d}\times
J\ni(x,z)\mapsto(x,\varrho(x,z))\in\Omega^{-}$, where $J=(-\infty,0)$ and

$\varrho(x,z)=z+H(x,z),\quad
H(x,z)=e^{z|D_{x}|}\eta(x),\quad(x,z)\in\mathbb{R}^{d}\times J.$ (3.3)

Clearly, $\varrho(x,0)=\eta(x)$ and $\varrho(x,z)\to-\infty$ as $z\to-\infty$.
We have $\partial_{z}\varrho(x,z)=1+e^{z|D_{x}|}|D_{x}|\eta(x)$ and

$\|e^{z|D_{x}|}|D_{x}|\eta\|_{L^{\infty}(\mathbb{R}^{d}\times J)}\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}$

where we have used Lemma 2.9. This implies

######  Lemma 3.1.

There exists a positive constant $c_{0}=c_{0}(d)\in(0,1)$ such that if

$\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\leq
c_{0}\quad\text{then}\quad\partial_{z}\varrho\geq\frac{1}{2}$ (3.4)

and thus the mapping $(x,z)\in\mathbb{R}^{d}\times J\mapsto\Omega^{-}$ is a
Lipschitz diffeomorphism.

A direct calculation shows that if $g:\Omega^{-}\to\mathbb{R}$ then
$\widetilde{g}(x,z)=g(x,\varrho(x,z))$ satisfies

$\operatorname{div}_{x,z}(\mathcal{A}\nabla_{x,z}\widetilde{g})(x,z)=\partial_{z}\varrho(\Delta_{x,y}g)(x,\varrho(x,z))$
(3.5)

with

$\mathcal{A}=\begin{bmatrix}\partial_{z}\varrho\text{I}&-\nabla_{x}\varrho\\\
-(\nabla_{x}\varrho)^{T}&\frac{1+|\nabla_{x}\varrho|^{2}}{\partial_{z}\varrho}\end{bmatrix}=\text{I}+\begin{bmatrix}(|D_{x}|H)\text{I}&-\nabla_{x}H\\\
-(\nabla_{x}H)^{T}&\frac{|\nabla_{x}H|^{2}-|D_{x}|H}{1+|D_{x}|H}\end{bmatrix}.$
(3.6)

Since $\phi$ is harmonic in $\Omega^{-}$, $v(x,z)=\phi(x,\varrho(x,z))$
satisfies

$\operatorname{div}_{x,z}(\mathcal{A}\nabla_{x,z}v)(x,z)=0\quad\text{in}~{}\mathbb{R}^{d}\times
J,$ (3.7)

or equivalently,

$\Delta_{x,z}v=\partial_{z}Q_{a}[v]+|D_{x}|Q_{b}[v]\quad\text{in}~{}\mathbb{R}^{d}\times
J,$ (3.8)

where

$\displaystyle
Q_{a}[v]=\nabla_{x}H\cdot\nabla_{x}v-\frac{|\nabla_{x}H|^{2}-|D_{x}|H}{1+|D_{x}|H}\partial_{z}v,$
(3.9) $\displaystyle
Q_{b}[v]=\mathcal{R}_{x}\big{(}\nabla_{x}H\partial_{z}v-|D_{x}|H\nabla_{x}v\big{)},\quad\mathcal{R}_{x}=|D_{x}|^{-1}\operatorname{div}_{x}.$

Then the Dirichlet-Neumann operator can be expressed in terms of $v$ as

$\displaystyle G(\eta)f$
$\displaystyle=\Big{(}\frac{1+|\nabla_{x}\varrho|^{2}}{\partial_{z}\varrho}\partial_{z}v-\nabla_{x}\varrho\cdot\nabla_{x}v\Big{)}|_{z=0}$
(3.10)
$\displaystyle=\Big{[}\Big{(}1+\frac{|\nabla_{x}H|^{2}-|D_{x}|H}{1+|D_{x}|H}\Big{)}\partial_{z}v-\nabla_{x}H\cdot\nabla_{x}v\Big{]}|_{z=0}$
$\displaystyle=(\partial_{z}v-Q_{a}[v])|_{z=0}.$

By factorizing $\Delta_{x,z}v=(\partial_{z}+|D_{x}|)(\partial_{z}-|D_{x}|)v$
and setting

$w=(\partial_{z}-|D_{x}|)v-Q_{a}[v],$ (3.11)

we obtain from (3.8) that

$\displaystyle(\partial_{z}+|D_{x}|)w=|D_{x}|(Q_{b}[v]-Q_{a}[v]).$ (3.12)

In terms of $w$, (3.10) becomes

$G^{-}(\eta)\eta=|D_{x}|f+w|_{z=0}.$ (3.13)

Since $\nabla_{x,y}\phi\to 0$ as $y\to-\infty$, so are $\nabla_{x,z}v$,
$Q_{a}[v]$ and $Q_{b}[v]$. Then $v$ and $w$ as solutions of (3.11) and (3.12)
are given by

$\displaystyle
v(x,z)=e^{z|D_{x}|}f(x)+\int_{0}^{z}e^{(z-z^{\prime})|D_{x}|}\\{w(x,z^{\prime})+Q_{a}[v](x,z^{\prime})\\}dz^{\prime},\quad
z\leq 0,$ (3.14) $\displaystyle
w(x,r)=\int_{-\infty}^{r}e^{-(r-\tau)|D_{x}|}|D_{x}|\\{Q_{b}[v](x,\tau)-Q_{a}[v](x,\tau)\\}d\tau,\quad
r\leq 0.$ (3.15)

Therefore, $v$ is a fixed point of the operator

$\displaystyle\mathcal{T}[v](x,z)$
$\displaystyle=e^{z|D_{x}|}f(x)+\int_{0}^{z}e^{(z-z^{\prime})|D_{x}|}Q_{a}[v](x,z^{\prime})dz^{\prime}$
(3.16)
$\displaystyle+\int_{0}^{z}e^{(z-z^{\prime})|D_{x}|}\int_{-\infty}^{z^{\prime}}e^{-(z^{\prime}-\tau)|D_{x}|}|D_{x}|\\{Q_{b}[v](x,\tau)-Q_{a}[v](x,\tau)\\}d\tau
dz^{\prime}.$

Note that $\mathcal{T}$ depends on $\eta$. We shall prove that for small
$\eta$, $\mathcal{T}$ has a unique fixed point in suitable Chemin-Lerner
spaces. Thus it is convenient to define

######  Definition 3.2.

Let $\eta\in\dot{B}^{1}_{\infty,1}$ satisfy
$\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}<c_{0}$. If $v$ is the fixed point of
$\mathcal{T}$, then the Dirichlet-Neumann operator $G(\eta)f$ is defined by

$G(\eta)f=|D_{x}|f+R^{-}(\eta)f:=|D_{x}|f+\int_{-\infty}^{0}e^{\tau|D_{x}|}|D_{x}|\\{Q_{b}[v](x,\tau)-Q_{a}[v](x,\tau)\\}d\tau.$
(3.17)

In order to obtain the existence of a unique fixed point of $\mathcal{T}$, we
shall appeal to the following elemetary lemma.

######  Lemma 3.3.

Let $E_{1}$ and $E_{2}$ be two norm spaces such that $E_{1}$ is complete.
Assume that $E_{2}$ has the Fatou property: if $(u_{n})$ is a bounded sequence
in $E_{2}$ then there exist $u\in E_{2}$ and a subsequence $(u_{n_{k}})$ such
that $u_{n_{k}}\to u$ in a sense weaker than norm convergence in $E_{1}$ and
that

$\|u\|_{E_{2}}\leq C\liminf_{k\to\infty}\|u_{n_{k}}\|_{E_{2}}.$

Assume that $K:E_{1}\to E_{1}$ is a linear such that $K:E_{1}\cap E_{2}\to
E_{2}$ and the following property holds. There exist
$(\alpha_{1},\alpha_{2})\in(0,1)^{2}$ and $A>0$ such that for all $u\in
E_{1}\cap E_{2}$,

$\displaystyle\|K(u)\|_{E_{1}}\leq\alpha_{1}\|u\|_{E_{1}},$ (3.18)
$\displaystyle\|K(u)\|_{E_{2}}\leq\alpha_{2}\|u\|_{E_{2}}+A\|u\|_{E_{1}}.$
(3.19)

Then for any $u_{0}\in E_{1}\cap E_{2}$ there exists a unique fixed point
$u_{*}\in E_{1}\cap E_{2}$ of the mapping $u\mapsto u_{0}+K(u)$.

###### Proof.

By the Banach contraction principle, (3.18) implies that $u_{0}+K(u)$ has a
unique fixed point $u_{*}\in E_{1}$. Moreover, $u_{*}=\lim_{n\to\infty}u_{n}$
in $E_{1}$, where $u_{n+1}=u_{0}+K(u_{n})$. For
$\beta_{1}>(1-\alpha_{1})^{-1}$, using (3.18) and induction we find that
$\|u_{n}\|_{E_{1}}<\beta_{1}\|u_{0}\|_{E_{1}}$ for all $n$. Next, for

$\beta_{2}>\frac{\alpha_{2}}{1-\alpha_{2}},\quad
M>\frac{A\beta_{1}}{1-\alpha_{2}},\quad
R=\beta_{2}\|u_{0}\|_{E_{2}}+M\|u_{0}\|_{E_{1}},$

we claim that $\|u_{n}-u_{0}\|_{E_{2}}<R$ for all $n$. Indeed, this is obvious
when $n=0$ and if it is true for $u_{n}$ then by (3.19),

$\displaystyle\|u_{n+1}-u_{0}\|_{E_{2}}$
$\displaystyle\leq\alpha_{2}\|u_{n}\|_{E_{2}}+A\|u_{n}\|_{E_{1}}$
$\displaystyle\leq\alpha_{2}(R+\|u_{0}\|_{E_{2}})+A\beta_{1}\|u_{0}\|_{E_{1}}$
$\displaystyle=\alpha_{2}(1+\beta_{2})\|u_{0}\|_{E_{2}}+(\alpha_{2}M+A\beta_{1})\|u_{0}\|_{E_{1}}$
$\displaystyle\leq\beta_{2}\|u_{0}\|_{E_{2}}+M\|u_{0}\|_{E_{1}}=R.$

Thus $(u_{n})$ is bounded in $E_{2}$, so that the Fatou property imposed on
$E_{2}$ guarantees $u_{*}\in E_{1}\cap E_{2}$. ∎

Given $I\subset\mathbb{R}\cup\\{\pm\infty\\}$, $\mu\in\mathbb{R}$ and
$(p,q)\in[1,\infty]^{2}$, we define the interpolation space

$\displaystyle X^{\mu}(I)$
$\displaystyle=\widetilde{L}^{\infty}(I;\dot{B}^{\mu}_{\infty,1})\cap\widetilde{L}^{1}(I;\dot{B}^{\mu+1}_{\infty,1}).$
(3.20)

Recall that the product rules (2.7) and (2.8) fail in
$\dot{B}_{\infty,1}^{0}$. As a substitute for $X^{0}(I)$ we define

$X_{*}(I)=L^{\infty}(J;L^{\infty})\cap\widetilde{L}^{1}(I;\dot{B}_{\infty,1}^{1}).$
(3.21)

Note that

$\|\cdot\|_{L^{\infty}(I;L^{\infty})}\leq\|\cdot\|_{\widetilde{L}^{\infty}(I;\dot{B}_{\infty,1}^{0})},\quad\|\cdot\|_{X_{*}(I)}\leq\|\cdot\|_{X^{0}(I)}.$
(3.22)

######  Proposition 3.4.

1) Let $\eta\in\dot{B}^{1}_{\infty,1}$ and $f\in\dot{B}^{1}_{\infty,1}$. There
exists $c_{1}=c_{1}(d)<c_{0}<1$ such that if

$\||D|_{x}\eta\|_{\dot{B}^{0}_{\infty,1}}<c_{1},$ (3.23)

then $\mathcal{T}$ has a unique fixed point $v$ in

$\mathcal{V}_{*}=\\{v\in\mathcal{D}^{\prime}(\mathbb{R}^{d}\times
J):(|D_{x}|v,\partial_{z}v)\in X_{*}(J)\\}/\mathbb{R}.$ (3.24)

Moreover, there exists $C=C(d)$ such that

$\|v\|_{\mathcal{V}_{*}}\leq C\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}$ (3.25)

and

$\|R^{-}(\eta)f\|_{\dot{B}^{0}_{\infty,1}}\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}.$
(3.26)

2) Let $r>0$, $\eta\in\dot{B}^{1}_{\infty,1}\cap\dot{B}^{1+r}_{\infty,1}$ and
$f\in\dot{B}^{1}_{\infty,1}\cap\dot{B}^{1+r}_{\infty,1}$. There exists
$c_{*}=c_{*}(d,r)<c_{1}$ such that if

$\||D|_{x}\eta\|_{\dot{B}^{0}_{\infty,1}}<c_{*},$ (3.27)

then the unique fixed point $v\in\mathcal{V}_{*}$ of $\mathcal{T}$ belongs to

$\mathcal{V}^{r}=\\{v\in\mathcal{D}^{\prime}(\mathbb{R}^{d}\times
J):(|D_{x}|v,\partial_{z}v)\in X^{r}(J)\\}/\mathbb{R}.$ (3.28)

Moreover, there exists $C=C(d,r)$ such that

$\|v\|_{\mathcal{V}^{r}}\leq
C\||D_{x}|f\|_{\dot{B}^{r}_{\infty,1}}+C\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}$
(3.29)

and

$\displaystyle\|R^{-}(\eta)f\|_{\dot{B}^{r}_{\infty,1}}$ $\displaystyle\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{r}_{\infty,1}}+C\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}.$
(3.30)

###### Proof.

For notational simplicity we shall write $Q_{a}=Q_{a}[v]$ and
$Q_{b}=Q_{b}[v]$. With $w$ given by (3.15), we have

$\mathcal{T}[v](x,z)=e^{z|D_{x}|}f(x)+K[u](x,z):=e^{z|D_{x}|}f(x)+\int_{0}^{z}e^{(z-z^{\prime})|D_{x}|}\\{w(x,z^{\prime})+Q_{a}(x,z^{\prime})\\}dz^{\prime}.$
(3.31)

Throughout this proof, condition (3.23) is always assumed, so that in many
instances a nondecreasing function of
$\||D_{x}|\eta\|_{\dot{B}_{\infty,1}^{0}}$ is simply controlled by an absolute
constant.

1. Assume that $\eta,~{}f\in\dot{B}_{\infty,1}^{1}$. We shall prove that $\mathcal{T}$ has a unique fixed point in $\mathcal{V}_{*}$. Indeed, the estimate (2.11) (with $\nu=1$) yields

$\|w\|_{X^{0}(J)}\leq
C\||D_{x}|(Q_{b}-Q_{a})\|_{\widetilde{L}^{1}(J;\dot{B}^{0}_{\infty,1})}\leq
C\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}.$ (3.32)

Upon changing the variables $z\mapsto-z$, (2.11) and (3.32) imply

$\displaystyle\||D_{x}|(K[v])\|_{X^{0}(J)}$ $\displaystyle\leq
C\|w\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}+C\|Q_{a}\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}\leq
C\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}.$ (3.33)

Now, since

$\partial_{z}K[u](x,z)=\int_{0}^{z}e^{(z-z^{\prime})|D_{x}|}|D_{x}|\\{w(x,z^{\prime})+Q_{a}(x,z^{\prime})\\}dz^{\prime}+w(x,z)+Q_{a}(x,z),$
(3.34)

using (2.11) and (3.32) we estimate

$\|\partial_{z}K[u](x,z)\|_{\widetilde{L}^{1}(J;\dot{B}_{\infty,1}^{1})}\leq
C\|w\|_{\widetilde{L}^{1}(J;\dot{B}_{\infty,1}^{1})}+C\|Q_{a}\|_{\widetilde{L}^{1}(J;\dot{B}_{\infty,1}^{1})}\leq
C\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}.$ (3.35)

On the other hand, in view of (3.22), (2.11) and (3.32), we have

$\displaystyle\|\partial_{z}K[u]\|_{L^{\infty}(J;L^{\infty})}$
$\displaystyle\leq
C\|w\|_{\widetilde{L}^{1}(J;\dot{B}_{\infty,1}^{1})}+C\|Q_{a}\|_{\widetilde{L}^{1}(J;\dot{B}_{\infty,1}^{1})}+\|w\|_{\widetilde{L}^{\infty}(J;\dot{B}_{\infty,1}^{0})}+\|Q_{a}\|_{L^{\infty}(J;L^{\infty})}$
(3.36) $\displaystyle\leq
C\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}+\|Q_{a}\|_{L^{\infty}(J;L^{\infty})}.$

A combination of (3.33), (3.35), (3.36) and (3.22) leads to

$\|\big{(}|D_{x}|K[v],\partial_{z}K[v]\big{)}\|_{X_{*}(J)}\leq
C\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}+\|Q_{a}\|_{L^{\infty}(J;L^{\infty})}.$
(3.37)

Note that if we choose $\mathcal{V}^{0}$ in place of $\mathcal{V}_{*}$, then
we have to estimate $\partial_{z}K[u]$ in
$\widetilde{L}^{\infty}(J;\dot{B}_{\infty,1}^{0})$. This in turn requires a
bound for the nonlinear term $Q_{a}$ in
$\widetilde{L}^{\infty}(J;\dot{B}_{\infty,1}^{0})$. However, the product rule
(2.8) does not hold when $s=0$. On the other hand, it is readily seen from the
definition of $Q_{a}$ in (3.9) that

$\displaystyle\|Q_{a}\|_{L^{\infty}(J;L^{\infty})}$
$\displaystyle\leq\|\nabla_{x}H\|_{L^{\infty}(J;L^{\infty})}\|\nabla_{x}v\|_{L^{\infty}(J;L^{\infty})}+C\|(|D_{x}|H,\nabla_{x}H)\|_{L^{\infty}(J;L^{\infty})}\|\partial_{z}v\|_{L^{\infty}(J;L^{\infty})}$
(3.38) $\displaystyle\leq
C\||D_{x}|\eta\|_{\dot{B}_{\infty,1}^{0}}\|\nabla_{x,z}v\|_{L^{\infty}(J;L^{\infty})}.$

Consequently, in view of (3.37) and (3.38), we obtain

$\|\big{(}|D_{x}|(K[v]),\partial_{z}(K[v])\big{)}\|_{X_{*}(J)}\leq
C\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}+C\||D_{x}|\eta\|_{\dot{B}_{\infty,1}^{0}}\|\nabla_{x,z}v\|_{L^{\infty}(J;L^{\infty})}.$
(3.39)

For later use, we now estimate
$\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}$ for $r\geq
0$. Let us treat the most difficult term in $Q_{a}$

$I=\frac{|\nabla_{x}H|^{2}-|D_{x}|H}{1+|D_{x}|H}\partial_{z}v$

which can be decomposed as

$I=|\nabla_{x}H|^{2}\partial_{z}v-|\nabla_{x}H|^{2}F(|D_{x}|H)\partial_{z}v-F(|D_{x}|H)\partial_{z}v:=I_{1}+I_{2}+I_{3},\quad
F(z)=z/(1+z).$

Again, we only consider the most difficult term $I_{2}$. Since
$\||D_{x}|H\|_{L^{\infty}}<c_{1}<1$, we can modify $F$ to a $C^{\infty}$
function taking same values on the range of $|D_{x}|H$. Let $r\geq 0$. Since
$1+r>0$, using the tame estimate (2.8) gives

$\displaystyle\|I_{2}\|_{\widetilde{L}^{1}(I;\dot{B}^{1+r}_{\infty,1})}$
$\displaystyle\leq
C\left\||\nabla_{x}H|^{2}F(|D_{x}|H)\right\|_{L^{\infty}(J;L^{\infty})}\|\partial_{z}v\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}$
$\displaystyle\quad+C\left\||\nabla_{x}H|^{2}F(|D_{x}|H)\right\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}\|\partial_{z}v\|_{L^{\infty}(J;L^{\infty})}.$

Since $|F(z)|\leq 1$, we have

$\left\||\nabla_{x}H|^{2}F(|D_{x}|H)\right\|_{L^{\infty}(J;L^{\infty})}\leq\|\nabla_{x}H\|^{2}_{L^{\infty}(J;L^{\infty})}\leq\||D_{x}|\eta\|^{2}_{\dot{B}^{0}_{\infty,1}}.$

Applying (2.8) again yields

$\displaystyle\left\||\nabla_{x}H|^{2}F(|D_{x}|H)\right\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}$
$\displaystyle\leq
C\||\nabla_{x}H|^{2}\|_{L^{\infty}(J;L^{\infty})}\left\|F(|D_{x}|H)\right\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}$
$\displaystyle\quad+C\||\nabla_{x}H|^{2}\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}\left\|F(|D_{x}|H)\right\|_{L^{\infty}(J;L^{\infty})}$
$\displaystyle\leq
C\|\nabla_{x}\eta\|_{L^{\infty}}^{2}\left\|F(|D_{x}|H)\right\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}+C\||\nabla_{x}H|^{2}\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}.$

To estimate $F(|D_{x}|H)$ we appeal to the nonlinear estimate (2.9) to have

$\left\|F(|D_{x}|H)\right\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}\leq
C\||D_{x}|H\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}\leq
C\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}},\quad C=C(d,r).$

By virtue of (2.8) and (2.10),

$\||\nabla_{x}H|^{2}\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}\leq
C\|\nabla_{x}\eta\|_{L^{\infty}}\|\nabla_{x}\eta\|_{\dot{B}^{r}_{\infty,1}}.$

We have proved that

$\left\||\nabla_{x}H|^{2}F(|D_{x}|H)\right\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}\leq
C\||D_{x}|\eta\|_{B^{r}_{\infty,1}}.$

Putting together the above considerations leads to

$\|I_{2}\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\|\partial_{z}v\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}+C\||D_{x}|\eta\|_{B^{r}_{\infty,1}}\|\partial_{z}v\|_{L^{\infty}(J;L^{\infty})},\quad
C=C(d,r).$

Other terms in $Q_{a}$ and $Q_{b}$ can be estimated similarly, yielding

$\displaystyle\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\|\nabla_{x,z}v\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}+C\||D_{x}|\eta\|_{B^{r}_{\infty,1}}\|\nabla_{x,z}v\|_{L^{\infty}(J;L^{\infty})}$
(3.40)

for all $r\geq 0$, where $C=C(d,r)$.

Applying (3.40) with $r=0$ then combining it with (3.39), we arrive at a
closed estimate for $K[v]$ in $\mathcal{V}_{*}$

$\|K[v]\|_{\mathcal{V}_{*}}\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\|v\|_{\mathcal{V}_{*}},\quad
C=C(d).$ (3.41)

We choose $c_{1}<c_{0}$ sufficiently small such that $Cc_{1}<\frac{1}{4}$ and
consider $\eta$ satisfying (3.23). Then by virtue of (3.41), the Banach
contraction principle yields the existence of a unique fixed point
$v\in\mathcal{V}_{*}$ of $\mathcal{T}=e^{z|D_{x}|}f+K$. Since

$\|e^{z|D_{x}|}f\|_{\mathcal{V}_{*}}\leq\|e^{z|D_{x}|}f\|_{\mathcal{V}^{0}}\leq
C\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}},$

it follows from (3.41) that

$\|v\|_{\mathcal{V}_{*}}\leq 2C\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}},\quad
C=C(d).$ (3.42)

Thus (3.25) holds. Finally, since
$\|\cdot\|_{L^{1}\dot{B}^{r}_{\infty,1}}=\|\cdot\|_{\widetilde{L}^{1}\dot{B}^{r}_{\infty,1}}$,
the left-hand side of (3.26) is bounded by

$\displaystyle\|e^{\tau|D_{x}|}|D_{x}|(Q_{b}(x,\tau)-Q_{a}(x,\tau))\|_{L^{1}_{\tau}(J;\dot{B}^{0}_{\infty,1})}\leq
C\|Q_{b}-Q_{a}\|_{\widetilde{L}^{1}_{\tau}(J;\dot{B}^{1}_{\infty,1})},$ (3.43)

so that (3.26) follows from (3.40) (with $r=0$) and (3.25).

2. Assume that $\eta,~{}f\in\dot{B}_{\infty,1}^{1}\cap\dot{B}_{\infty,1}^{1+r}$ with $r>0$. As in (3.32) and (3.33), upon using the formula (3.34) for $\partial_{z}K[v]$ we have

$\|w\|_{X^{r}(J)}\leq
C\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}$ (3.44)

and

$\displaystyle\|(|D_{x}|K[v],\partial_{z}K[v])\|_{X^{r}(J)}$
$\displaystyle\leq
C\|w\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}+\|Q_{a}\|_{\widetilde{L}^{\infty}(J;\dot{B}^{r}_{\infty,1})}$
(3.45) $\displaystyle\leq
C\|(Q_{a},Q_{b})\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}+\|Q_{a}\|_{\widetilde{L}^{\infty}(J;\dot{B}^{r}_{\infty,1})}.$

Unlike part 1), since $r>0$ the nonlinear term $Q_{a}$ can be handled using
the product rule (2.8) and the nonlinear estimate (2.9) as in the proof of
(3.40). More precisely,

$\displaystyle\|Q_{a}\|_{\widetilde{L}^{\infty}(J;\dot{B}^{r}_{\infty,1})}\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\|\nabla_{x,z}v\|_{\widetilde{L}^{\infty}(J;\dot{B}^{r}_{\infty,1})}+C\||D_{x}|\eta\|_{B^{r}_{\infty,1}}\|\nabla_{x,z}v\|_{L^{\infty}(J;L^{\infty})}.$
(3.46)

In conjunction with (3.40) and (3.46), (3.45) implies that
$K:\mathcal{V}_{*}\cap\mathcal{V}^{r}\to\mathcal{V}^{r}$ and

$\|K[v]\|_{\mathcal{V}^{r}}\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\|v\|_{\mathcal{V}^{r}}+C\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}\|v\|_{\mathcal{V}_{*}},\quad
C=C(d,r)$ (3.47)

for all $v\in\mathcal{V}_{*}\cap\mathcal{V}^{r}$. We choose
$c_{*}=c_{*}(d,r)<c_{1}$ sufficiently small so that $Cc_{*}<\frac{1}{4}$ and
consider $\eta$ satisfying (3.27). Then in view of (3.41) and (3.47), $K$
satisfies the conditions (3.18) and (3.19) of Lemma 3.3 with
$E_{1}=\mathcal{V}_{*}$ and $E_{2}=\mathcal{V}^{r}$. Therefore, the unique
fixed point $v\in\mathcal{V}_{*}$ of $\mathcal{T}$ belongs to
$\mathcal{V}^{r}$. Inserting (3.42) into (3.47) we find

$\|v\|_{\mathcal{V}^{r}}\leq
C||D_{x}|f\|_{\dot{B}^{r}_{\infty,1}}+C\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}},\quad
C=C(d,r),$

whence (3.29) follows. We have used the fact that
$\|e^{z|D_{x}|}f\|_{\mathcal{V}^{r}}\leq
C(d)\||D_{x}|f\|_{\dot{B}_{\infty,1}^{r}}$ which in turn follows from (2.10).
Finally, (3.30) can be obtained as in (3.26). ∎

Recalling the linearization $G^{-}(\eta)f=|D_{x}|f+R^{-}(\eta)f$, we deduce
the following from (3.26) and (3.30).

######  Corollary 3.5.

Let $r\geq 0$, $\eta\in\dot{B}^{1}_{\infty,1}\cap\dot{B}^{1+r}_{\infty,1}$ and
$f\in\dot{B}^{1}_{\infty,1}\cap\dot{B}^{1+r}_{\infty,1}$. Assume that $\eta$
satisfies (3.27). Then there exists $C=C(d,r)>0$ such that

$\|G^{-}(\eta)f\|_{\dot{B}^{r}_{\infty,1}}\leq
C\||D_{x}|f\|_{\dot{B}^{r}_{\infty,1}}+C\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}.$
(3.48)

###  3.2. Contraction

We prove contraction estimates for the Dirichlet-Neumann operator

$G^{-}(\eta_{1})-G^{-}(\eta_{2})=R^{-}(\eta_{1})-R^{-}(\eta_{2}).$

######  Proposition 3.6.

Let $r\geq 0$, $f\in\dot{B}^{1}_{\infty,1}\cap\dot{B}^{r+1}_{\infty,1}$ and
$\eta_{j}\in\dot{B}^{1}_{\infty,1}\cap\dot{B}^{r+1}_{\infty,1}$, $j=1,2$. If
$\eta_{1}$ and $\eta_{2}$ satisfy (3.27), then there exists $C=C(d,r)$ such
that

$\displaystyle\|[R^{-}(\eta_{1})-R^{-}(\eta_{2})]f\|_{\dot{B}^{r}_{\infty,1}}$
(3.49) $\displaystyle\leq
C\mathcal{A}_{r}\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{r}_{\infty,1}}+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{r}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}},$

where $\eta_{\delta}=\eta_{1}-\eta_{2}$ and

$\mathcal{A}_{s}=\||D_{x}|\eta_{1}\|_{\dot{B}^{s}_{\infty,1}}+\||D_{x}|\eta_{2}\|_{\dot{B}^{s}_{\infty,1}}.$
(3.50)

###### Proof.

We recall from (3.17) that

$R^{-}(\eta_{j})f_{j}(x)=\int_{-\infty}^{0}e^{\tau|D_{x}|}|D_{x}|\\{Q_{b,j}[v_{j}](x,\tau)-Q_{a,j}[v_{j}](x,\tau)\\}d\tau,$

where $v_{j}$ is the unique fixed point of $\mathcal{T}_{j}$ associated to
$\eta_{j}$. Here we use the same notation as in subsection 3.1 but with the
index $j\in\\{1,2\\}$. We use in addition the notation
$g_{\delta}=g_{1}-g_{2}$.

Step 1. Since
$\|\cdot\|_{L^{1}\dot{B}^{r}_{\infty,1}}=\|\cdot\|_{\widetilde{L}^{1}\dot{B}^{r}_{\infty,1}}$,
we deduce

$A_{r}:=\|R^{-}(\eta_{1})f_{1}-R^{-}(\eta_{2})f_{2}\|_{\dot{B}^{r}_{\infty,1}}\leq
C\|(Q_{b}[v])_{\delta}(x,\tau)-(Q_{a}[v])_{\delta}\|_{\widetilde{L}^{1}(J;\dot{B}^{r+1}_{\infty,1})}.$

Let us consider the following typical term under the norm on the right-hand
side

$T=\nabla_{x}H_{1}\cdot\nabla_{x}v_{1}-\nabla_{x}H_{2}\cdot\nabla_{x}v_{2}=\nabla_{x}H_{1}\cdot\nabla_{x}v_{\delta}+\nabla_{x}H_{\delta}\cdot\nabla_{x}v_{2}.$

By the tame estimate (2.7), for $r\geq 0$ we have

$\displaystyle\|T\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}$
$\displaystyle\leq
C\|\nabla_{x}H_{1}\|_{L^{\infty}(J;L^{\infty})}\|\nabla_{x}v_{\delta}\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}+C\|\nabla_{x}H_{1}\|_{\widetilde{L}^{1}(J;\dot{B}^{r+1}_{\infty,1})}\|\nabla_{x}v_{\delta}\|_{L^{\infty}(J;L^{\infty})}$
$\displaystyle\quad+C\|\nabla_{x}H_{\delta}\|_{L^{\infty}(J;L^{\infty})}\|\nabla_{x}v_{2}\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}+C\|\nabla_{x}H_{\delta}\|_{\widetilde{L}^{1}(J;\dot{B}^{r+1}_{\infty,1})}\|\nabla_{x}v_{2}\|_{L^{\infty}(J;L^{\infty})}.$

When $r=0$, this yields

$\displaystyle\|T\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}$
$\displaystyle\leq
C\mathcal{A}_{0}\|v_{\delta}\|_{\mathcal{V}_{*}}+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\|v_{2}\|_{\mathcal{V}_{*}}$
(3.51) $\displaystyle\leq
C\big{(}\mathcal{A}_{0}\|v_{\delta}\|_{\mathcal{V}_{*}}+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}_{\infty,1}^{0}}\big{)}:=E_{*},$

where we have used (3.25) for $v_{2}$. On the other hand, when $r>0$ we obtain

$\displaystyle\|T\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}$ (3.52)
$\displaystyle\leq
C\Big{\\{}\mathcal{A}_{0}\|v_{\delta}\|_{\mathcal{V}^{r}}+\mathcal{A}_{r}\|v_{\delta}\|_{\mathcal{V}_{*}}+\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{r}_{\infty,1}}+\mathcal{A}_{r}\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}$
$\displaystyle\quad+\||D_{x}|\eta_{\delta}\|_{\dot{B}^{r}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}\Big{\\}}:=E_{r},$

where we have used (3.25) and (3.29) for $v_{2}$. We have proved that

$\displaystyle A_{r}\leq
C\|(Q_{b}[v])_{\delta}(x,\tau)-(Q_{a}[v])_{\delta}\|_{\widetilde{L}^{1}(J;\dot{B}^{r+1}_{\infty,1})}\leq\begin{cases}E_{*}\quad\text{when
}r=0,\\\ E_{r}\quad\text{when }r>0.\end{cases}$ (3.53)

Step 2. In view of (3.31), we have

$v_{\delta}=(\mathcal{K}[v])_{\delta}=\int_{0}^{z}e^{(z-z^{\prime})|D_{x}|}\\{w_{\delta}(x,z^{\prime})+(Q_{a}[v])_{\delta}(x,z^{\prime})\\}dz^{\prime}.$

The proof of (3.37) also gives

$\|v_{\delta}\|_{\mathcal{V}_{*}}\leq
C\|((Q_{a}[v])_{\delta},(Q_{b}[v])_{\delta})\|_{\widetilde{L}^{1}(J;\dot{B}^{1}_{\infty,1})}+\|(Q_{a}[v])_{\delta}\|_{L^{\infty}(J;L^{\infty})}.$

Using the definition of $Q_{a}[v]$ we estimate

$\|(Q_{a}[v])_{\delta}\|_{L^{\infty}(J;L^{\infty})}\leq
C\||D_{x}|\eta_{\delta}\|_{\dot{B}_{\infty,1}^{0}}\|\nabla_{x,z}v_{1}\|_{L^{\infty}(J;L^{\infty})}+C\||D_{x}|\eta_{2}\|_{\dot{B}_{\infty,1}^{0}}\|\nabla_{x,z}v_{\delta}\|_{L^{\infty}(J;L^{\infty})}\leq
E_{*}.$

Then upon recalling (3.53), we deduce from the two preceding estimates that

$\|v_{\delta}\|_{\mathcal{V}_{*}}\leq
E_{*}=C\mathcal{A}_{0}\|v_{\delta}\|_{\mathcal{V}^{0}}+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}_{\infty,1}^{0}}.$

For $\eta_{j}$ satisfying (3.27) with $c_{*}$ smaller if necessary so that
$C\mathcal{A}_{0}\leq 2Cc_{*}<1$, $\|v_{\delta}\|_{\mathcal{V}^{0}}$ on the
right-hand side can be absorbed by the right-hand side, giving

$\|v_{\delta}\|_{\mathcal{V}_{*}}\leq
C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}.$
(3.54)

Plugging this back in $E_{*}$, we conclude in view of (3.53) that

$A_{0}\leq
C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}$

which is the estimate (3.49) for $r=0$.

Next, let $r>0$. Analogously to (3.45) we have

$\|v_{\delta}\|_{\mathcal{V}^{r}}\leq
C\|((Q_{a}[v])_{\delta},(Q_{b}[v])_{\delta})\|_{\widetilde{L}^{1}(J;\dot{B}^{1+r}_{\infty,1})}+\|(Q_{a}[v])_{\delta}\|_{\widetilde{L}^{\infty}(J;\dot{B}^{r}_{\infty,1})},$
(3.55)

where

$\displaystyle\|(Q_{a}[v])_{\delta}\|_{\widetilde{L}^{\infty}(J;\dot{B}^{r}_{\infty,1})}$
$\displaystyle\leq
C\||D_{x}|\eta_{\delta}\|_{\dot{B}_{\infty,1}^{r}}\|\nabla_{x,z}v_{1}\|_{L^{\infty}(J;L^{\infty})}+C\||D_{x}|\eta_{2}\|_{\dot{B}_{\infty,1}^{0}}\|\nabla_{x,z}v_{\delta}\|_{\widetilde{L}^{\infty}(J;\dot{B}_{\infty,1}^{r})}$
$\displaystyle\leq E_{r}.$

Therefore, it follows in view of (3.53) that

$\|v_{\delta}\|_{\mathcal{V}^{r}}\leq E_{r}.$ (3.56)

Inserting (3.54) into $E_{r}$ on the right-hand side of (3.56), then absorbing
$\|v_{\delta}\|_{\mathcal{V}^{r}}$, we find

$\displaystyle\|v_{\delta}\|_{\mathcal{V}^{r}}$ $\displaystyle\leq
C\mathcal{A}_{r}\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{r}_{\infty,1}}$
$\displaystyle\quad+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{r}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}.$

Using this bound for $\|v_{\delta}\|_{\mathcal{V}^{r}}$ in $E_{r}$, we deduce
that $E_{r}$ obeys the same bound as $\|v_{\delta}\|_{\mathcal{V}^{r}}$.
Finally, (3.49) follows from (3.53). ∎

##  4\. Proof of Theorem 1.2

We first prove a simple fixed-point lemma tailored to our problem.

######  Lemma 4.1.

Let $(E,\|\cdot\|)$ be a Banach space, and let $\nu>0$. Denote by $B_{\nu}$
the closed ball of radius $\nu$ centered at $0$ in $E$. Assume that
$\mathcal{B}:E\times E\to E$ and there exists
$\mathcal{F}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that

$\nu\mathcal{F}(2\nu)\leq\frac{1}{2}$ (4.1)

and the following conditions hold.

  * •

For all $x\in B_{\nu}$, $\mathcal{B}(x,\cdot)$ is linear and

$\|\mathcal{B}(x,y)\|\leq\mathcal{F}(\|x\|)\|x\|\|y\|\quad\forall y\in
B_{\nu}.$ (4.2)

  * •

For all $x_{1},~{}x_{2},~{}y\in B_{\nu}$,

$\|\mathcal{B}(x_{1},y)-B(x_{2},y)\|\leq\mathcal{F}(\|x_{1}\|+\|x_{2}\|)\|x_{1}-x_{2}\|\|y\|.$
(4.3)

Then, there exists $\delta=\delta(\nu,\mathcal{F})>0$ small enough such for
all $x_{0}\in E$ with norm less than $\delta$, $x\mapsto
x_{0}+\mathcal{B}(x,x)$ has a unique fixed point $x_{*}$ in $B_{\nu}$ with
$\|x_{*}\|\leq 2\|x_{0}\|$.

###### Proof.

For all $x,~{}y\in B_{\nu}$, combining (4.2) and (4.3) gives

$\displaystyle\|\mathcal{B}(x,x)-\mathcal{B}(y,y)\|$
$\displaystyle\leq\|\mathcal{B}(x,x-y)\|+\|\mathcal{B}(x,y)-\mathcal{B}(y,y)\|$
(4.4)
$\displaystyle\leq\mathcal{F}(\|x\|)\|x\|\|x-y\|+\mathcal{F}(\|x\|+\|y\|)\|x-y\|\|y\|.$

Let $\delta\in(0,\frac{\nu}{2})$ be such that $2\delta\mathcal{F}(2\delta)<1$.
For $\|x_{0}\|<\delta$, we construct the fixed point of
$x_{0}+\mathcal{B}(x,x)$ by the iterative scheme
$x_{n+1}=x_{0}+\mathcal{B}(x_{n},x_{n})$, $n\geq 0$. Using (4.2) and induction
we find that $\|x_{n}\|<2\|x_{0}\|<\nu$ for all $n\geq 0$. Then, (4.4) implies

$\displaystyle\|x_{n+1}-x_{n}\|$
$\displaystyle=\|\mathcal{B}(x_{n},x_{n})-\mathcal{B}(x_{n-1},x_{n-1})\|$
$\displaystyle\leq
2\delta\mathcal{F}(2\delta)\|x_{n}-x_{n-1}\|+2\delta\mathcal{F}(4\delta)\|x_{n}-x_{n_{1}}\|.$

Choosing $\delta$ smaller so that
$2\delta\mathcal{F}(2\delta)+2\delta\mathcal{F}(4\delta)\leq\frac{1}{2}$, we
obtain $\|x_{n+1}-x_{n}\|\leq\frac{1}{2}\|x_{n}-x_{n_{1}}\|$. Thus the
sequence $(x_{n})$ converges to $x_{*}\in E$. In particular, $\|x_{*}\|\leq
2\|x_{0}\|<\nu$ and it follows from (4.4) that
$\|\mathcal{B}(x_{n},x_{n})-\mathcal{B}(x_{*},x_{*})\|\to 0$. Therefore,
letting $n\to\infty$ in the iterative scheme yields
$x_{*}=x_{0}+\mathcal{B}(x_{*},x_{*})$. If $x^{\prime}$ is another fixed point
in $B_{\nu}$ , then by (4.4) and (4.1),

$\|x_{*}-x^{\prime}\|=\|\mathcal{B}(x_{*},x_{*})-\mathcal{B}(x^{\prime},x^{\prime})\|\leq
2\delta\mathcal{F}(2\delta)\|x_{*}-x^{\prime}\|+\nu\mathcal{F}(2\nu)\|x-y\|\leq\frac{3}{4}\|x_{*}-x^{\prime}\|,$

provided that $2\delta\mathcal{F}(2\delta)\leq\frac{1}{4}$. Therefore, $x_{*}$
is the unique fixed point in $B_{\nu}$. ∎

###  4.1. The one-phase problem

From (1.11) and (3.17), the one-phase Muskat problem can be written as the
nonlinear fractional heat equation

$\partial_{t}\eta+\kappa|D_{x}|\eta=-\kappa
R^{-}(\eta)\eta,\quad\eta|_{t=0}=\eta_{0},$ (4.5)

or equivalently

$\eta(t)=e^{-\kappa t|D_{x}|}\eta_{0}-\mathcal{B}(\eta,\eta),$ (4.6)

where

$\mathcal{B}(\eta,f)(x,t)=\kappa\int_{0}^{t}e^{-\kappa(t-\tau)|D_{x}|}(R^{-}(\eta)f)(\tau)d\tau.$

Let $X^{1}_{\kappa}([0,T])$ denote the Banach space $X^{1}([0,T])$ (see
(3.20)) endowed with the equivalent norm

$\|\eta\|_{X^{1}_{\kappa}([0,T])}=\|\eta\|_{\widetilde{L}^{\infty}([0,T];\dot{B}_{\infty,1}^{1})}+\kappa\|\eta\|_{\widetilde{L}^{1}([0,T];\dot{B}_{\infty,1}^{2})}.$
(4.7)

By virtue of Lemma 2.6, $X^{1}_{k}([0,T])\subset
C([0,T];\dot{B}_{\infty,1}^{1})$. Then the global well-posedness statement in
Theorem 1.2 is a consequence of the following lemma.

######  Lemma 4.2.

There exists a small number $\delta>0$ such that if
$\|\eta_{0}\|_{\dot{B}_{\infty,1}^{1}}<\delta$, then $e^{-\kappa
t|D_{x}|}\eta_{0}-\mathcal{B}(\eta,\eta)$ has a unique fixed point $\eta$ in
$X^{1}_{k}([0,T])$ for any $T>0$ with norm less than
$C\|\eta_{0}\|_{\dot{B}_{\infty,1}^{1}}$, $C=C(d)$.

###### Proof.

Let $T>0$ be arbitrary. We appeal to Lemma 4.1 with $E=X^{1}_{\kappa}([0,T])$
and $\nu=c_{*}/2$, where $c_{*}$ is given in (3.27). Note that if
$\|\eta\|_{X^{1}_{\kappa}([0,T])}<c_{*}$, then
$\|\eta\|_{C([0,T];\dot{B}_{\infty,1}^{1})}<c_{*}$, so that for all
$t\in[0,T]$, $\eta(t,\cdot)$ satisfies the smallness condition (3.27). In
addition, in view of (2.10) we have

$\|e^{-\kappa t|D_{x}|}\eta_{0}\|_{X^{1}_{\kappa}([0,T])}\leq
C_{1}\|\eta_{0}\|_{\dot{B}_{\infty,1}^{1}},\quad C_{1}=C_{1}(d).$

Applying the parabolic estimate (2.11) with $(q_{1},q_{2})=(\infty,1)$ and
$(q_{1},q_{2})=(1,1)$, we obtain

$\|\mathcal{B}(\eta,f)\|_{X^{1}_{\kappa}([0,T])}\leq
C\kappa\|R^{-}(\eta)f\|_{\widetilde{L}^{1}([0,T];\dot{B}_{\infty,1}^{1})}=C\kappa\|R^{-}(\eta)f\|_{L^{1}([0,T];\dot{B}_{\infty,1}^{1})}.$

Recall from (3.30) that

$\|R^{-}(\eta)f\|_{\dot{B}_{\infty,1}^{1}}\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{1}_{\infty,1}}+C\||D_{x}|\eta\|_{\dot{B}^{1}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}.$

It follows that

$\displaystyle\|\mathcal{B}(\eta,f)\|_{X^{1}_{\kappa}([0,T])}$
$\displaystyle\leq
C\||D_{x}|\eta\|_{L^{\infty}_{t}\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{L^{1}_{t}\dot{B}^{1}_{\infty,1}}+C\||D_{x}|\eta\|_{L^{1}_{t}\dot{B}^{1}_{\infty,1}}\||D_{x}|f\|_{L^{\infty}_{t}\dot{B}^{0}_{\infty,1}}$
(4.8) $\displaystyle\leq
C\|\eta\|_{X^{1}_{\kappa}([0,T])}\|f\|_{X^{1}_{\kappa}([0,T])},\quad C=C(d),$

where we have used (2.6) in the second inequality.

Similarly, we have

$\|\mathcal{B}(\eta_{1},f)-\mathcal{B}(\eta_{2},f)\|_{X^{1}_{\kappa}([0,T])}\leq
C\kappa\|R^{-}(\eta_{1})f-R^{-}(\eta_{2})f\|_{\widetilde{L}^{1}([0,T];\dot{B}_{\infty,1}^{1})}.$

Recall the contraction estimate (3.49):

$\displaystyle\|R^{-}(\eta_{1})f-R^{-}(\eta_{2})f\|_{\dot{B}_{\infty,1}^{1}}$
$\displaystyle\leq
C\mathcal{A}_{1}\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}}+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{0}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{1}_{\infty,1}}+C\||D_{x}|\eta_{\delta}\|_{\dot{B}^{1}_{\infty,1}}\||D_{x}|f\|_{\dot{B}^{0}_{\infty,1}},$

where $C=C(d)$ and
$\mathcal{A}_{1}=\||D_{x}|\eta_{1}\|_{\dot{B}^{1}_{\infty,1}}+\||D_{x}|\eta_{2}\|_{\dot{B}^{1}_{\infty,1}}$.
We deduce

$\displaystyle\|\mathcal{B}(\eta_{1},f)-\mathcal{B}(\eta_{2},f)\|_{X^{1}_{\kappa}([0,T])}$
$\displaystyle\leq
C\Big{(}\|\eta_{1}\|_{X^{1}_{\kappa}([0,T])}+\|\eta_{2}\|_{X^{1}_{\kappa}([0,T])}+2\Big{)}\|\eta_{\delta}\|_{X^{1}_{\kappa}([0,T])}\|f\|_{X^{1}_{\kappa}([0,T])}.$
(4.9)

In view of (4.8) and (4.9), $\mathcal{B}$ satisfies the conditions (4.2) and
(4.3) in Lemma 4.1 with $\mathcal{F}(z)=C(z+2)$. Finally, choosing $c_{*}$
smaller if necessary so that $\nu=c_{*}/2$ satisfies (4.1), we can apply Lemma
4.1 to conclude the proof. ∎

Let $\eta_{0}^{j}\in\dot{B}_{\infty,1}^{1}$, $j=1,2$ be initial data with norm
less than $\delta$ given in Lemma 4.2. Denoting by $\eta^{j}$ the
corresponding solutions. Then the estimates (4.8) and (4.9) imply

$\displaystyle\|\eta^{1}-\eta^{2}\|_{X^{1}_{k}([0,T])}$
$\displaystyle\leq\|e^{-\kappa
t|D_{z}|}(\eta^{1}_{0}-\eta^{2}_{0})\|_{X^{1}_{\kappa}([0,T])}+\|\mathcal{B}(\eta^{1},\eta^{1})-\mathcal{B}(\eta^{2},\eta^{2})\|_{X^{1}_{\kappa}([0,T])}$
$\displaystyle\leq
C\|\eta^{1}_{0}-\eta^{2}_{0}\|_{\dot{B}_{\infty,1}^{1}}+C\|\eta^{1}\|_{X^{1}_{\kappa}([0,T])}\|\eta^{1}-\eta^{2}\|_{X^{1}_{\kappa}([0,T]}$
$\displaystyle\quad+C\|\eta^{1}-\eta^{2}\|_{X^{1}_{\kappa}([0,T])}\|\eta^{2}\|_{X^{1}_{\kappa}([0,T])},$

where we have used that $\|\eta^{j}\|_{X^{1}_{\kappa}([0,T])}\leq 2\delta<1$.
Therefore, choosing $\delta$ smaller if necessary we obtain the stability
estimate

$\|\eta^{1}-\eta^{2}\|_{X^{1}_{k}([0,T])}\leq
C\|\eta^{1}_{0}-\eta^{2}_{0}\|_{\dot{B}_{\infty,1}^{1}}\quad\forall T>0,\quad
C=C(d).$ (4.10)

This shows that the solution map is Lipschitz continuous in the topology of
$\dot{B}_{\infty,1}^{1}$, thereby establishing global well-posedness in the
strong sense of Hadamard.

###  4.2. The two-phase problem

Let us recall the Dirichlet-Neumann reformulation of the two-phase Muskat
problem:

$\displaystyle\partial_{t}\eta=-\frac{1}{\mu^{-}}G^{-}(\eta)f^{-},$ (4.11)
$\displaystyle\begin{cases}f^{+}-f^{-}=-\llbracket\rho\rrbracket\eta,\\\
\frac{1}{\mu^{+}}G^{+}(\eta)f^{+}-\frac{1}{\mu^{-}}G^{-}(\eta)f^{-}=0.\end{cases}$
(4.12)

Substituting $f^{+}$ from the first equation in (4.12), the second equation
becomes

$\frac{1}{\mu^{-}}G^{-}(\eta)f^{-}=\frac{1}{\mu^{+}}G^{+}(\eta)f^{-}-\frac{\llbracket\rho\rrbracket}{\mu^{+}}G^{+}(\eta)\eta.$

Then setting

$R^{\pm}(\eta)g=G^{\pm}(\eta)g\mp|D_{x}|g,$ (4.13)

we obtain after arranging terms

$|D_{x}|f^{-}=\frac{1}{\mu^{+}+\mu^{-}}(\mu^{-}R^{+}(\eta)-\mu^{+}R^{-}(\eta))f^{-}-\frac{\llbracket\rho\rrbracket\mu^{-}}{\mu^{+}+\mu^{-}}G^{+}(\eta)\eta.$
(4.14)

Plugging this into the linearization
$G^{-}(\eta)f^{-}=|D_{x}|f^{-}+R^{-}(\eta)f^{-}$, it follows in view of (4.11)
that

$\displaystyle\partial_{t}\eta$
$\displaystyle=-\frac{1}{\mu^{+}+\mu^{-}}(R^{+}(\eta)+R^{-}(\eta))f^{-}-\frac{\llbracket\rho\rrbracket}{\mu^{+}+\mu^{-}}|D_{x}|\eta+\frac{\llbracket\rho\rrbracket}{\mu^{+}+\mu^{-}}R^{+}(\eta)\eta.$

In the stable case $\rho^{+}<\rho^{-}$,
$\kappa=\frac{\llbracket\rho\rrbracket}{\mu^{+}+\mu^{-}}>0$ and $\eta$ obeys
the nonlinear fractional heat equation

$\partial_{t}\eta+\kappa|D_{x}|\eta=-\kappa(R^{+}(\eta)+R^{-}(\eta))\frac{f^{-}}{\llbracket\rho\rrbracket}+\kappa
R^{+}(\eta)\eta.$ (4.15)

In order for (4.15) to be a self-contained equation, it remains to determine
$f^{-}$ from $\eta$. From (4.14) we see that $f^{-}$ is a fixed point of

$\displaystyle\mathcal{K}(\eta)g=\frac{1}{\mu^{+}+\mu^{-}}(\mu^{-}|D_{x}|^{-1}R^{+}(\eta)-\mu^{+}|D_{x}|^{-1}R^{-}(\eta))g-\frac{\llbracket\rho\rrbracket\mu^{-}}{\mu^{+}+\mu^{-}}|D_{x}|^{-1}G^{+}(\eta)\eta.$
(4.16)

######  Proposition 4.3.

Let $r\geq 0$ and let
$\eta\in\dot{B}^{1}_{\infty,1}\cap\dot{B}^{r+1}_{\infty,1}$. If $\eta$
satisfies (3.27) then $\mathcal{K}(\eta)$ has a unique fixed point $f^{-}$ in
$\dot{B}^{1}_{\infty,1}\cap\dot{B}^{1+r}_{\infty,1}$ and

$\|f^{-}\|_{\dot{B}^{r}_{\infty,1}}\leq
C(d,r)\llbracket\rho\rrbracket\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}.$
(4.17)

###### Proof.

We apply Lemma 3.3 with $E_{1}=\dot{B}^{1}_{\infty,1}$,
$E_{2}=\dot{B}^{1+r}_{\infty,1}$ and

$u_{0}=-\frac{\llbracket\rho\rrbracket\mu^{-}}{\mu^{+}+\mu^{-}}|D_{x}|^{-1}G^{+}(\eta)\eta\in
E_{1}\cap E_{2}$

since, under condition (3.27), (3.48) gives

$\|u_{0}\|_{\dot{B}^{1+r}_{\infty,1}}\leq
C\llbracket\rho\rrbracket\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}},\quad
C=C(d,r).$

It then suffices to prove that the mapping

$g\mapsto\mathcal{K}_{1}(g):=\frac{1}{\mu^{+}+\mu^{-}}(\mu^{-}|D_{x}|^{-1}R^{+}(\eta)-\mu^{+}|D_{x}|^{-1}R^{-}(\eta))g$

satisfies (3.18) and (3.19). Indeed, under condition (3.27) with $c_{*}$
smaller if necessary, we can apply (3.30) to have

$\displaystyle\|\mathcal{K}_{1}(\eta)g\|_{\dot{B}^{1+r}_{\infty,1}}$
$\displaystyle\leq\||D_{x}|^{-1}R^{-}(\eta)\|_{\dot{B}^{1+r}_{\infty,1}}+\||D_{x}|^{-1}R^{+}(\eta)\|_{\dot{B}^{1+r}_{\infty,1}}$
$\displaystyle\leq
C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\|g\|_{\dot{B}^{1+r}_{\infty,1}}+C\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}\|g\|_{\dot{B}^{1}_{\infty,1}},\quad
C=C(d,r).$

Thus there exists a unique fixed point
$g\in\dot{B}^{1}_{\infty,1}\cap\dot{B}^{1+r}_{\infty,1}$ for $\mathcal{K}$;
moreover,

$\|g\|_{\dot{B}^{1+r}_{\infty,1}}\leq
C\llbracket\rho\rrbracket\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}+C\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}\|g\|_{\dot{B}^{1+r}_{\infty,1}}+C\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}\|g\|_{\dot{B}^{0}_{\infty,1}}.$
(4.18)

When $r=0$, combining (4.18) and (3.27) yields

$\|g\|_{\dot{B}^{0}_{\infty,1}}\leq
C(d)\llbracket\rho\rrbracket\||D_{x}|\eta\|_{\dot{B}^{0}_{\infty,1}}.$

Plugging this into (4.18) and using (3.27) again we find that for $r\geq 0$,

$\|g\|_{\dot{B}^{r}_{\infty,1}}\leq
C(d,r)\llbracket\rho\rrbracket\||D_{x}|\eta\|_{\dot{B}^{r}_{\infty,1}}$

which completes the proof. ∎

Assume that $\eta_{1}$ and $\eta_{2}$ satisfy (3.27), and $f^{-}_{j}$ is the
fixed point of $\mathcal{K}(\eta_{j})$. Set $\eta_{\delta}=\eta_{1}-\eta_{2}$
and $f_{\delta}=f_{1}-f_{2}$. It follows that

$\displaystyle\||D_{x}|f_{\delta}\|_{\dot{B}_{\infty,1}^{r}}$
$\displaystyle\leq\|R^{+}(\eta_{1})f^{-}_{1}-R^{+}(\eta_{2})f^{-}_{2}\|_{\dot{B}_{\infty,1}^{r}}+\|R^{-}(\eta_{1})f^{-}_{1}-R^{-}(\eta_{2})f^{-}_{2}\|_{\dot{B}_{\infty,1}^{r}}$
$\displaystyle\quad+\kappa\|G^{+}(\eta_{1})\eta_{1}-R^{+}(\eta_{2})\eta_{2}\|_{\dot{B}_{\infty,1}^{r}}.$

Using the boundedness (3.30) for $R^{\pm}(\eta_{j})$ and the contraction
(3.49) for $R^{\pm}(\eta_{1})-R^{\pm}(\eta_{2})$, we easily deduce

$\||D_{x}|f_{\delta}\|_{\dot{B}_{\infty,1}^{r}}\leq
C\llbracket\rho\rrbracket\big{(}\mathcal{A}_{r}\||D_{x}|\eta_{\delta}\|_{\dot{B}_{\infty,1}^{0}}+\||D_{x}|\eta_{\delta}\|_{\dot{B}_{\infty,1}^{r}}\big{)},$
(4.19)

where $\mathcal{A}_{r}$ is given by (3.50).

With $f^{-}$ given in Proposition 4.3 and satisfying the estimates (4.17) and
(4.19), the nonlinearities $R^{\pm}(\eta)f^{-}$ and $R^{+}(\eta)\eta$ in
(4.15) obey the same bounds as for $R^{-}(\eta)\eta$ in the one-phase equation
(4.5). Therefore, the proof of Lemma 4.2 applies, leading to the global well-
posedness for the the two-phase problem with small data in
$\dot{B}_{\infty,1}^{1}$.

Acknowledgment. H. Q. Nguyen was partially supported by NSF grant DMS-190777.
The author would like to thank Benoit Pausader for generously sharing many
ideas.

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