Abstract
Global well-posedness of the Navier–Stokes equations with a free boundary condition is considered in the scaling critical homogeneous Besov spaces \({\dot{B}}_{p,1}^{-1+n/p}({\mathbb {R}}^n_+)\) with \(n-1< p< 2n-1\). To show the global well-posedness, we establish end-point maximal \(L^1\)-regularity for the initial-boundary value problem of the Stokes equations. Such an estimate is obtained via related estimate for the initial-boundary value problem of the heat equation with the inhomogeneous Neumann data as well as the pressure estimate in the critical Besov space framework. The proof heavily depends on the explicit expression of the fundamental integral kernel of the Lagrange transformed linearized Stokes equations and the almost orthogonal estimates with the space-time Littlewood–Paley dyadic decompositions. Our result here improves the initial space and boundary state than previous results by Danchin–Hieber–Mucha–Tolksdorf (Free boundary problems via Da Prato–Grisvard theory. arXiv:2011.07918v2) and ourselves (Ogawa and Shimizu in J Evol Equ 22(30):67, 2022; Ogawa and Shimizu in J Math Soc Jpn. arXiv:2211.06952v3).
Similar content being viewed by others
1 Introduction
1.1 The free boundary problem of the Navier–Stokes system
We consider the initial boundary value problem of the incompressible Navier–Stokes equations with free boundary condition. Let \(\Omega (t)\subset {\mathbb {R}}^n\) be a domain that is occupied by the fluid in the n-dimensional Euclidean space \({\mathbb {R}}^n\) with \(n\ge 2\) and let the initial domain be described by the upper region of a graph of the unknown function \(\bar{\eta }(t, y'):{\mathbb {R}}_+\times {\mathbb {R}}^{n-1}\rightarrow {\mathbb {R}}\) as
where \({\mathbb {R}}^{n-1}\) denotes the \(n-1\)-dimensional Euclidean space. The velocity of the fluid \(\bar{u}(t,y)\) and the pressure \(\bar{p}(t,y)\) for \(y\in \Omega (t)\) satisfy the incompressible Navier–Stokes equations:
Here, \(\partial \Omega (t)\) denotes the boundary of \(\Omega (t)\), \(\nu _t\) is the unit outward normal at a point \(y\in \partial \Omega (t)\) given by
\(T(\bar{u},\bar{p})\) is the stress tensor defined by \(T(\bar{u},\bar{p})=(\nabla \bar{u}+(\nabla \bar{u})^\textsf{T})-\bar{p}I\), where I is the \(n\times n\) identity matrix, \((\nabla _y\bar{u})_{i,j}=\big (\partial \bar{u}_j/\partial y_i\big )_{(1\le i,j\le n)}\), \((\nabla \bar{u})^\textsf{T}\) denotes the transposed matrix of \(\nabla \bar{u}\), where \(\nabla =\nabla _y=(\partial _{y_1},\partial _{y_2}, \ldots , \partial _{y_n})^\textsf{T}\) and \(\nabla '=\nabla '_y=(\partial _{y_1},\partial _{y_2}, \ldots , \partial _{y_{n-1}})^\textsf{T}\). \({\bar{u}}_0\) and \({\bar{\eta }}_0\) are given initial velocity and initial surface, respectively. Our basic assumption of the dynamics of the boundary of the fluid region \(\Omega (t)\) is governed by the kinematic condition (cf. Solonnikov [54]) which is shown from (1.2) by
In our setting (1.1), we do not take into account of the gravity force nor the surface tension.Footnote 1
Free boundary problems for incompressible fluids were first considered by Solonnikov [54] in the space-time \(L^2\) setting and he proved the time local well-posedness of the initial boundary value problem (1.1). It was generalized by Tani–Solonnikov [60], Tani [61, 62], Tani–Tanaka [63], Mucha–Zaja̧czkowski [33], Shibata–Shimizu [51, 52] (see also [41, 48, 49, 55,56,57,58,59]). Beale [5, 6] considered the free surface problem in a semi-infinite domain and Prüss–Simonett [43, 44] proved the local of (1.1) whose initial state \(\Omega =\Omega (0)\) is close to the half-space \({\mathbb {R}}^n_+\) in the class of Sobolev space \(W^{1,2}_p((0,T)\times \Omega )\) with \(p>n+2\). There are many other contributions on this direction, for instance, [1, 7, 8, 16,17,18, 25,26,27, 33, 34, 43,44,46, 51, 53] and references therein.
It is well-known that the incompressible Navier–Stokes equations are invariant under the scaling transform: For any \(\lambda >0\),
Subsequently the Cauchy problem of the Navier–Stokes equations can be solved globally in the Bochner class \(L^{\rho }\big ({\mathbb {R}}_+;{\dot{H}}^{s}_p({\mathbb {R}}^n;{\mathbb {R}}^n)\big )\)
by Fujita–Kato [23] (see also the relevant regularity criterion (cf., [40, 42, 47]). Setting \(\rho =\infty \), \(s=-1+n/p\) in (1.4), and the critical class at \(s=0\) is given, in particular, Kato [29] by \(C_b([0,T);L^n({\mathbb {R}}^n))\) and the scaling critical Besov spaces \({\dot{B}}^{-1+n/p}_{p,\sigma }({\mathbb {R}}^n)\), where \(1\le p<\infty \) and \(1\le \sigma \le \infty \) ( [3, 11,12,13, 30]). Meanwhile, ill-posedness of the Cauchy problem was shown in [10, 66, 70], namely the continuous dependence on the initial data in the classes \(u_0\in {\dot{B}}^{-1}_{\infty ,\sigma }({\mathbb {R}}^n)\), \(1\le \sigma \le \infty \) breaks down. It is then natural to ask if the free surface problem can also be solvable in such a scaling critical function class.
When \(\Omega (0)\equiv {\mathbb {R}}^n_+\), the problem (1.1) was considered by Danchin–Hieber–Mucha–Tolksdorf [15] in a scaling critical Besov space \({\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+)\) for \(n\ge 3\) with \(n-1<p<n\) via maximal \(L^1\)-regularity of the linear problem corresponding to (1.8). Their result is based on the Da Prato–Grisvard theory [19] and applied the result for the initial boundary value problem by Danchin–Mucha [16]. Independently the authors consider the free surface problem in [39] for the scaling critical space \({\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+)\) for \(n\le p<2n-1\) with \(n\ge 2\) using an explicit form of the Fourier image of the fundamental solutions to the linearized Stokes equations corresponding to (1.8) which has been obtained in Shibata–Shimizu [52]. The argument in the both proofs seems very different from each other and the results are compensated each other when \(n\ge 3\).
Under the kinematic boundary condition (1.3), the solution of the Cauchy problem
induces the problem into a fixed boundary value problem. Namely, the Euler coordinates \(y=y_{\bar{u}}(t)\in \Omega (t)\) are transformed into the Lagrangian coordinates \(\tilde{x}\in \Omega (0)\) connected by (1.5). If \(\bar{u}(t,y)\) is Lipschitz continuous with respect to y, then (1.5) can be solved uniquely by
where \(\tilde{x}\in \Omega (0)\) and \(\nu \) denotes the outer normal at the boundary \(\partial \Omega (0)\). Setting
and applying the Lagrangian coordinate to the original problem (1.1), the system is transformed into a fixed domain problem. We first notice that the kinematic condition (1.3) with (1.6) implies
which ensures us that the transformed domain does not move in time \(t>0\), i.e.,
and the fluid region is given by
Hence the dynamics of fluids is governed by the the following intermediate system:
where \(\nu \) denotes the outward normal at a point in \(\partial \Omega \), \(\mathrm{div\,}\bar{u}_0=0\) in the sense of distribution and the nonlinear terms of (1.8) are given by
Here \(\mathrm{div\,}K\) denotes \([\nabla ^\textsf{T}K]^\textsf{T} \equiv \Big (\sum _{k=1}^n \partial _{x_k} K_{kj}(x)\Big )^\textsf{T}\) for an \(n\times n\) matrix valued function \(K=[K_{kj}(x)]_{1\le k,j\le n}\), \(J(D(\tilde{u}))^{-1}\) denotes the inverse of the Jacobi matrix of the transform. We invoke the divergence-curl structure in (1.10) and (1.11) (see Solonnikov [56], see also (10.2) of Corollary 10.2 in Appendix). By applying the Lagrangian transformation, the free surface problem (1.1) is transformed into the fixed boundary problem and the system is transformed into the quasilinear parabolic equation (1.8) (see e.g., [57]).
In this paper, we discuss the time global existence of a solution of the transformed free surface problem (1.8) with non-flat initial surface. We need to discuss the corresponding maximal \(L^1\)-regularity for initial-boundary value problems of the Stokes equations with the associated non-stress boundary condition. We extend former results in the homogeneous Besov spaces \({\dot{B}}^{s}_{p,1}({\mathbb {R}}^n_+)\) with \(-1+1/p<s<1/p\) and \(1< p<\infty \) (see for the definition of the homogeneous Besov spaces below) and it naturally extends the well-posedness result to the free boundary problem for the Navier–Stokes equations in the scaling critical setting including both the results in [15] and [39]. Furthermore, we generalize the result into a non-flat initial surface \(\partial \Omega =\partial \Omega (0)\), where \(\partial \Omega \) is assumed to be described by the graph of a given small function \(y_n=\eta _0(y')\). Such an extension enable us to conclude the range of exponent p for the global well-posedness of the free surface problem of the Navier–Stokes equations into \(n-1<p<2n-1\) and hence our result includes former results [15] and [39].
Let us introduce an extension function of the boundary function \(\eta _0(\tilde{x}')\) into the whole domain \(\Omega \).
Definition. Let \(1\le q<\infty \). For \(\eta _0\in {\dot{B}}^{1+\frac{n-1}{q}}_{q,1}({\mathbb {R}}^{n-1})\), set
so that
where the operator \(\textrm{sech}(x_n|\nabla '|)\) is given by the Fourier multiplier
and \(\mathcal {F}^{-1}_{\xi '}\) denotes the Fourier inverse transform from \(\xi '\in {\mathbb {R}}^{n-1}\rightarrow x'\in {\mathbb {R}}^{n-1}\). We introduce the domain deformation (flattening) transform \(\mathcal {E}:\tilde{x}\in \Omega \mapsto x\in {\mathbb {R}}^n_+ =\{x=(x',x_n)\in {\mathbb {R}}^n; x'\in {\mathbb {R}}^{n-1},\, x_n>0\} \) given by
and the Jacobi matrix \(J(DE)=\partial \tilde{x}/\partial x\) of (1.16) with its determinant \(1+\partial _{x_n}E\). Since \(\partial _{x_n} E(x',x_n)=\textrm{sech}(x_n |\nabla '|)|\nabla '|\eta _0(x')\), under the smallness condition on \(|\nabla '|\eta _0\), \(\partial _{x_n}E>-1\) everywhere (cf. Lemma 8.1 below) and the deformation \(\mathcal {E}\) is bijective. If we set \(\phi (x_n)=x_n+E(\cdot ,x_n)\), then \(\partial _{x_n}\phi =1+\partial _{x_n} E\) and is strictly positive under the smallness condition for \(|\nabla '|\eta _0\), it means that \(\phi (x_n)\) is invertible and monotone increasing with respect to \(x_n\). Noting that \(\phi (0)=E(x',0)=\eta _0(x')\), we know that \(\mathcal {E}\) maps the domain \(\{ (\tilde{x}', \tilde{x}_n); \tilde{x}_n\gtrless \eta _0(\tilde{x}')\}\) into
(cf. [53]), and the boundary \(\partial \Omega =\{(\tilde{x}',\tilde{x}_n)\in {\mathbb {R}}^n;\,\tilde{x}_n=\eta _0(\tilde{x}')\}\) is transformed into a new boundary \(\partial {\mathbb {R}}^n_+ =\{(x',x_n)\in {\mathbb {R}}^n;\,x_n=0\}\). The component of the transposed inverse of the Jacobi matrix is given by using \(\partial _j=\partial _{x_j}\) (\(j=1,2,\ldots ,n\)) that
The covariant derivatives for a function \(K(x)=\tilde{K}(\tilde{x})\) ( \( 1\le j,k\le n\)) i.e., \(\nabla K=(\partial _1K, \partial _2K, \ldots \partial _{n}K)^\textsf{T}\) and a vector field \(F(x',x_n)=\tilde{F}(\tilde{x}',\tilde{x}_n):{\mathbb {R}}^n_+\rightarrow {\mathbb {R}}^n\), are expressed from (1.17) by
We also denote \(( \partial _{E} K)_j\) and \(D_EK\) the corresponding covariant derivatives and the Jacobi matrix form for any function K, respectively. If E is sufficiently smooth then it follows from (1.19) that
and
where the first and the second terms of the right hand side of (1.20) maintain their divergence form.
Introducing new unknown functions;
the Jacobi matrix is denoted by
Hence applying the boundary flattening operation \(\mathcal {E}\) in (1.16) to the problem (1.8), the system is transformed into the following problem on the flat boundary region \({\mathbb {R}}^n_+\):
where \(\nu _n=(0,\ldots , 0, -1)\) denotes the outward normal at a point in \(\partial {\mathbb {R}}^n_+\), the linear variable coefficient terms are given (cf. (1.20)) by y
The nonlinear terms of (1.22) are given by (1.9)–(1.13) and divergence-curl structure by
Here \(\nu _E={(\nabla ' E, -1)^\textsf{T}}/{\sqrt{1+|\nabla ' E|^2}}\). The initial data \(u_0\) satisfies the natural condition \(\mathrm{div\,}u_0=g(u,E)\big |_{t=0}\) in the sense of distributions. The notations \(\nabla _E\), \(\mathrm{div\,}_E\) (and hence \(D_E\)) are defined by (1.18) and (1.19), respectively and \(J(D_Eu)^{-1}\) denotes the inverse of the Jacobi matrix and \(J(DE)^{-1}\) is given by (1.17). Hereafter, we denote \(\Pi ^{m}_*(A)\) as a polynomial of A of order at most \( m=n-1\) or \(2n-2\) with \(*\) being either u, p, div or bu or bp which indicates the nonlinear terms (1.28)–(1.32). At the above stage, the problem (1.1) is transformed into the fixed and the flat boundary domain with the quasilinear variable coefficient problem.
2 Main results
Before stating our results, we define the Besov space and the Lizorkin–Triebel space in the half-space and on the half-line, respectively (see for details Peetre [42], Triebel [65]).
Definition (The Besov spaces). Let \(s\in {\mathbb {R}}\), \(1\le p, \sigma \le \infty \). Let \(\{\phi _j\}_{j\in \mathbb {Z}}\) be the Littlewood–Paley dyadic decomposition of unity for \(x\in {\mathbb {R}}^n\), i.e., \({\widehat{\phi }}\) is the Fourier transform of a smooth radial function \(\phi \) satisfying \({\widehat{\phi }}(\xi )\ge 0\), \(\textrm{supp}\, {\widehat{\phi }}\subset \{\xi \in {\mathbb {R}}^n\mid 2^{-1}<|\xi |<2\}\), and \({\widehat{\phi }}_j(\xi )={\widehat{\phi }}(2^{-j}\xi )\), \(\sum _{j\in \mathbb {Z}}{\widehat{\phi }}_j(\xi )=1\) for any \(\xi \in {\mathbb {R}}^n\setminus \{0\}\) for \(j\in \mathbb {Z}\), and \( \widehat{\phi }_{\widehat{0}}(\xi ) +\sum _{j\ge 1}{\widehat{\phi }}_j(\xi )=1\) for any \(\xi \in {\mathbb {R}}^n\), where \(\widehat{\phi }_{\widehat{0}}(\xi ) \equiv \widehat{\zeta }(|\xi |)\) with a low frequency cut-off \(\widehat{\zeta }(r)=1\) for \(0\le r<1\) and \(\widehat{\zeta }(r)=0\) for \(2<r\) (see [9]). For \(s\in {\mathbb {R}}\) and \(1\le p,\sigma \le \infty \), let \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n)\) be the homogeneous Besov space with the norm
and \({B}^s_{p,\sigma }({\mathbb {R}}^n)\) be the inhomogeneous Besov space with the norm
We introduce the homogeneous Besov space on the half-Euclidean space \({\mathbb {R}}^n_+=\{x\in {\mathbb {R}}^n; x=(x',x_n), x_n>0, x'\in {\mathbb {R}}^{n-1}\}\): \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n_+)\) as the set of all measurable functions f in \({\mathbb {R}}^n_+\) satisfying
where \(c_n^{-1}=(2\pi )^{n/2}\). The inhomogeneous version \({B}^s_{p,\sigma }({\mathbb {R}}^n_+)\) is analogously defined.
Definition (The Bochner–Lizorkin–Triebel spaces). Let \(s\in {\mathbb {R}}\) and \(X({\mathbb {R}}^n_+)\) be a Banach space on \({\mathbb {R}}^n_+\) with the norm \(\Vert \cdot \Vert _{X}\). Let \(\{\psi _k\}_{k\in \mathbb {Z}}\) be the Littlewood–Paley dyadic decomposition of unity for \(t\in {\mathbb {R}}\). For a Banach space X, let \({\dot{F}}^s_{1,1}({\mathbb {R}};X)\) be the Bochner–Lizorkin–Triebel space ( [31, 64]) with the norm
Analogously as above, we define the Bochner–Lizorkin–Triebel spaces \({\dot{F}}^s_{1,1}(I;X)\) for an interval \(I=(0, T)\) (\(T\le \infty \)) as the set of all measurable functions f on X satisfying
We should like to notice that \({\dot{F}}^s_{1,1}({\mathbb {R}}_+;X)\) is equivalent to \({\dot{B}}^s_{1,1}({\mathbb {R}}_+;X)\) from its definition.
Let \(\mathcal {D}'(\Omega )\) denote the distributions over \(\Omega \) and let \(C_b(I;X)\) be a set of all bounded continuous functions from an interval I to a Banach space X. We also use \(C_v({\mathbb {R}}^n_+)\) (or \(C_v({\mathbb {R}}^{n-1})\)) as a set of all continuous functions vanishing at \(|x|\rightarrow \infty \).
Theorem 2.1
(Global well-posedness of the transformed problem) Let \(n\ge 2\), \(n-1< p<2n-1\) and \(1\le q\le p(n-1)/(n-p)\) \((n-1<p< n)\) and \(1\le q< p(n-1)/ (p-n)\) \((n\le p<2n-1)\). If the initial data \(u_0\in {\dot{B}}^{-1+ n/p}_{p,1}({\mathbb {R}}^n_+)\) with the condition \(\mathrm{div\,}u_0=g(u,E)\big |_{t=0}\) in \(\mathcal {D}'({\mathbb {R}}^n_+)\) and the initial boundary \(\eta _0\in {\dot{B}}^{1+(n-1)/q}_{q,1}({\mathbb {R}}^{n-1})\) satisfy for some small \(\varepsilon _0>0\) that
then the initial boundary value problem (1.22) admits a unique global solution
with the estimate
where \(D^2 u=\partial _i\partial _j u\) \((i,j=1,\ldots n)\) and \(\varepsilon _1=\varepsilon _1(n,p,\varepsilon _0)\) is a constant.
Remark
Since our regularity class is the scaling invariant, the Fujita–Kato principle (cf. [23]) implies that the local well-posedness for the problem (1.22) also holds for the condition (2.2) being assumed only for the surface function \(\eta _0\). We also note that our result above includes the case \(n=2\) that does not seem to be included in the earlier result [15] on the free surface problem in an unbounded domain.
For the regularity of the initial surface \(\eta _0\in {\dot{B}}^{1+(n-1)/q}_{q,1}({\mathbb {R}}^{n-1})\), the exponent q can be taken independently of the regularity exponent p for the velocity field and the pressure and restricted by the limitation of the boundary bilinear estimate (see Proposition 10.3 (3) in Appendix). Under the restriction of \(\eta _0\), its mean value over \({\mathbb {R}}^{n-1}\) of \(\eta _0\) is vanishing if it is integrable and \(\nabla '\eta _0\in {\dot{B}}^{(n-1)/q}_{q,1}({\mathbb {R}}^{n-1})\subset C_v({\mathbb {R}}^{n-1})\) (cf. [15]).
Accordingly the original problem is considered to be solvable in the corresponding critical space if we introduce the space of a pull back of functions by observing the ordinary differential equation (1.5) is uniquely solvable. Let \(\mathcal {E}\) be defined from \(\Omega \rightarrow {\mathbb {R}}^n_+\) by (1.16).
Corollary 2.2
(Global well-posedness of the non-flat fixed boundary problem) Let \(n\ge 2\) and \(n-1< p<2n-1\). For the same \(\varepsilon _0\) in Theorem 2.1 and \(\bar{u}_0\circ \mathcal {E}^{-1}= u_0\in \dot{B}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+)\) with \(\mathrm{div\,}\bar{u}_0=0\) in \(\mathcal {D}'(\Omega )\) satisfying (2.2), let (u, p) be the global solution of (1.22) obtained in Theorem 2.1. Then the pull-back \((\tilde{u},\tilde{p})\) of (u, p) via the transformation (1.16) with the estimate (2.3) satisfies (1.8).
Corollary 2.3
(Global well-posedness of the free boundary problem (1.1)) Let \(n\ge 2\) and \(n-1< p<2n-1\). For the same \(\varepsilon _0\) in Theorem 2.1 and \(\bar{u}_0\circ \mathcal {E}^{-1}\in \dot{B}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+)\) with \(\mathrm{div\,}\bar{u}_0=0\) in \(\mathcal {D}'(\Omega )\) satisfying (2.2), let \((\tilde{u},\tilde{p}, \tilde{\eta })\) be the global solution of (1.8) obtained in Corollary 2.2. Then the pull-back \((\bar{u},\bar{p},\bar{\eta })\) of \((\tilde{u},\tilde{p},\tilde{\eta })\) given in (1.7) via the transformation (1.6) uniquely solves the original problem (1.1).
2.1 Maximal \(L^1\)-regularity for the linearized Stokes equations
In order to show the global well-posedness of the Navier–Stokes equations by the Lagrange coordinate form (1.22), maximal \(L^1\)-regularity for the heat equation with the Neumann boundary condition plays a crucial role. We consider a corresponding regularity estimate to the initial-boundary value problem of the Stokes equations with free stress boundary condition:
where \(u_0\), f, g and h are given initial, external and boundary data, respectively and \(\nu _n\) denotes the outer normal on \(\partial {\mathbb {R}}^n_+\). The following theorem improves the former result on maximal \(L^1\)-regularity with a free boundary value problem in Ogawa–Shimizu [39].
Theorem 2.4
(Maximal \(L^1\)-regularity for the Stokes system) Let \( 1< p< \infty \) and \(-1+ 1/p<s< 1/p\). The problem (2.4) admits a unique solution (u, p) with
if and only if the data in (2.4) satisfy
where \((-\Delta )^{-1}\) denotes the inverse operator of the Laplacian with 0-Dirichlet boundary condition on \(\partial {\mathbb {R}}^n_+\). Besides the solution (u, p) satisfies the following estimate for some constant \(C_M>0\) depending only on p, s and n
The above theorem is that the range of the differentiability exponent s is enlarged than our previous results [38] and [39]. Namely Theorem 2.4 includes the case \(0<s<1/p\). Such an extension is established by reconsidering the detailed estimate for the linear heat equations. Indeed, there is no limitation for the upper bound of s from our explicit analysis in the subsequent theorems (Theorems 2.5, 4.1). The limitation is posed in order to make clear the condition of the homogeneous Besov setting (see Proposition 3.2 below). After establishing maximal \(L^1\)-regularity in the range \(-1+1/p<s<1/p\), the proof of the global well-posedness Theorem 2.1 follows by a reasonable structure of the Besov space that \({\dot{B}}^{n/p}_{p,1}({\mathbb {R}}^n_+)\) is a Banach algebra and all the nonlinear terms can be estimated by such a structure, which can be seen in [15] and [39]. Note that the upper range of \(p<2n-1\) is caused by the worst nonlinear estimate which arose from the boundary nonlinearity.
To establish maximal regularity on the half-space problem (2.4), we decompose the problem (2.4) into several partial components of the data and reduce the problem into the inhomogeneous problem with only boundary data being provided as we presented in the previous works [38, 39]. First we remove the divergence data g as in the proof of Theorem 2.1 in [39]. Introducing properly extended data \(\widetilde{f} =(\bar{f}_1^{o}, \bar{f}_2^{o}, \ldots , \bar{f}_n^{e})\) with \(\bar{f}\) being the divergence term correction and \(\bar{f}_{\ell }^{o}\), \(\bar{f}_n^{e}\) \((\ell =1,2,\ldots n-1)\) denote the odd and even extension to \({\mathbb {R}}^n\), respectively, and \(\widetilde{u_0}\) into \({\mathbb {R}}^n\) in the similar manner, we consider the Cauchy problem of the Stokes flow:
Thanks to extension of \(\widetilde{f}\) and \(\widetilde{u_0}\) we notice that \(\widetilde{p}(x',0)=0\) by the setting of the problem (2.6) (cf. [50, (4.21)]). Then by subtracting the solution of (2.6) from the the original problem (2.4), one can reduce the problem to the following initial boundary value problem for (v, q) with inhomogeneous boundary data:
where we set
In order to prove Theorem 2.4, it is essential to show maximal \(L^1\)-regularity for (2.7). The following estimate is obtained partially in [15] and [39].
Theorem 2.5
Let \( 1< p< \infty \) and \(-1+1/p<s < 1/p\). The problem (2.7) admits a unique solution
if and only if the data in (2.7) satisfy
Besides the solution (v, q) satisfies the following estimate for some constant \(C_M>0\) depending only on p, s and n:
The function class connected to the x-variable in Theorem 2.5 is restricted in \({\dot{B}}^s_{p,1}({\mathbb {R}}^n_+)\subsetneq {\dot{W}}^{s,p}({\mathbb {R}}^n_+)\) and such a restriction is necessary for maximal \(L^1\)-regularity; maximal \(L^1\)-regularity fails for the Lebesgue spaces \(L^p\) even over the whole space \({\mathbb {R}}^n\) in general (see [35]. See also a possible estimate Giga–Saal [24]). On the other hand, the condition (2.9) is sufficient to conclude the estimate (2.10) in Theorem 2.5 even for the end-point spatial exponent \(p=1\). This end-point is excluded because the trace estimate fails when \(p=1\).
The rest of this paper is organized as follows. After preparing basic relations in the Besov space in the half-space \({\mathbb {R}}^n_+\) in the next section, we present a basic formulation for the proof in particular the reduction to the boundary value problems of the heat equations with the Neumann boundary condition and the Stokes equations with the non-stress boundary condition in Sect. 4. We construct an explicit solution formula of the fundamental solutions in Sect. 5. In Sect. 6, we recall the linear boundary estimate of inhomogeneous Neumann type and in Sect. 7 maximal \(L^1\) regularity for the Stokes system is shown. Section 8 is devoted to the linear and nonlinear perturbation estimates and Sect. 9 shows the proof of the global well-posedness for the transformed Navier–Stokes equations. The final section Appendix includes various bilinear estimates.
Throughout this paper, we use the following notations. For \(x\in {\mathbb {R}}^n\), \(\langle x\rangle \equiv (1+|x|^2)^{1/2}\). The boundary \(\partial {\mathbb {R}}^n_+\) is denoted by \({\mathbb {R}}^{n-1}\) for the variables \(x'=(x_1,x_2,\ldots , x_{n-1})\). The transpose of a matrix A is denoted by \(A^\textsf{T}\). The Fourier and the inverse Fourier transforms of \(f\in \mathcal {S}({\mathbb {R}}^n)\) are defined with \(c_n=(2\pi )^{-n/2}\) by
For any functions \(f=f(t,x',x_n)\) and \(g=g(t,x',x_n)\), \(f\underset{(t)}{*}g \), \(f\underset{(t,x')}{*}g \) and \(f\underset{(x_n)}{*}g\) stand for the convolution between f and g with respect to the variable indicated under \(*\), respectively. If both f and g are vector field functions, \(f\underset{(t,x')}{\cdot *}g\) denotes the convolution in \(x'\) as well as the inner product of f and g, i.e.,
In the summation \(\sum _{k\in \mathbb {Z}}\), the parameter k runs for all integers \(k\in \mathbb {Z}\) and for \(\sum _{k\le j}\), k runs for all integers less than or equal to \(j\in \mathbb {Z}\). We denote the norm of \(L^p({\mathbb {R}}^{n-1})\) with \(x'\in {\mathbb {R}}^{n-1}\) variable by \(\Vert \cdot \Vert _{L^p_{x'}}\). Let \(L^{\rho }(I;X)\) denotes the \(\rho \)-th powered Lebesgue–Bochner space upon a Banach space X. The norm for the Bochner–Lizorkin–Triebel spaces on \({\dot{F}}^s_{p,\rho }\big (I;X({\mathbb {R}}^{n-1})\big )\) we use
unless it may cause any confusion. For the Besov spaces, we abbreviate \({\mathbb {R}}^n\) for \({\dot{B}}^s_{p,\sigma } ={\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n)\) and its norm \(\Vert \cdot \Vert _{{\dot{B}}^s_{p,\sigma }}\). For \(a\in {\mathbb {R}}^n\), we denote \(B_R(a)\) as the open ball centered at a with its radius \(R>0\). We also denote the complement of \(B_R(0)\) by \(B_R^c\). \(\Gamma (\cdot )\) denotes the Gamma function. Various constants are simply denoted by C unless otherwise stated.
3 The homogeneous Besov space in the half-space
3.1 The homogeneous Besov spaces on the half-space
We recall the summary for the Besov spaces over a domain \(\Omega \) near the half-Euclidean space \({\mathbb {R}}^n_+\). Let \(\ell _0=\{\{a_k\}_k; k\in \mathbb {Z}, \; a_k\in {\mathbb {R}}, \; \lim _{|k|\rightarrow \infty }|a_k|=0,\; \Vert \{a_k\}_k\Vert _{\ell _0}=\max _k |a_k|\}\subsetneq \ell _{\infty }\). It is well known that \((\ell _0)^*\simeq \ell _1\).
Definition Let \(\sigma =0\) or \(1\le \sigma < \infty \) with \(s\in {\mathbb {R}}\). Let
Definition Let \(1\le p<\infty \) and \(1\le \sigma <\infty \) with \(s\in {\mathbb {R}}\).
by the Besov norm \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n_+)\) (see Bahouri-Chemin-Danchin [4] and Bergh–Löfström [9]). It is shown that the above defined space coincides the space \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n_+)\) defined by the restriction in (2.1). Namely, the following proposition is shown by Triebel [65] and Danchin–Mucha [16] (see also [28, 38]).
Proposition 3.1
[16, 65] Let \(1\le p<\infty \). (1) For \(0\le s\), \(1\le \sigma <\infty \),
(2) For \(-\infty <s\le 1/p\) and for \(1<\sigma <\infty \),
(3) For \(-\infty<s< 1/p\) and \(\sigma =1\),
We consider the restriction operator \(R_0\) by multiplying a cut-off function \( \chi _{{\mathbb {R}}^n_+}(x)= 1 \) over \({\mathbb {R}}^n_+\) and otherwise 0, i.e., for \(f\in {\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n)\) with setting \(R_0f=\chi _{{\mathbb {R}}^n_+}(x)f(x)\) in \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n)\) if \(s>0\) and it is understood in a distributional sense. Let \(E_0\) be the zero extension operator from \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n_+)\) to \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n)\). Using Proposition 3.1, the following statement is a variant introduced by Triebel [65, p. 228].
Proposition 3.2
Let \(1\le p<\infty \), \(1\le \sigma <\infty \) and \(-1+1/p<s<1/p\). It holds that
are linear bounded operators. Besides it holds that
where Id denotes the identity operator. Namely \(E_0\) and \(R_0\) are a retraction and a co-retraction, respectively.
The proof of Proposition 3.2 is along the same line of the proof in [65] (cf. [38]). Note that the spaces are homogeneous Besov spaces and then the arrangement appears in Proposition 3 in Danchin–Mucha [16] is required. Furthermore, Triebel [65, Theorem 2.9.1] states that
Proposition 3.3
(cf. [15, 65]) Let \( 1\le p<\infty \) and \(s\in {\mathbb {R}}\), \(f\in {\dot{B}}^{s+1}_{p,1}({\mathbb {R}}^n_+)\) then \(\nabla f\in {\dot{B}}^{s}_{p,1}({\mathbb {R}}^n_+)\) and hence \(\nabla f\in \overset{\circ \quad }{B^{s}_{p,1}}({\mathbb {R}}^n_+)\) if \(s<1/p\). Conversely if \(\nabla f\in {\dot{B}}^{s}_{p,1}({\mathbb {R}}^n_+)\) then \(f\in {\dot{B}}^{s+1}_{p,1}({\mathbb {R}}^n_+)\) if \(-1+ 1/p<s\le -1+ n/p\).
In what follows, we restrict ourselves to the regularity range of the Besov spaces \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n_+)\) in \(-1+1/p<s<1/p\) for \(1<p<\infty \) unless otherwise stated. According to Proposition 3.2, we may regard that any distribution in \({\dot{B}}^s_{p,\sigma }({\mathbb {R}}^n_+)\) under such restriction on s and p can be extended into a distribution over whole space \({\mathbb {R}}^n\) and conversely.
3.2 The L-P decomposition with a separation of variables
In order to split the variables \(x'\in {\mathbb {R}}^{n-1}\) and \(x_n\in {\mathbb {R}}_+\), we introduce an \(x'\)-parallel decomposition and an \(x_n\)-parallel decomposition by Littlewood–Paley type. We introduce \(\{\overline{\Phi _m}\}_{m\in \mathbb {Z}}\) as a Littlewood–Paley dyadic frequency decomposition of unity in separated variables \((\xi ',\xi _n)\in {\mathbb {R}}^{n-1}\times {\mathbb {R}}\).
Definition (The Littlewood–Paley decomposition of separated variables). For \(m\in \mathbb {Z}\), let
(one can choose \(\widehat{\zeta _m}(r) =\sum _{\ell \le m-1}\widehat{\phi _{\ell }}(r)+\widehat{\phi _{-\infty }}(r)\) with a correction distribution \(\widehat{\phi _{-\infty }}(r)\) at \(r=0\)) and set
Then it is obvious from Fig. 1 (restricted on the upper half region in \({\mathbb {R}}^n\)) that
The support of Littlewood–Paley decomposition \(\{\overline{\Phi _m}\}_{m\in \mathbb {Z}}\)
Definition (Various kinds of the Littlewood–Paley dyadic decompositions).
Let \((\tau ,\xi ',\xi _n)\in {\mathbb {R}}\times {\mathbb {R}}^{n-1}\times {\mathbb {R}}\) be Fourier adjoint variables corresponding to \((t,x',x_n)\in {\mathbb {R}}_+\times {\mathbb {R}}^{n-1}\times {\mathbb {R}}_+\). Let \(\{\Phi _m(x)\}_{m\in \mathbb {Z}}\) be the standard (supported in annulus) Littlewood–Paley dyadic decomposition by \( x=(x',x_n)\in {\mathbb {R}}^n_+\).
-
\(\{ \overline{\Phi _m}(x) \}_{m\in \mathbb {Z}}\): the Littlewood–Paley dyadic decomposition over \( x=(x',x_n)\in {\mathbb {R}}^n_+\) given by (3.3).
-
\(\{\psi _k(\tilde{t})\}_{k\in \mathbb {Z}}\): the Littlewood–Paley dyadic decompositions in \({\tilde{t}}\in {\mathbb {R}}\).
-
\(\{\phi _j(x')\}_{j\in \mathbb {Z}}\) and \(\{\phi _j(\tilde{x}_n)\}_{j\in \mathbb {Z}}\): the standard (annulus type) Littlewood–Paley dyadic decompositions in \(x'\in {\mathbb {R}}^{n-1}\) and \(\tilde{x}_n\in {\mathbb {R}}\), respectively.
-
\(\{\zeta _m(x')\}_{m\in \mathbb {Z}}\) and \(\{\zeta _m(\tilde{x}_n)\}_{m\in \mathbb {Z}}\): the lower frequency smooth cut-off given by (3.2), respectively.
-
For the Littlewood–Paley decompositions \(\{\phi _j(x')\}_{j\in \mathbb {Z}}\) and \(\{\psi _k(t)\}_{k\in \mathbb {Z}}\), we set
$$\begin{aligned} \left\{ \begin{aligned}&\widetilde{\phi _j}=\phi _{j-1}+\phi _j+\phi _{j+1}, \\&\widetilde{\psi _k}=\psi _{k-1}+\psi _k+\psi _{k+1} \end{aligned}\right. \end{aligned}$$(3.5)that stands for the j-neighborhood of \(\phi _j(x')\) and the k-neighborhood of \(\psi _k(t)\), respectively.
Since all the above defined decompositions are even functions, we identify \(\tilde{t}\in {\mathbb {R}}\) and \(\tilde{x}_n\in {\mathbb {R}}\) with \(|{\tilde{t}}|=t>0\) and \(|\tilde{x}_n|=x_n>0\), respectively. Then it is easy to see that the norm of the Besov spaces on \({\mathbb {R}}^n_+\) defined by \(\{\Phi _m\}_m\) is equivalent to the one from the Littlewood–Paley decomposition of direct sum type \(\{\overline{\Phi _m}\}_m\).
4 The initial boundary value problem for the Stokes equations
In this section, we study the solution formula for the initial boundary value problem of the heat equation with the Neumann boundary value problem. The formula is a basis to consider the solution formula to the initial boundary value problem of the Stokes equations.
4.1 The initial Neumann boundary value problem for the heat equation
For \(I=(0,T)\) with \(0<T\le \infty \), let u be a solution of the initial-boundary value problem of the second-order parabolic equation with variable coefficients and the inhomogeneous Neumann boundary condition in the half-space \({\mathbb {R}}^n_+=\{ x=(x',x_n); x'\in {\mathbb {R}}^{n-1}, x_n>0\}\):
where \(\partial _t\) and \(\partial _i=\partial _{x_i}\) are partial derivatives with respect to t and \(x_i\), \(u=u(t,x)\) denotes the unknown function, \(u_0=u_0(x)\), \(f=f(t,x)\) and \(g=g(t,x')\) are given initial, external force and boundary data, respectively.
In this context, Weidemaier [69] and Denk–Hieber–Prüss [20, 21] obtained maximal regularity in general settings. Let \(I=(0,T)\) for \(T<\infty \), \(1<\rho ,p<\infty \) and \(1/2-1/(2p)\ne 1/\rho \). The initial-boundary value problem (4.1) has a unique solution u in \(W^{1,\rho }({\mathbb {R}}_+;L^p({\mathbb {R}}^n_+))\cap L^{\rho }({\mathbb {R}}_+;W^{2,p}({\mathbb {R}}^n_+))\) with the compatibility condition
and the solution fulfills the estimate:
where \(B^{2-1/p}_{p,p}({\mathbb {R}}^{n-1})\) and \(F^{1-1/2p}_{\rho ,p}(I;X)\) denote the interpolation spaces of the Besov and Lizorkin–Triebel type, respectively. The end-point case \(\rho =1\) is considered in Ogawa–Shimizu [38] both with Dirichlet and Neumann boundary value problems in \(-1+1/p<s\le 0\).
Theorem 4.1
(The Neumann boundary condition) Let \(1< p< \infty \), \(-1+1/p< s<1/p\). Then the problem (4.1) admits a unique solution
if and only if the external, initial and boundary data in (4.1) satisfy
respectively. Moreover end-point maximal \(L^1\)-regularity holds:
where \(C_M\) is depending only on p, s and n.
Remarks. (i) The linear evolution generated by the Laplacian generates \(C_0\)-semigroup in \(\dot{B}^s_{p,1}({\mathbb {R}}^n_+)\) for \(1\le p<\infty \) and the estimate of maximal \(L^1\)-regularity ensures that the absolute continuity of the solution in t-variable.
(ii) Since \( 1/2-1/(2p)<1 \) for all \(1< p<\infty \), the pointwise compatibility condition (4.2) is not required.
(iii) If \(p=\infty \), the corresponding result holds for the homogeneous Besov space
instead of the Besov space \({\dot{B}}^s_{\infty ,1}({\mathbb {R}}^n_+)\). Note that \(\dot{\mathcal {B}}^0_{\infty ,1}({\mathbb {R}}^n_+)\subset C_{v,0}({\mathbb {R}}^n_+) \equiv \{f\in C({\mathbb {R}}^n_+); \text {supp }f \subset {\mathbb {R}}^n_+, |f(x)|\rightarrow 0,\,\text {as}\, |x|\rightarrow \infty , x\in {\mathbb {R}}^n_+\}\) for the endpoint case \((s,p)=(0,\infty )\).
We only show the estimate for the full time interval \({\mathbb {R}}_+\) but a similar estimate for the finite time interval \(I=(0,T)\) with \(T<\infty \) is also available. In such a case, the restriction on the initial data \(u_0\) can be relaxed into the inhomogeneous Besov space \(B^s_{p,1}({\mathbb {R}}^n_+)\supset {\dot{B}}^s_{p,1}({\mathbb {R}}^n_+)\) and the constant appeared in the estimate can be estimated as \(C_M\simeq O(\log T)\) as \(T\rightarrow \infty \).
For the proof of Theorem 4.1, the principal arugment is reduced into the following problem:
Then the following result yields our main result for the Neumann problem Theorem 4.1.
Theorem 4.2
(Maximal \(L^1\)-regularity by the Neumann boundary data) Let \(1< p<\infty \) and \(-1+1/p<s< 1/p\). There exists a unique solution
to (4.3) if and only if
Besides it holds the estimate:
where C is depending only on p, s and n.
When \(p=\infty \), the analogous result holds under arranging the function class as in the remark after Theorem 4.1.
To show Theorem 4.2, we extend the boundary data \(h(t,x')\) into \(t<0\) by the zero extension and apply the Laplace transform with respect to t, the partial Fourier transform with respect to \(x'\) and we obtain the solution formula of (4.3) as
by using the boundary potential term:
where \(c_{n+1}=(2\pi )^{-(n+1)/2}\) and \(\Gamma \) is a pass parallel to the imaginary axis.
4.2 The solution formula for the Stokes equations
We construct the solution formula of (2.7) following the method by Shibata–Shimizu [50] and [52].
Let \(H=H(t,x')\equiv (H'(t,x'),H_n(t,x'))\) be the boundary data extended into \(t\le 0\) by the zero extension. Besides we assume that they are smooth and decay sufficiently fast at \(|x'|\rightarrow \infty \). The solution formula for the \(\ell \)-th component of the velocity and the pressure is obtained by Shibata–Shimizu [52, (5.19)] as follows: Letting
For any smooth rapidly decreasing boundary data \((\widehat{H'},\widehat{H_n})\) in both \((\tau ,\xi ')\) variables, we consider that
where we take a limit of the integral pass avoiding the singularity at \((\tau ,\xi ')=(0,0)\). All the other components of the velocity fields \(v_{\ell }(t,x)\) (\(\ell =1,2,\ldots , n-1\)) are given by the above two components \((v_n, q)\) and the boundary data \(H=(H',H_n)\) from the Eq. (2.7) (see [52] for the detail of their derivation).
Our main task is to prove maximal \(L^1\)-regularity of the velocity \(v_n\) and the pressure term q of (2.7) which is directly obtained from the inhomogeneous boundary data. We also set symbols of the singular integral operator by the following Fourier multipliers: \(m_*(\tau ,\xi '):{\mathbb {R}}\times {\mathbb {R}}^{n-1}\rightarrow {\mathbb {R}}^n\) as
Using the potential expression (4.8), we obtain a desired pressure estimate by the boundary data H. For any smooth data \((\widehat{H'},\widehat{H_n})\) in both \((\tau ,\xi ')\) variables, we see the explicit expression of \(\nabla q\) and the n-th component \(\partial _t v_n\) can be expressed as
where \(B=B(\tau ,\xi ')\) and \(D(\tau , \xi ')\) are defined by (4.6) and (4.7), respectively. Hence from (4.10), (4.12), (4.13), the term operated by the Laplacian is given by
For the construction of the explicit expression of the solution of (2.7) in [50, (4.24), (4.25)] and [52, (5.19)], the other components of the velocity fields \(v'=(v_1(t,x), v_2(t,x), \ldots , v_{n-1}(t,x))\) satisfy the initial boundary value problem of the heat equations as the pressure and the n-th component velocity as the external force and boundary condition as follows: For \(\ell =1,2\ldots ,n-1\),
Here we remark that \(v'\) in (4.15) and \(v_n\) in (4.8) satisfy the divergence free condition \(\mathrm{div\,}v=0\). Since \(\Delta q(t,x',x_n)=0\) by (4.9), we see from the problem (4.15) that
where we use the formulas (4.8)–(4.9) with a view of (4.5). Hence maximal \(L^1\)-regularity for the velocity \(v_{\ell }\) can be reduced to the maximal regularity estimate for the initial Neumann boundary value problem of the heat equation in the half-space (4.1). We then turn into our attention to the initial boundary value problem of the heat equation with the Neumann boundary condition (cf. [38] and [37]).
5 The potential of boundary term and almost orthognality
5.1 The boundary potential
In this subsection, we derive the exact solution formula of (4.3) which is a bases of the solution formula to the velocity \(v_n(t)\). Let \(h=h(t,x')\) be the boundary data extended into \(t<0\) by the zero extension. The solution to the problem (4.3) is expressed by
where \(G_N\) denotes the Green’s function of the initial-boundary value problem (4.3) identified by
where \(B(\tau ,\xi ')=\sqrt{i\tau +|\xi |^2}\) (cf. [38]). From (4.5) \(\Psi _N(t, x',x_n)=\partial _t G_N(t,x',x_n)\), we regard \(x_n\) as if it is a spectral parameter like \((\lambda , \xi ')\), we then decompose this boundary potential (4.5) by a combination of two families of the Littlewood–Paley dyadic decomposition of unity. Here we notice that from (5.1)–(5.2), the potential \(\Psi _N\) represents the solution operated by the Laplace operator as in (4.4).
5.2 Almost orthogonality of the Neumann boundary potential
In this section we recall the almost orthogonality estimates that are shown in Ogawa–Shimizu [38] and [39] and are mentioned in Sect. 4.1. The estimate is in between the boundary potential term \(\Psi _N\) for the Neumann boundary problem of the heat equation and the time and space Littlewood–Paley decompositions \(\{\psi _k\}_{k\in \mathbb {Z}}\) and \(\{\phi _j\}_{ j\in \mathbb {Z}}\). For the symbol of the gradient of the pressure, we introduce the useful notation for a part of the symbol defined by (4.6); \(B(\tau ,\xi ')=\sqrt{i\tau +|\xi '|^2}\).
Lemma 5.1
(Almost orthogonality I [38]) For \(k,j,\ell \in \mathbb {Z}\) let \(\{\psi _k(t)\}_{k\in \mathbb {Z}}\) and \(\{\phi _j(x')\}_{j\in \mathbb {Z}}\) be the time and the space Littlewood–Paley dyadic decomposition and let \(\Psi _N(t,x',x_n)\) be the boundary potential defined in (4.5). Set
where \(\Psi _N(t,x',x_n)\) is given by (4.5). Then there exists a constant \(C_n>0\) depending only on the dimension n satisfying
For estimating the term involving the grand Littlewood–Paley decomposition, the proof involves an \(x_n\)-convolution between the potential \(\Psi _N\) and \(\phi _m(x_n)\). Concerning the related estimate, we show the following second orthogonal estimate:
Lemma 5.2
(Almost orthogonality II [38]) Let \(k,j,m\in \mathbb {Z}\) and assume \(j\le m+1\). Let \(\Psi _N(t,x',x_n)\) be the potential of the solution for the Neumann data defined by (4.5) and let \(\{\psi _k(t)\}_{k\in \mathbb {Z}}\) and \(\{\phi _j(x')\}_{j\in \mathbb {Z}}\) be a spatial and time Littlewood–Paley decomposition. Let \(\Psi _{N,k,j}(t,x',x_n)\) be defined by (5.3). Then for any \(N\in {\mathbb {N}}\), there exists a constant \(C_N>0\) such that for \(\{\phi _m(x_n)\}_{m\in \mathbb {Z}}\),
The proof of Lemmas 5.1 and 5.2 are very similar to the case for the Dirichlet boundary condition obtained in [38]. Indeed, the estimate for the Neumann boundary condition is stated in [38]. The only difference between those two boundary condition is the factor (4.6) in the formula (4.5) and the difference simply reflects the difference of regularity. See [38] for the details.
5.3 Almost orthogonality of the pressure potential
We derive almost orthogonality concerning the pressure term which is shown in Ogawa–Shimizu [39].
Definition (The pressure potentials). For \(j,k\in \mathbb {Z}\), let \(\{\psi _k(t)\}_{k\in \mathbb {Z}}\), \(\{\phi _j(x')\}_{j\in \mathbb {Z}}\) be the Littlewood–Paley decompositions for \(t\in {\mathbb {R}}\) and \(x'\in {\mathbb {R}}^{n-1}\) valuables, respectively. We set for \(x_n=x_n>0\),
where \(m_{\pi }:{\mathbb {R}}\times {\mathbb {R}}^{n-1}\rightarrow {\mathbb {R}}^n\) is defined in (4.11). We extend the potential \(\pi (t,x',x_n)\) into all \(x_n\in {\mathbb {R}}\) by the even extension (i.e. exchange \(x_n\) into \(|x_n|\)).
Recalling the notation \(\widetilde{\phi _j}\), \(\widetilde{\psi _k}\) defined in (3.5) and noting that
we have for \(x_ n>0\) that
where \((\pi '_{k,j}, \pi _{n,k,j})\) denote the potential for the derivative of the pressure with \({\mathbb {R}}^{n-1}\) direction and \(x_n\) direction given in (5.6), respectively and we use the notion of the inner product-convolution (2.11) and the data is extended by the zero extension for \(t\le 0\). We show the almost orthogonality and its variation in the following.
Lemma 5.3
(Pressure almost orthogonality I [39]) For \(k,j\in \mathbb {Z}\), let \(\pi _{k,j}(t, x',x_n)\) be the pressure potentials defined by (5.6) and let \(\{\psi _k(t)\}_{k\in \mathbb {Z}}\) and \(\{\phi _j(x')\}_{j\in \mathbb {Z}}\) be the Littlewood–Paley decompositions for time and space, respectively.
-
(1)
For the time-dominated region \(k\ge 2j\), there exists \(C_n>0\) such that for any \(x_n\in {\mathbb {R}}_+\) and \(t\in {\mathbb {R}}\),
$$\begin{aligned}&\big \Vert \pi _{k,j}(t,\cdot ,x_n)\big \Vert _{L^1_{x'}} \le C_n 2^{j}\big (1+(2^jx_n)^{n+2}\big ) e^{-2^{(j-1)}x_n}\frac{2^k}{\langle 2^k t\rangle ^{2}}, \end{aligned}$$(5.8)where \(\Vert \cdot \Vert _{L^1_{x'}}\) denotes the \(L^1({\mathbb {R}}^{n-1})\) norm in \(x'\)-variable.
-
(2)
For the space-dominated region \(k<2j\), there exists \(C_n>0\) such that for any \(x_n\in {\mathbb {R}}_+\) and \(t\in {\mathbb {R}}\),
$$\begin{aligned} \Big \Vert \sum _{k< 2j}\pi _{k,j}(t,\cdot ,x_n) \Big \Vert _{L^1_{x'}} \le C_n 2^{j}\big (1+(2^jx_n)^{n+2}\big ) e^{-2^{(j-1)}x_n} \frac{2^{2j}}{\langle 2^{2j} t\rangle ^{2}}. \end{aligned}$$(5.9)
The estimates are extended to \(x_n\in {\mathbb {R}}\) by the even extensions.
We consider the almost orthogonality estimate of second type which will be used for the triumphal arch type Littlewood–Paley dyadic decomposition.
Lemma 5.4
(Pressure almost orthogonality II [39]) Let \(k,j,m\in \mathbb {Z}\) and \(\pi _{k,j}(t, x',x_n)\) be the pressure potential given by (5.6) and let \(\{\psi _k(t)\}_{k\in \mathbb {Z}}\) and \(\{\phi _j(x')\}_{j\in \mathbb {Z}}\) be the Littlewood–Paley decompositions for time and space, respectively. Assume that \(j\le m\), then for any \(N\in {\mathbb {N}}\) and for \(\{\phi _m(x_n)\}_{m\in \mathbb {Z}}\), there exists a constant \(C_{n,N}>0\) depending on n and N such that the following estimates hold:
-
(1)
For the time-dominated region \(k\ge 2j\),
$$\begin{aligned} \big \Vert \phi _m\underset{(x_n)}{*}\ \pi _{k,j}(t,\cdot ,x_n) \big \Vert _{L^1_{x'}} \le&\, C_{n,N}\frac{2^j 2^{-(m-j)} }{\langle 2^j x_n\rangle ^N} \frac{2^k }{\langle 2^k t\rangle ^{2}}. \end{aligned}$$(5.10) -
(2)
For the space-dominated region \(k< 2j\), it holds that
$$\begin{aligned} \Big \Vert \sum _{k<2j} \phi _m\underset{(x_n)}{*}\pi _{k,j}(t,\cdot ,x_n) \Big \Vert _{L^1_{x'}} \le C_{n,N} \frac{2^j 2^{-(m-j)} }{\langle 2^j x_n\rangle ^N} \frac{2^{2j}}{\langle 2^{2j}t\rangle ^{2}}. \end{aligned}$$(5.11)
6 Estimates for the inhomogeneous Neumann boundary condition
6.1 The space-time splitting argument
For \(k,j\in \mathbb {Z}\) let \(\{\psi _k\}_{k\in \mathbb {Z}}\) and \(\{\phi _j(x')\}_{j\in \mathbb {Z}}\) be the Littlewood–Paley dyadic decomposition of time and space variables, respectively and we introduce the decomposed boundary potential defined by (5.3). Since the support of the Fourier image of \(\Phi _m\) only survives where \(m\simeq j\), we see that
where \(\Psi _N\) and \(\Psi _{N,k,j}\) are defined by (4.5) and (5.3), respectively. The \(L^p({\mathbb {R}}^{n}_+)\) norm of the first term of the right hand side of (6.1) is estimated by the Hausdorff–Young inequality of \(x_n\)-variable and the term \(\zeta _{m-1}(|x_n|)\) can be treated as the following:
The term \(\zeta _m(|x'|)\) in the second term of the right hand side of (6.1) is canceling by the \(\phi _j(x')\)-convolution in \(\Psi _{N,k,j}\) (cf. (5.3)).
Regarding the relation (4.4) and applying the Hausdorff–Young inequality to the right hand side of (6.1), it follows that
where the first term of the right hand side of (6.2) includes \(\phi _m(x')\), once the outer decomposition \(\sum _{m\in \mathbb {Z}}\) is fixed then the inner decomposition \(\{\phi _j(x')*\}_{j\in \mathbb {Z}}\) is restricted into only \(|j-m|\le 1\) and the summation for j disappears. This is the one of the main differences from the result shown in [38] and [39].
We separate the estimate of (6.2) into two regions; one is time-dominated area and the other is space-dominated area. The relation between each variables is illustrated in Fig. 2.
The space-time splitting
In order to prove Theorem 4.2, it is enough to prove the following lemma.
Lemma 6.1
Let \(1\le p< \infty \) and \(s\in {\mathbb {R}}\). The terms \(P_1^N\) and \(P_2^N\) defined in (6.2) are estimated as follows:
Remark The above estimates are crucial to extend the regularity range of maximal regularity into higher range \(s>0\).
The most of the estimates are very similar to the case of the proof of the Dirichlet boundary case appeared in Ogawa–Shimizu [38, Lemma 4.3]. However, the detailed proof for the Neumann boundary case was not given there. Since the above estimates are crucial for showing our new result, we give a full proof of the estimates.
Proof of Lemma 6.1
We split the boundary data h into the time-dominated region and the space-dominated region. Let \(\widetilde{\psi _k}\) and \(\widetilde{\phi _j}\) be defined in (3.5). Since
and letting \(h_m(t,x')\equiv \widetilde{\phi _m}(x')\underset{(x')}{*}h(t,x')\) (\(m\in \mathbb {Z}\)), we proceed
where \( \Psi _{N,k,m}(t,x',x_n) \equiv \psi _k(t)\underset{(t)}{*} \phi _m(x')\underset{(x')}{*}\ \Psi _N(t,x',x_n)\). We see that \( L_1\) is the time-dominated region and applying the Minkowski and the Hausdorff–Young inequality with using (5.3), we have
Then by the almost orthogonal estimate between the boundary potential \(\Psi _{N}\) and the Littlewood–Paley decomposition \(\psi _k\) in time, namely we invoke Lemma 5.1. Noting the restriction \(k\ge 2m\) on the time-dominated region and \(\psi _k(s)\underset{(s)}{*}\widetilde{\psi _k}(s)=\psi _k(s)\), we apply the first estimate in (5.4) to (6.7) and obtain that
Meanwhile the second term \(L_2\) is the space-dominated region and letting \(h_m(t,x')\equiv \widetilde{\phi _m}(x')*h(t,x')\), we apply again the Minkowski inequality, the Hausdorff–Young inequality and (5.3),
From (6.6), (6.8) and (6.9), the estimate (6.3) is shown.Footnote 2
We then prove (6.4). Similar way to (6.6) from (6.5), we split \(P_2^N\) into the time-like region and the space-like region;
The first term \(M_1\) of the right hand side is the time-dominated part, letting \(h_j(t,x')\equiv \widetilde{\phi _j}(x')*h(t,x')\), we apply the Minkowski inequality and the Hausdorff–Young inequality with (5.3) as well as the almost orthogonal estimate (5.5) between \(\phi _m\) and \(\Psi _{N,k,j}\) in Lemma 5.2 with \(m\simeq j\). Then setting \(2^mx_n=\tilde{x}_n\), the first term of the right hand side of (6.10) can be estimated as follows:
where the estimate is valid even for \(p=1\).
On the other hand for the estimate \( M_2\), we proceed a similar way to treat \( M_1\). Exchanging the order of the summation of j and k and setting \( h_j(t,x')\equiv \phi _j(x')*h(t,x'), \) it follows by changing \(m-j\rightarrow m\) and (6.10) that
Here we notice that there is no restriction on p nor s. In the last estimate, we exchange the order of the integration in time and the summation of m and k and use the Hausdorff–Young inequality to remove the convolution with the time potential term and then recovers the time integration out side. From (6.10), (6.11) and (6.12) the estimate (6.4) is shown. This completes the proof of Lemma 6.1.\(\square \)
6.2 The boundary trace estimates
We show the optimality for the boundary trace estimate which is required for establishing maximal regularity. This shows that the condition on the boundary data in those theorems are not only sufficient but also a necessary condition (see for more detailed estimates for the boundary trace [28, 32])
Proposition 6.2
(Sharp boundary derivative trace) For \(1< p< \infty \) and \(-1+1/p<s\), there exists a constant \(C>0\) such that for all function \( u=u(t,x',x_n)\in {\dot{W}}^{1,1}({\mathbb {R}}_+; {\dot{B}}^s_{p,1}({\mathbb {R}}^{n}_+))\), \( \Delta u\in L^1({\mathbb {R}}_+; {\dot{B}}^{s}_{p,1}({\mathbb {R}}^{n}_+))\) with \(\partial _{x_n}u(0,x',x_n)=0\), it holds for all \(\ell =1,2,\ldots , n\) that
Remark If a frequency projection upon the time-dominated region
is operated to the left hand side of (6.13), then the spatial end-point exponent \(p=1\) is included in the above statement, while the estimate (6.14) is valid for \(p=1\) (cf. [39]). Hence if we combine the both regularities of the trace side, \(p=1\) is available.
7 Maximal regularity for the Stokes equations
To show Theorem 2.5, we show maximal \(L^1\)-regularity for the pressure term.
7.1 The estimate for the pressure
We recall the notations for the potential (5.6) for the pressure \(\nabla q\) that is shown in Ogawa–Shimizu [39].
Proposition 7.1
[39] Let \(1\le p< \infty \) and \(s \in {\mathbb {R}}\). For given data
there exists \(C>0\) independent of H such that the pressure part q of the problem (2.7) satisfies the estimate
To show the pressure estimate (7.1), we use the potential expression \(\pi _{k,j}(t,x',x_n)\) in (5.6) and the Littlewood–Paley decomposition of unity (3.3);
Concerning the first term of the right-hand side of (7.2), we estimate that the convolution with \(\zeta _{m-1}(x_n)\) can be treated by the Hausdorff–Young inequality in \(x_n\)-variable. Note that the potential \(\pi (t,x',x_n)\) has the even extension in \(x_n\in {\mathbb {R}}\) and hence the \(L^p({\mathbb {R}}^{n}_+)\) norm of the term is estimated as follows:
and we apply Lemma 5.3. Concerning the second term of the right-hand side of (7.2), the number of overlapping supports of the kernel \(\zeta _m(x')\underset{(x')}{*}\phi _j(x')\) is limited in finite numbers, i.e., \(|m-j|\le 1\) and we apply the almost orthogonality of the second type stated in Lemma 5.4.
Proof of Proposition 7.1
Let the boundary data \(H(t,x')=(H'(t,x'),H_n(t.x'))\) is extended into \(t<0\) by the zero extension. From (6.1),
and observing the estimate (7.3) we divide the term into two terms.
where we use the inner product-convolution \(\cdot *\) defined by (2.11). Noting that the data H is divided into the time-dominated region \(k\ge 2j\) and the space-dominated region \(k<2j\), respectively, as
where we set
and we use \(\widetilde{\phi _j}=\phi _{j-1}+\phi _j+\phi _{j+1}\) and \(\widetilde{\psi _k}\) with a similar arrangement. Then noticing \(\widetilde{\phi _j}\underset{(x')}{*}\phi _j=\phi _j\), \(\widetilde{\psi _k}\underset{(t)}{*}\psi _k=\psi _k\), and Proposition 3.2, we divide \(P_1(t)\) into \(L_1\) and \(L_2\) to have the following:
where \(\{\pi _{k,m}\}_{k,m\in \mathbb {Z}}\) is defined in (5.6). For the time-dominated part \(L_1\), since \(k\ge 2m\), we apply the almost orthogonality estimate (5.8) in Lemma 5.3, and the estimate can be obtained in a very similar way to (6.8) and (6.9). By the change of variable \(2^mx_n=\bar{x}_n\), it holds that
On the other hand, when \(k<2m\), for the space-dominated part \(L_2\), applying the almost orthogonality estimate (5.9) in Lemma 5.3 with using the Minkowski inequality, the Hausdorff–Young inequality, we obtain
In the same way for \(P_1(t)\), we decompose \(P_2(t)\) into the time-dominated region and the space-dominated region.
For the time-dominated part \( M_1\), using the Minkowski inequality, the Hausdorff–Young inequality, and also using (5.10) in Lemma 5.4 (1) (the second almost orthogonality), we have
The space-dominated part \( M_2\) is estimated in the similar way as \( M_1\). Applying the Minkowski inequality, the Hausdorff–Young inequality, and using the almost orthogonality (5.11) in Lemma 5.4 (2) for \(k< 2m\), we have
Combining all the estimates (7.4)–(7.10), we obtain
The restriction on the regularity exponent s stems from the structure of the homogeneous Besov space stated in Propositions 3.1–3.3.\(\square \)
The following estimate is the sharp trace estimate and it is required for showing maximal regularity for the velocity part of the Stokes equation.
Proposition 7.2
Let \(1\le p< \infty \) and \(-(n-1)/p'<s\le (n-1)/p\). Given boundary data
let q be the pressure term defined by (4.9). Then there exists a constant \(C>0\) such that the following estimates hold:
Proof of Proposition 7.2
Let \(\{\psi _k\}_{k\in \mathbb {Z}}\) and \(\{\phi _j\}_{j\in \mathbb {Z}}\) be the Littlewood–Paley dyadic decomposition of the unity in \(t\in {\mathbb {R}}\) and \(x'\in {\mathbb {R}}^{n-1}\) variables, respectively. For simplicity, we assume that \(q\in \mathcal {S}_0({\mathbb {R}}^{n-1})\) and show the estimates (7.11) and (7.12). The results follows by the density \(\mathcal {S}_{0}({\mathbb {R}}^{n-1})\subset {\dot{B}}^s_{p,1}({\mathbb {R}}^{n-1})\), where \(\mathcal {S}_{0}({\mathbb {R}}^{n-1})\) denotes the rapidly decreasing functions with vanishing at the origin of their Fourier images. Then the resulting estimates follows from the following bounds.
Indeed, admitting the above estimate (7.13), the Minkowski inequality yields
which implies (7.11). The estimate (7.12) also follows from (7.14) in the similar way as
To see (7.13), from (4.9), it follows
where symbols \(B(\tau ,\xi ')\) and \(D(\tau ,\xi ')\) are given in (4.6) and (4.7) and the support of the symbol on the right hand side is in an annulus domain and hence there is no singular point in both \(\tau \), \(|\xi '|\)-variables and it gives a smooth symbol.
For the symbol of the gradient of the pressure, we recall the symbol \(B(\tau ,\xi ')=\sqrt{i\tau +|\xi '|^2}\) defined by (4.6).\(\square \)
Lemma 7.3
Let \(\sigma \in {\mathbb {R}}\), \(\zeta '\in {\mathbb {R}}^{n-1}\) and k, j, \(\ell \in \mathbb {Z}_+\).
(1) For the time-dominated region \(k-2j\ge 0\),
(2) For the space-dominated region \(k-2j<0\), there exist constants \(1<C\) independent of j and k such that
In particular, there exists a constant \(c>0\) such that
(3) Let \(D(\tau ,\xi ')\) be given by (4.7) and let \(k,j\in \mathbb {Z}\) and \(2^{-1}<\sigma ,|\zeta '|<2\). Then it holds that
Proof of Lemma 7.3
(1) In the case when \(k- 2j\ge \ell \ge 2\), by using \(2^{-1}<|\sigma |,\,|\zeta '|<2\), it holds that
and (7.16) follows immediately.
(2) In the case when \( 2j-k>\ell \ge 1\), it holds that
The constants c and C can be taken as \(c=1/2\) and \(C=\sqrt{5}\).
(3) In particular, the argument of \(B(\tau ,\xi ')\) is less than \(\frac{\pi }{4}\), (7.19) follows immediately.
\(\square \)
In (4.10) and (4.11), we see that the common factor of the both symbols contains
and the only zero-point of the denominator is \(\tau =\xi '=0\) and properly away from 0 under the support of Littlewood–Paley cut-off functions (see [50, Lemma 4.4]). Hence in \(k-2j\ge 0\), we see that
Analogously for the space-like region \(k-2j<0\), we see from (7.17) that
Those bounds enable us to treat the operator given by (7.29) is \(L^p({\mathbb {R}}^{n-1})\) bounded in \(x'\) and \(L^1\) bound in t-variable. Thus the estimate (7.13) holds for all \(1\le p\le \infty \). This completes the proof of Proposition 7.2. \(\square \)
7.2 Estimate for the velocity
Once we obtain the estimates for the pressure \(\nabla q\) to (2.7), the required estimates for the velocity \(v_n\) of the solution to (2.7) can be obtained by establishing the bounded estimate for the singular integral part of the fundamental solution in (4.12) and then applying maximal regularity in Theorem 2.1 for the initial boundary value of the heat equations (4.1). Then the estimates for the rest of the velocity components \(v_{\ell }\) follow from the estimate for (4.14) and the pressure with (4.15) (cf. [37,38,39]). To this end, we prepare the following estimate.
Proposition 7.4
Let \(1\le p<\infty \) and \(s\in {\mathbb {R}}\). Let \(m_{\Psi }(\tau ,\xi ')\) be the symbol defined in (4.10) and let \(M_{\Psi }\) be the Fourier multiplier operator defined by
for any \(H\in {\dot{F}}^{1/2-1/(2p)}_{1,1}({\mathbb {R}}_+;{\dot{B}}^s_{p,1}({\mathbb {R}}^{n-1})) \cap L^1({\mathbb {R}}_+;{\dot{B}}^{s+1-1/p}_{p,1}({\mathbb {R}}^{n-1}))\). Then it satisfies the following estimates:
Proof of Proposition 7.4
The proof is shown in an analogous way seen in the proof of Proposition 7.2. Noting
where \(\widetilde{\psi }\) and \(\widetilde{\phi }\) are defined by (3.5), it suffices to show that
then immediately by the Hausdorff–Young inequality, we obtain
and the estimates (7.23) and (7.24) follow. To see (7.27), it follows from (4.10) that
where symbols \(B=B(\tau ,\xi ')\) and \(D=D(\tau ,\xi ')\) are given by (4.6) and (4.7) and the support of the symbol on the right hand side is in an annulus domain and hence there is no singular point in both \(\tau \), \(|\xi '|\)-variables and it gives a smooth symbol. For \(\sigma \in {\mathbb {R}}\) and \(\zeta '\in {\mathbb {R}}^n\) with \(1/2<|\sigma |,|\zeta '|<2\). For \(a>0\), \(\sigma \in {\mathbb {R}}\) and \(\zeta '\in {\mathbb {R}}^{n-1}\), the estimates (7.16) and (7.17) give the bounds when \(k\ge 2j+4\) that
for \(|\alpha |\le 2\). Analogously for the space-like region, we see from (7.17) that
for \(|\beta |\le n\). Those bounds enable us to obtain the estimates (7.25) and (7.26) by integration by parts and (7.30)–(7.31), we see that \(|x'|\ge 1\) and \(|t|\ge 1\),
Thus the estimates (7.27) hold for all \(1\le p\le \infty \). A similar argument implies the estimate and (7.28) also follows. This shows the proof of Proposition 7.4.\(\square \)
Proof of Theorem 2.5
Let the boundary data satisfy the regularity assumption (2.9). First we consider the n-th component of the unknown velocity that satisfies the initial boundary value problem (4.14). The direct application of Proposition 7.1–7.4 and Theorem 4.1 yields that the solution \(v_n(t,x)\) to the problem (4.12) (and hence (4.14)) fulfills the following estimate:
The other components of the velocity fields \(v'=(v_1(t,x), v_2(t,x), \ldots , v_{n-1}(t,x))\) satisfy the initial boundary value problem (4.15) by the pressure and the n-th component velocity as the external force and boundary condition. Similarly to the above estimate, we have from Proposition 7.1, 6.2, Theorem 4.1 and the estimate (7.33) that the solution \(v_\ell (t,x)\) to the problem (4.15) has the estimate
In fact, one can apply the analogous arugment to obtain the above estimate for the velocity \(v_{\ell }\) as the way of \(v_n\) with using the expression (4.16). Combining the estimates (7.33) and (7.34) for all \(\ell =1,2,\ldots , n-1\) as well as the pressure estimate (7.1) in Proposition 7.1, we conclude that the desired estimate (2.10) holds.
Conversely, if the solution (v, q) to the problem (2.7) exists, then it holds by letting f by v in the trace estimate (6.13) of Proposition 6.2 that
This shows regularity for the boundary data is necessary. This proves Theorem 2.5.
\(\square \)
Proof of Theorem 2.4
Applying the maximal \(L^1\)-regularity result to the initial-boundary value problem of the Stokes equations with the boundary condition, we obtain end-point maximal \(L^1\)-maximal regularity from (7.33), (7.34). Hence by combining the maximal regularity estimates for the problems (2.6) in [36] (see also [35]), (2.7), (2.8) and the estimate (2.10) in Theorem 2.5, we obtain (2.5).
Conversely, by using (7.11)–(7.12) in Proposition 7.2, (6.13)–(6.14) in Proposition 6.2, we conclude that regularity for data is necessary for the existence of the solution (u, p) to the Stokes system (2.4).
Concerning the uniqueness, we invoke the standard argument (see [52, Theorem 4.3 and 5.7]) for the half-space. Under the assumption \(-1+1/p<s<1/p\), let (v, q) be a solution of the Stokes system (2.4) with vanishing data with regularity given in Theorem 2.4. Let \(\phi \in C^\infty _0({\mathbb {R}}\times {\mathbb {R}}^n_+)\) be supported in \((-1,T)\times {\mathbb {R}}^n_+\) with its extention \(\widetilde{\phi }\) to \({\mathbb {R}}\times {\mathbb {R}}^n\) satisfying \(\widehat{\widetilde{\phi }}(t,0)=0\) and let \((v_*,q_*)\) be a solution of the adjoint Stokes system except the pressure sign with external force \(\phi \) with the regularity class
The regularity (7.35) is ensured by our existence proof. Here we note that \({\dot{B}}^{-s}_{p',1}({\mathbb {R}}^n_+)\subset {\dot{B}}^{-s}_{p',\infty }({\mathbb {R}}^n_+)\simeq ({\dot{B}}^s_{p,1}({\mathbb {R}}^n_+))^*\), where \( -1+1/p'< -s < 1/p'\) with the subset of the dual space
Let \(\chi (x)\) be a smooth non-negative cut-off function with \(\text {supp }\chi (x)\subset \{x=(x',x_n)\in {\mathbb {R}}^n_+, x'\in {\mathbb {R}}^{n-1}, 1<x_n<2\} \) and set \(\chi _R(x)\equiv R^{-1}\chi (R^{-1}x)\) for \(R>0\). Let \(I=(-1,T)\) and \(-1+1/p<s<1/p\) (i.e., \(-1+1/p'<-s<1/p'\)). Using the mean value theorem we see that
and similarly
We then claim that
and the left hand side of (7.38) vanishes as \(R\rightarrow \infty \), since \(-s<\frac{1}{p'}\), \(v_*\) can be approximated by \(C^{\infty }_{0,0}({\mathbb {R}}^n_+)=\big \{f\in C_0^{\infty }({\mathbb {R}}^n_+); \tilde{f}(x)=f(x) (x_n>0)\), properly extended into \(x_n\le 0\), \(\widehat{\tilde{f}(}0)=0\big \}\) functions \(\{v_{*k}\}_k\) in the norm \({\dot{B}}^{-s}_{p',1}({\mathbb {R}}^{n}_+)\), pointwisely over I. Maximal \(L^1\)-regularity for the solution \(v_*\in {\dot{W}}^{1,1}(I;{\dot{B}}^{-s}_{p',1}({\mathbb {R}}^n_+))\) gives translation invariant in t-variable and it provides that \(v_*\) is uniformly continuous, the approximation and the convergence can be uniform on I. Hence the right hand side of (7.36)–(7.37) converges to 0 as \(R\rightarrow \infty \) which justify the integration by parts (see [39] for the case \(-1+1/p<s\le 0\)). Analogously, one can find that
for all range of \(-1+1/p<s<1/p\) and the dual coupling of the boundary trace is also justified. The above relations ensure the following argument remains valid: Using (7.35) –(7.40),
from which and the Hahn–Banach extension theorem, we conclude \(v=0\) and hence \(q=0\) by \(\nabla q=0\) in \({\mathbb {R}}^n_+\) and \(q(\cdot , 0)=0\) by (2.7).
This completes the proof of Theorem 2.4.\(\square \)
8 The linear and nonlinear perturbation estimates
8.1 Estimate for the extension function of initial surface
First we give an auxiliary estimate for the extension function given by the initial surface \(\eta _0\).
First we show the estimate for the extension function E defined in (1.15).
Lemma 8.1
Let \(1\le q<\infty \) and \(\eta _0\in {\dot{B}}^{1+(n-1)/q}_{q,1}({\mathbb {R}}^{n-1})\). Then there exists a constant \(C>0\) such that
The above estimate is one of maximal regularity estimates for the half Laplacian heat semi-group in view of (1.15).
Proof of Lemma 8.1
Let us extend \(\eta _0(x')\) into the whole space \({\mathbb {R}}^n\) by regarding \(x_n\le 0\) as
where \(\epsilon _0>0\) is chosen properly small, which is one of a proper extension in \(x_n\in {\mathbb {R}}\). Then the above estimate can be proven by the restriction of the estimate for \(\tilde{E}\). To see the estimate (8.1), we employ maximal trace regularity. Since \(\nabla ' \eta _0\in {\dot{B}}^{\frac{n-1}{q}}_{q,1}({\mathbb {R}}^{n-1})\), \(\nabla '\eta _0=\sum _{m\in \mathbb {Z}}\phi _m*\nabla '\eta _0\) holds in \(\mathcal {S}'\), where \(\phi _m\) denotes the Littlewood–Paley dyadic decomposition in \({\mathbb {R}}^{n-1}\) and it follows by the relation between the supports of the Fourier images of \(\overline{\Phi _j}\) and \(\phi _m\) that
Then for the \(\ell \)-th component of the first term of the right hand side of (8.2) can be seen for all \(\ell =1,2\ldots , n-1\) that
where we set \(\widetilde{\phi _j}=\phi _{j-1}+\phi _j+\phi _{j+1}\). The estimate for the second term II is along the similar way.
Finally, we confirm that
which is justified by the argument found in [35, Proposition 2.1]. Indeed, noticing \(\mathcal {F}[\textrm{sech}ax](\xi )=a^{-1}\textrm{sech}a^{-1}\xi \) for \(a>0\) and \(\textrm{sech}a^{-1}\xi \) is bounded and converging to 0 around \(a\simeq 0\), by making a coupling with \(\varphi \in \mathcal {S}\) that
\(\square \)
8.2 Estimates for the linear perturbation
We now consider the estimates for the linear variable coefficient terms defined in (1.24)–(1.27). All the estimate is based on the bilinear estimate in the homogeneous Besov space Proposition 10.3. See Appendix below.
To show the estimates for the linear variable coefficient terms, we prepare the following basic lemma.
Lemma 8.2
For \(1\le q < \infty \), let \(E(x,x_n)\) is given by (1.15) and assume that for some small \(\varepsilon _0>0\)
Then there exists a constant \(C>0\) such that
where \(\nabla '=(\partial _1,\partial _2,\ldots , \partial _{n-1})^\textsf{T}\).
Proof of Lemma 8.2
To see (8.3), we use the Taylor expansion of
and noticing that \({\dot{B}}^{\frac{n}{q}}_{q,1}({\mathbb {R}}^n_+)\) is the Banach algebra, it follows from Lemma 8.1 that
The second and third estimates follow in a similar way.\(\square \)
Proposition 8.3
(Estimates for linear variable coefficient terms) Let \(n\ge 2\) and \(1\le p<2n\). For \(u\in C(\overline{{\mathbb {R}}_+}; {\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+))\), \(\partial _t u\), \(D^2 u\), \(\nabla p\in L^1({\mathbb {R}}_+;{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+))\) and E defined in (1.15), let f(u, p, E) \(\equiv f(u,E)+f(p,E)\), g(u, E) and \(h(u,p,E)\equiv h(u,E)+h(p,E)\) be the terms defined in (1.23), (1.24), (1.25), (1.26) and (1.27) respectively. Under the assumption \(\Vert \nabla ' \eta _0\Vert _{{\dot{B}}^{(n-1)/{q}}_{q,1}({\mathbb {R}}^{n-1})}\) is small enough, the following estimates hold: For \(1\le q < pn/|p-n|\),
For \(1\le q\le p(n-1)/(n-p)\) \((1\le p< n)\) and \(1\le q< p(n-1)/ (p-n)\) \((n\le p<\infty )\),
Proof of Proposition 8.3
Recalling the definition of f(u, E) and f(p, E), and the covariant derivative (1.18), we show (8.4) by
where we apply Lemma 8.1 and notice that \({\dot{B}}^{n/p}_{p,1}({\mathbb {R}}^n_+)\) is the Banach algebra and no restriction on the exponents p nor q. The estimates (8.5) and (8.6) follow in a similar way.
To see the boundary terms, we recall the term into the velocity part and the pressure part such as (1.26) and (1.27).
For those terms, we prepare the boundary bilinear estimate of space-time type. We introduce auxiliary norms of the Chemin–Lerner type (cf. [14]).
Definition. For \(1\le p, \rho \le \infty \) and \(r,s\in {\mathbb {R}}\), the Besov space and the Bochner space of Chemin–Lerner type \(\widetilde{{\dot{B}}^r_{\rho ,1}\big ({\mathbb {R}}_+};{\dot{B}}^{s}_{p,1}({\mathbb {R}}^{n-1})\big )\) and \(\widetilde{L^{\rho }\big ({\mathbb {R}}_+};{\dot{B}}^{s}_{p,1}({\mathbb {R}}^{n-1})\big )\) are defined by the following norms:
\(\square \)
Lemma 8.4
(Multiple estimates for boundary terms) Let \(n\ge 2\), \(1\le p<2n-1\), \(1\le q\le p(n-1)/(n-p)\) \((1\le p< n)\) and \(1\le q< p(n-1)/ (p-n)\) \((n\le p<2n-1)\) and assume that functions F and G over \({\mathbb {R}}_+\times {\mathbb {R}}^{n-1}\) satisfy \( F\in {\dot{F}}^{1/2-1/2p}_{1,1}({\mathbb {R}}_+;{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^{n-1})) \cap L^1({\mathbb {R}}_+;{\dot{B}}^{(n-1)/p}_{p,1}({\mathbb {R}}^{n-1}))\) and \( G\in \widetilde{{\dot{B}}^{1/2-1/2p}_{\infty ,1}({\mathbb {R}}_+};{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^{n-1})) \cap \widetilde{L^{\infty }({\mathbb {R}}_+};{\dot{B}}^{(n-1)/q}_{q,1}({\mathbb {R}}^{n-1}))\). Then the following estimate holds:
The proof of Lemma 8.4 directly follows from Proposition 10.5 shown in Appendix below (cf. [39]).
Since \(\nabla E\) is independent of t (using the fact that the average of \(\psi _k\) vanishes) we notice that
and hence applying the bilinear estimate (8.10), we obtain the following estimates: Since \(\partial _n E \big |_{x_n=0}=0\) at the boundary \(\partial {\mathbb {R}}^n_+\),
In very much similar way, we find that
The estimates (8.11) and (8.12) yield the resulting estimate (8.7).
The other estimate (8.8) can be shown in much straightforward way: Because \({\dot{B}}^{(n-1)/{q}}_{q,1}({\mathbb {R}}^{n-1})\) is the Banach algebra, it follows directly that
The case for h(p, E) also follows in a similar way. This shows the proof of Proposition 8.3. \(\square \)
8.3 The nonlinear estimates
The perturbation terms for the Navier–Stokes equations in the Lagrangian coordinate, it holds that the following multilinear estimates.
Proposition 8.5
(Nonlinear estimates for \(F_{p}(u,p,E)\) and \(G_{\mathrm{div\,}}(u,E)\)) Let \(n\ge 2\), \(1\le p<2n\) and \(1\le q< np/|p-n|\). For \(u\in C(\overline{{\mathbb {R}}_+}; {\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+))\), \(\partial _t u\), \(D^2 u\), \(\nabla p\in L^1({\mathbb {R}}_+;{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+))\) and E defined in (1.15), let \(F_{p}(u,p,E)\) and \(G_{\mathrm{div\,}}(u,E)\) be the nonlinear terms defined in (1.10) and (1.11), respectively. Then the following estimates hold provided \(\Vert \nabla ' \eta _0\Vert _{{\dot{B}}^{(n-1)/q}_{q,1}({\mathbb {R}}^{n-1})}\) is small enough,
and
Proof of Proposition 8.5
To show the estimate (8.13), we see the form (1.29) that for any \(1\le p<\infty \),
Here we estimate \(D_Eu\) term by its definition and it follows that
Thus we conclude from (8.16) and (8.17) that
provided \(\Vert \nabla ' \eta _0\Vert _{{\dot{B}}^{\frac{n-1}{q}}_{q,1} ({\mathbb {R}}^{n-1})}\) is small enough.
Secondly, we proceed in a similar manner for (8.14) by observing (1.30), we have for all \(1\le p<\infty \) that
where \(\sigma _k(\Vert \nabla E\Vert _{{\dot{B}}^{n/q}_{q,1}({\mathbb {R}}^{n}_+)})\) denotes a term involving \(\Vert \nabla E\Vert _{{\dot{B}}^{n/q}_{q,1}({\mathbb {R}}^{n}_+)}\) of order at most 1. Using Lemmas 8.1, 8.2 and (8.17), we conclude from (8.18),
The proof of (8.15) can be done by a quite analogous way.\(\square \)
Proposition 8.6
(Nonlinear estimate for \(F_u(u,E)\)) Let \(n\ge 2\), \(1\le p<\infty \). For \(D^2 u\in L^1({\mathbb {R}}_+;{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+))\), let \(F_{u}(u,E)\) be defined by (1.9). Then the following estimate holds:
The proof of Proposition 8.6 is very similar to the proof of Proposition 8.5 (cf. [39, Proposition 5.6]). Since
those terms are divergence form and the estimates are reduced into the multilinear estimate over the Banach algebra \({\dot{B}}^{n/p}_{p,1}({\mathbb {R}}^n_+)\).
We finally treat the boundary nonlinearities as follows.
Proposition 8.7
(Multiple estimates for boundary nonlinearity) Let \(n\ge 2\), \(1\le p<2n-1\), \(1\le q\le p(n-1)/(n-p)\) \((1\le p< n)\) and \(1\le q< p(n-1)/ (p-n)\) \((n\le p<\infty )\), and assume that functions u and p satisfy \(u\in C(\overline{{\mathbb {R}}_+}; {\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+))\), \(\partial _t u\), \(D^2 u\) \(\nabla p \in L^1({\mathbb {R}}_+;{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+))\) and \(\nabla ' \eta _0\in {\dot{B}}^{(n-1)/q}_{q,1}({\mathbb {R}}^{n-1})\), \(p|_{x_n=0} \in {\dot{F}}^{1/2-1/2p}_{1,1}({\mathbb {R}}_+;{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^{n-1})) \cap L^1({\mathbb {R}}_+;{\dot{B}}^{(n-1)/p}_{p,1}({\mathbb {R}}^{n-1}))\). Let \(H_{u}(u,E)\) and \(H_p(u,p,E)\) be the boundary terms defined by (1.31) and (1.32), respectively. Then the following estimates hold provided \(\Vert \nabla ' \eta _0\Vert _{{\dot{B}}^{(n-1)/q}_{q,1}({\mathbb {R}}^{n-1})}\) is small enough:
In order to prove of Proposition 8.7, we prepare some lemmas. First we introduce auxiliary norms of the Chemin–Lerner type (cf. [14]) for the proof Proposition 8.7.
Lemma 8.8
For any \(1\le p<\infty \),
Proof of Lemma 8.8
The estimates are shown in [39, Lemma 5.9]. We give an outlined proof here. The first estimate (8.24) follows by using \(\widetilde{\psi }_k(t)=\psi _{k-1}(t)+\psi _k(t)+\psi _{k+1}(t)\) and noticing \(\Vert \partial _t^{-1}\psi _k\Vert _{L^{\infty }({\mathbb {R}}_+)} \le \Vert \psi _k\Vert _{L^1({\mathbb {R}}_+)}\), where \(\partial _t^{-1}\psi _k\) is defined as
that
The second inequality (8.25) follows from the following estimate:
\(\square \)
Proof of Proposition 8.7
From (1.12) and (1.13) and from the regularity assumptions; we notice that the sharp trace estimate implies
We first prove the estimate (8.22) holds. Setting
in Lemma 8.4 with regarding (8.26), we find that
The polynomial terms can be estimated as the following way: First the space \(\widetilde{L^{\infty }({\mathbb {R}}_+;}{\dot{B}}^{(n-1)/p}_{p,1}({\mathbb {R}}^{n-1}))\) is the Banach algebra (see (10.9) in Lemma 10.6 below) and by the estimate (8.25), we have for \(Du|_{x_n=0}\in L^1({\mathbb {R}}_+;{\dot{B}}^{(n-1)/p}_{p,1}({\mathbb {R}}^{n-1}))\) that
Secondly by Proposition 10.5 in Appendix, Lemma 8.8 and the boundary bilinear estimate (10.3) in Proposition 10.3, we see that
by the sharp trace estimate Proposition 6.2. Combining the estimates (8.27)–(8.29), we obtain (8.22). The estimate (8.23) can also be done in a very similar way. This completes the proof of Proposition 8.7.\(\square \)
9 The global well-posedness
In this section, we show an outlined proof of Theorem 2.1 (cf. [39]).
Proof of Theorem 2.1
We define the complete metric space
where
The constant \(M>0\) is chosen to be small enough depending on the norm of the initial data. Given \((\tilde{u},\tilde{p})\in X\), we consider the liner inhomogeneous initial boundary value problem:
where \(u_0(x)=\bar{u}_0(x', x_n-E(x',x_n))\), the linear variable coefficient terms are given by (1.24)–(1.27) and the nonlinear terms are (1.28)–(1.32).
We define the map \( \Phi : X\rightarrow X \) by \( (\tilde{u},\tilde{p}) \rightarrow (u,p)\equiv \Phi \big [\tilde{u}, \tilde{p}\big ] \) and prove that \(\Phi \) is contraction on X.
First we show that a priori estimate of \(\Phi [{u}, {p}]\) in \(L^1({\mathbb {R}}_+;{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n))\). Let (u, p) solve (9.1). Applying Theorem 2.4 to the Eq. (9.1), we have by (2.5), Propositions 8.5–8.7 to the nonlinear terms that
Therefore if we choose the initial data small enough
then we obtain from (9.2) that
Moreover, for all \((u_1,p_1)\), \((u_2,p_2)\in X\), we know that the difference
satisfy the same estimate (9.2) without \(\Vert u_0\Vert _{{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n_+)}\), i.e.,
Therefore if we choose
then it holds that
which shows the map
is contraction. By the fixed point theorem of Banach–Caccioppoli, there exists a unique fixed point (u, p) of the map \(\Phi \) in X.
Then the unique fixed point (u, p) satisfies (9.1) with the all right members changed into (u, p) and it is a time global strong solution of (1.22). This completes the proof of Theorem 2.1.\(\square \)
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Notes
Practically the natural setting is \(\Omega (0)={\mathbb {R}}^n_-\) under the gravity circumstance.
Up to this level there is no restriction on p nor s.
References
Abels, H.: The initial-value problem for the Navier–Stokes equations with a free surface in \(L^q\)-Sobolev spaces. Adv. Differ. Equ. 10, 45–64 (2005)
Abidi, H., Paicu, M.: Existence globale pour un fluide inhomogène. Ann. Inst. Fourier (Grenoble) 57, 883–917 (2007)
Amann, H.: On the strong solvability of the Navier–Stokes equations. J. Math. Fluid Mech. 2, 16–98 (2000)
Bahouri, H., Chemin, J-Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. Grundlehren der mathematische Wisssenshaften vol. 343. Springer, Berlin (2011)
Beale, J.T.: The initial value problem for the Navier–Stokes equations with a free surface. Commun. Pure Appl. Math. 34, 359–392 (1981)
Beale, J.T.: Large-time regularity of viscous surface waves. Arch. Ration. Mech. Anal. 84, 307–352 (1984)
Beale, J. T., Nishida, T.: Large-time behavior of viscous surface waves, Recent topics in nonlinear PDE, II (Sendai, 1984), 1–14, North-Holland Mathematical Studies, vol. 128. Lecture Notes Numerical Applied Analysis, 8. North-Holland, Amsterdam (1985)
Beale, J.T., Nishida, T., Teramoto, Y.: Decay of solutions of the Stokes system arising in free surface flow on an infinite layer. RIMS Kokyuroku Bessatsu B 82, 137–157 (2020)
Bergh, J., Löfström, J.: Interpolation Spaces; An Introduction. Springer, Berlin (1976)
Bourgain, J., Pavlović, N.: Ill-posedness of the Navier–Stokes equations in a critical space in 3D. J. Funct. Anal. 255, 2233–2247 (2008)
Cannone, M.: Ondelettes. Paraproduits et Navier–Stokes. Diderot Editeur, Arts et Sciences, Paris (1995)
Cannone, M., Planchon, F.: Self-similar solutions for Navier–Stokes equations in \(\mathbb{R} ^{3}\). Commun. P.D.E. 21, 179–193 (1996)
Chemin, J.-Y.: Théorèmes d’unicité pour le systéme de Navier–Stokes tridimensionnel. J. Anal. Math. 77, 27–50 (1999)
Chemin, J.-Y., Lerner, N.: Flot de champ de vecteurs non lipschitziens et équations de Navier–Stokes. J. Differ. Equ. 121, 314–328 (1995)
Danchin, R., Hieber, M., Mucha, P., Tolksdorf, P.: Free boundary problems via Da Prato–Grisvard theory. Preprint arXiv:2011.07918v2
Danchin, R., Mucha, P.B.: A critical functional framework for the inhomogeneous Navier–Stokes equations in the half-space. J. Funct. Anal. 256, 881–927 (2009)
Danchin, R., Mucha, P.B.: A Lagrangian approach for the incompressible Navier–Stokes equations with variable density. Commun. Pure Appl. Math. 65, 1458–1480 (2012)
Danchin, R., Mucha, P.B.: Critical functional framework and maximal regularity in action on system of incompressible flows. Memoirs of the Society of Science, France, vol. 143. Société mathématique de France (2015)
Da Prato, G., Grisvard, P.: Sommes d’opérateurs linéaires et équations différentielles opérationelles. J. Math. Pure Appl. 54, 305–387 (1975)
Denk, R., Hieber, M., Prüss, J.: \({\cal{R}}\)-boundedness. In: Fourier Multipliers and Problems of Elliptic and Parabolic Type. Memoirs of AMS, 166, No. 788 (2003)
Denk, R., Hieber, M., Prüss, J.: Optimal \(L_p\)-\(L_q\)-regularity for parabolic problems with inhomogeneous boundary data. Math. Z. 257, 193–224 (2007)
Evans, C.L.: Partial Differential Equations. American Mathematical Society, Providence (2000)
Fujita, H., Kato, T.: On Navier–Stokes initial value problem 1. Arch. Ration. Mech. Anal. 46, 269–315 (1964)
Giga, Y., Saal, J.: \(L^1\) maximal regularity for the Laplacian and applications. Discrete Contin. Dyn. Syst. I, 495–504 (2011)
Giga, Y., Sohr, H.: Abstract \(L^p\) estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)
Gui, G.: Lagrangian approach to global well-posedness of the viscous surface wave equations without surface tension. Peking Math. J. 4, 1–82 (2021)
Guo, Y., Tice, I.: Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6, 287–369 (2013)
Johnsen, J., Sickel, W.: On the trace problem for Lizorkin–Triebel spaces with mixed norms. Math. Nachr. 281, 669–696 (2008)
Kato, T.: Strong \(L^p\) - solution of the Navier–Stokes equation in \({\mathbb{R} }^m\) with applications to weak solutions. Math. Z. 187, 471–480 (1984)
Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data. Commun. Partial Differ. Equ. 19, 959–1014 (1994)
Lizorkin, P.I.: Properties of functions of class \(\Lambda ^r_{p,\theta }\). Trudy Mat. Inst. Steklov 131, 158–181 (1974)
Meyries, M., Veraar, M.C.: Traces and embeddings of anisotropic function spaces. Math. Ann. 360, 571–606 (2014)
Mucha, P.B., Zajaczkowski, W.: On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion. Appl. Math. (Warsaw) 27, 319–333 (2000)
Nishida, T.: Equations of fluid dynamics-Free surface problems. Commun. Pure Appl. Math. 39, 221–231 (1986)
Ogawa, T., Shimizu, S.: End-point maximal \(L^1\)-regularity for a Cauchy problem to parabolic equations with variable coefficient. Math. Ann. 365, 661–705 (2016)
Ogawa, T., Shimizu, S.: Global well-posedness for the incompressible Navier–Stokes equations in the critical Besov space under the Lagrangean coordinate. J. Differ. Equ. 274, 613–651 (2021)
Ogawa, T., Shimizu, S.: Maximal \(L^1\)-regularity of the heat equation and application to a free boundary problem of the Navier–Stokes equations near the half space. J. Elliptic Parabol. Equ. 7(2), 571–587 (2021)
Ogawa, T., Shimizu, S.: Maximal \(L^1\)-regularity for parabolic initial-boundary value problems with inhomogeneous data. J. Evol. Equ. 22(30), 67 (2022)
Ogawa,T., Shimizu, S.: Maximal \(L^1\)-regularity and free boundary problem for the incompressible Navier–Stokes equations in critical spaces. J. Math. Soc. Jpn. https://doi.org/10.2969/jmsj/88288828 (to appear)
Ohyama, T.: Interior regularity of weak solutions of the time-dependent Navier–Stokes equations. Proc. Jpn. Acad. 36, 273–277 (1960)
Padula, M., Solonnikov, V.A.: On the global existence of nonsteady motions of a fluid drop and their exponential decay to a uniform rigid rotation. Quad. Mat. 10, 185–218 (2002)
Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pure. Appl. 48, 173–182 (1959)
Prüss, J., Simonett, G.: On the two-phase Navier–Stokes equations with surface tension. Interface Free Bound. 12, 311–345 (2010)
Prüss, J., Simonett, G.: Moving Interfaces and Quasi-linear Parabolic Differential Equations. Monographs in Mathematics 105, Birkhäuser, Basel (2016)
Saito, H.: Global solvability of the Navier–Stokes equations with a free surface in the maximal \(L_p\)-\(L_q\) class. J. Differ. Equ. 264, 1475–1520 (2018)
Schweizer, B.: Free boundary fluid systems in a semigroup approach and oscillatory behavior. SIAM J. Math. Anal. 28, 1135–1157 (1997)
Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)
Shibata, Y.: Local well-posedness of free surface problem for the Navier–Stokes equations in a general domain. Discrete Contin. Dyn. Syst. Ser. S 9, 315–342 (2016)
Shibata, Y.: \({\cal{R} }\)-Boundedness, Maximal Regularity and Free Boundary Problems for the Navier–Stokes Equations. Lecture Notes in Mathematics 2254, pp. 193–462. Springer, Berlin (2020)
Shibata, Y., Shimizu, S.: On a resolvent estimate for the Stokes system with Neumann boundary condition. Differ. Integr. Equ. 16, 385–426 (2003)
Shibata, Y., Shimizu, S.: On the free boundary problem for the Navier–Stokes equations. Differ. Integr. Equ. 20(3), 241–276 (2007)
Shibata, Y., Shimizu, S.: On the \(L_p\)-\(L_q\) maximal regularity of the Neumann problem for the Stokes equations in a bounded domain. J. Reine Angew. Math. 615, 157–209 (2008)
Shimizu, S.: Local solvability of free boundary problems for the two-phase Navier–Stokes equations with surface tension in the whole space. Prog. Nonlinear Differ. Equ. Appl. 80, 547–686 (2011)
Solonnikov, V.A.: Solvability of the problem of the motion of a viscous incompressible fluid bounded by a free surface. Izv. Akad. Nauk SSSR Ser. Math. 41, 1388–1424 (1977) (in Russian). English transl.: Math. USSR Izv.11, 1323–1358 (1977)
Solonnikov, V.A.: Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid. Zap. Nauchn. Sem. (LOMI) 140, 179–186 (1984) (in Russian); English transl.: J. Sov. Math. 32, 223–238 (1986)
Solonnikov, V.A.: Unsteady motion of a finite mass of fluid, bounded by a free surface. Zap. Nauchn. Sem. (LOMI), 152, 137–157 (1986) (in Russian); English transl.: J. Sov. Math. 40, 672–686 (1988)
Solonnikov, V.A.: On the transient motion of an isolated volume of viscous incompressible fluid. Math. USSR Izv. 31, 381–405 (1988)
Solonnikov, V.A.: On nonstationary motion of a finite isolated mass of self-gravitating fluid. Algebra i Analiz 1, 207–249 (1989) (in Russian); English transl.: Leningr. Math. J. 1, 227–276 (1990)
Solonnikov, V.A.: Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. Algebra i Analiz 3, 222–257 (1991) (in Russian); English transl.: St. Petersburg Math. J. 3, 189–220 (1992)
Solonnikov, V.A., Tani, A.: Free boundary problem for a viscous compressible flow with a surface tension. In: Rassias, kTh. M. (ed.) Constantin Carathéodory: An international Tribute, pp. 1270–1303 (1991)
Tani, A.: On the free boundary problem for compressible viscous fluid motion. J. Math. Kyoto Univ. 24, 839–859 (1981)
Tani, A.: Small-time existence for the three-dimensional Navier–Stokes equations for an incompressible fluid with a free surface. Arch. Ration. Mech. Anal. 133, 299–331 (1996)
Tani, A., Tanaka, N.: Large time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Ration. Math. Mech. 130, 303–314 (1995)
Triebel, H.: Spaces of distributions of Besov type in Euclidean \(n\)-space, duality, interpolation. Ark. Mat. 11, 13–64 (1973)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)
Wang, B.: Ill-posedness for the Navier–Stokes equations in critical Besov spaces \({\dot{B}}^{-1}_{\infty , q}\). Adv. Math. 268, 350–372 (2015)
Weidemaier, P.: On the trace theory for functions in Sobolev spaces with mixed \(L_p\)-norm. Czech. Math. J. 44, 7–20 (1994)
Weidemaier, P.: Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed \(L_p\)-norm. Electron. Res. Announc. Am. Math. Soc. 8, 47–51 (2002)
Weidemaier, P.: Vector-valued Lizorkin–Triebel spaces and sharp trace theory for functions in Sobolev spaces with mixed \(L_p\)-norm for parabolic problem. Sb. Math. 196, 777–790 (2005)
Yoneda, T.: Ill-posedness of the 3D Navier–Stokes equations in a generalized Besov space near \(BMO^{-1}\). J. Funct. Anal. 258, 3376–3387 (2010)
Acknowledgements
The authors would like to thank the referees for careful reading and valuable comments. The first author is partially supported by JSPS Grant-in-Aid for Scientific Research (S) JP19H05597, Scientific Research (B) JP18H01131 and Challenging Research (Pioneering) JP20K20284. The second author is partially supported by JSPS Grant-in-Aid for Scientific Research (B) JP21H00992 and Fostering Joint International Research (B) JP18KK0072.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
10 Appendix
10 Appendix
1.1 10.1 Null-Lagrangian structure
According to Evans [22, section 8.1], we recall the null Lagrangian structure for the Jacobi of a Lipschitz continuous function u.
For \(n\in {\mathbb {N}}\), let A be a \(n\times n\) matrix whose components are denoted by \(\{a_{kj}\}\) and consider its \(\ell \times \ell \) sub-matrix \(A^{[\ell ]}\) given by
where \(\sigma _k,\tau _j\in \{1,2,\ldots , n\}\) with \(1\le \sigma _1<\sigma _2<\cdots <\sigma _{\ell }\le n\) and \(1\le \tau _1<\tau _2<\cdots <\tau _{\ell }\le n\).
Lemma 10.1
[22] Let \(1\le \ell \le n\) and let \(u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) be a Lipschitz continuous function and \(J(Du)^{[\ell ]}\) denotes the \(\ell \times \ell \) sub-matrix of the Jacobi matrix J(Du) defined by (10.1), \(\textrm{cof }\,(J(Du)^{[\ell ]})_{kj}\) denotes the (k, j)- cofactor and \(\textrm{cof }\,(J(Du)^{[\ell ]})\) be the cofactor matrix. Then for any \(x\in {\mathbb {R}}^n\) with \(\det (J(Du)(x)\ne 0\) it holds that
Naturally the divergence-curl structure leads to the following corollary.
Corollary 10.2
Let \(1\le p<\infty \) and \(1\le q<\infty \), For \(D\tilde{u}\), \(\tilde{p}\in L^1({\mathbb {R}}_+;{\dot{B}}^{\frac{n}{p}}_{p,1}(\Omega ))\), let \(F_u (\tilde{u})\), \( F_p(\tilde{u},\tilde{p})\), \(G_{\mathrm{div\,}}(\tilde{u})\), \(H_{u}(\tilde{u})\) and \(J(D\tilde{u})\) be given by (1.10)–(1.13). Then the following identities hold:
Proof of Corollary 10.2
Since \(\mathrm{div\,}\bar{u}(t,y)=0\), the Liouville theorem implies \(\det J(D\tilde{u})=1\) and hence \(\big (J(D\tilde{u})^{-1}\big )^\textsf{T} =\big ((\det J(D\tilde{u}))^{-1}\textrm{cof }\,J(D\tilde{u})^\textsf{T}\big )^\textsf{T} =\textrm{cof }\,J(D\tilde{u})\). From Lemma 10.1,
The other terms follow in a similar observation.\(\square \)
1.2 10.2 Bilinear estimates
The following bilinear estimates over the whole space \({\mathbb {R}}^n\) are obtained by Abidi–Paicu [2] (cf. [39]).
Proposition 10.3
[2] Let \(1\le p,\, p_1,\, p_2,\, \sigma ,\, \lambda _1,\, \lambda _2 \le \infty \), \(1/p\le 1/p_1+1/p_2\), \(p_1\le \lambda _{2}\), \(p_2\le \lambda _{1}\) and
and \(s_1+s_2+n\inf (0,1-1/p_1-1/p_2)>0\) with \(s_1+n/\lambda _2\le n/p_1\), \(s_2+n/\lambda _1\le n/p_2\) (and hence \(r=s_1+s_2-n\big (1/p_1+1/p_2-1/p\big )>-n+n/p\)).
-
(1)
There exists \(C>0\) such that for any \(f\in {\dot{B}}^{s_1}_{p_1,1}\) and \(g\in {\dot{B}}^{s_2}_{p_2,1}\) the following estimate holds
$$\begin{aligned} \Vert fg\Vert _{{\dot{B}}^{r}_{p,1}} \le C \Vert f\Vert _{{\dot{B}}^{s_1}_{p_1,1}} \Vert g\Vert _{{\dot{B}}^{s_2}_{p_2,1}}. \end{aligned}$$ -
(2)
If \(1\le p<\infty \) and \(1\le q\le pn/(n-p)\) \((1\le p< n)\) and \(1\le q< pn/ (p-n)\) \((n\le p<\infty )\), then for any \(f\in {\dot{B}}^{\frac{n}{q}}_{q,1}({\mathbb {R}}^n_+)\) and \(g\in {\dot{B}}^{-1+\frac{n}{p}}_{p,1}({\mathbb {R}}^n_+)\),
$$\begin{aligned} \Vert f\; g\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}({\mathbb {R}}^n_+)} \le C\Vert f\Vert _{{\dot{B}}^{ \frac{n}{q}}_{q,1}({\mathbb {R}}^n_+)} \Vert g\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}({\mathbb {R}}^n_+)}. \end{aligned}$$(10.4)In particular, for any \(f\in {\dot{B}}^{\frac{n}{p}}_{p,1}({\mathbb {R}}^n_+)\) and \(g\in {\dot{B}}^{-1+\frac{n}{p}}_{p,1}({\mathbb {R}}^n_+)\) with \(1\le p<2n\), the following estimate holds
$$\begin{aligned} \Vert f\;g\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}({\mathbb {R}}^n_+)} \le C \Vert f\Vert _{{\dot{B}}^{\frac{n}{p}}_{p, 1}({\mathbb {R}}^n_+)} \Vert g\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}({\mathbb {R}}^n_+)}. \end{aligned}$$(10.5) -
(3)
For the boundary case, \(n\ge 2\), \(1\le p<\infty \) and \(1\le q\le p(n-1)/(n-p)\) \((1\le p< n)\) and \(1\le q< p(n-1)/ (p-n)\) \((n\le p<\infty )\). Then it holds that
$$\begin{aligned}&\Vert f\; g \Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}({\mathbb {R}}^{n-1})} \le C \Vert f\Vert _{{\dot{B}}^{\frac{n-1}{q}}_{q,1}({\mathbb {R}}^{n-1})} \Vert g\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}({\mathbb {R}}^{n-1})}.\end{aligned}$$(10.6)
See the possible range of exponents (p, q) to the estimate (10.4) in Fig. 4 below. Hence regarding the region of possible choice of 1/q, we see that
The possible range of exponents (p, q) in the bilinear estimate (10.4) (the green area)
On the other hand, the restriction of the exponent (p, q) for the boundary estimate is given by
The possible range of exponents (p, q) at the boundary (the green area)
The red lined area in both Figs. 3 and 4 are the possible range of the exponents with boundary and inner nonlinear estimates. If \(p=q\) then the possible range for q is limited \(p=q<2n-1\) as is seen in the Fig. 4.
Since Danchin–Mucha [17] treats the equations depending on the density, the restriction on the exponent p in the solution space \({\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^n)\) stems from the restriction on \(1\le p<2n\) for the above bilinear estimate (10.5). One may improve the restriction by using the divergence-curl free structure of nonlinear terms.
The bilinear estimates as above hold for the case when the two functions \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) and \(g: {\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) has the divergence structure condition:
where \(D_x\) denotes any combination of partial derivatives by \(x=(x_1,x_2,\ldots , x_n)\) of the first order. A typical case is given by the form when f and g satisfies divergence free-rotation free structure as \(\mathrm{div\,}f=0\) and \(\textrm{rot }g=0\).
Proposition 10.4
(Bilinear estimate under divergence structure) Let \(1\le p < \infty \) and \(f\in {\dot{B}}^{-1+n/p}_{p,1}\) and \(g\in {\dot{B}}^{n/p}_{p,1}\).
-
(1)
If there exists \(F=F(x)\) such that \(f\cdot g =D_{x}(F\cdot g)\) with \(f=D_x F(x)\) in the sense of distribution, where \(D_x\) is any combination of the first differentiation in x. Then
$$\begin{aligned} \Vert f\cdot g\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}} \le C\Vert f\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}} \Vert g\Vert _{{\dot{B}}^{\frac{n}{p}}_{p,1}}. \end{aligned}$$(10.7) -
(2)
In particular, with additional conditions \(\mathrm{div\,}f=0\), \(\textrm{rot }g=0\) in the distribution sense, it holds
$$\begin{aligned} \Vert f\cdot g\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}} \le C\Vert f\Vert _{{\dot{B}}^{-1+\frac{n}{p}}_{p,1}} \Vert g\Vert _{{\dot{B}}^{\frac{n}{p}}_{p,1}}. \end{aligned}$$(10.8)
Proposition 10.5
(The space-time bilinear estimate) Let \(1\le \rho \le \infty \) and \(1\le p<2n-1\) and \(1\le q\le p(n-1)/(n-p)\) \((1\le p< n)\) and \(1\le q< p(n-1)/ (p-n)\) \((n\le p<2n-1)\). Then for \(F\in \widetilde{{\dot{B}}^{1/2-1/(2p)}_{\rho ,1}({\mathbb {R}}_+;}{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^{n-1})) \cap \widetilde{L^{\rho }({\mathbb {R}}_+;}{\dot{B}}^{(n-1)/p}_{p,1}({\mathbb {R}}^{n-1}))\) and \(G\in \widetilde{{\dot{B}}^{1/2-1/(2p)}_{\infty ,1} ({\mathbb {R}}_+;}{\dot{B}}^{-1+n/p}_{p,1}({\mathbb {R}}^{n-1})) \cap \widetilde{L^{\infty }({\mathbb {R}}_+;} {\dot{B}}^{(n-1)/q}_{q,1}({\mathbb {R}}^{n-1}))\), it holds that
where we recall the definition of the function class defined in (8.9).
The proof of Proposition 10.5 can be shown in a very similar way to the proof of the related boundary bilinear estimate shown in [39, Proposition 7.6] with an aid of the estimate (10.3) above.
Lemma 10.6
\(\widetilde{L^{\infty }({\mathbb {R}}_+;}{\dot{B}}^{(n-1)/p}_{p,1}({\mathbb {R}}^{n-1}))\) is the Banach algebra, namely for any \(f, g\in \widetilde{L^{\infty }({\mathbb {R}}_+;}{\dot{B}}^{(n-1)/p}_{p,1}({\mathbb {R}}^{n-1}))\) it holds
See for the proof of above Proposition 10.5 and Lemma 10.6,[39].
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ogawa, T., Shimizu, S. Free boundary problems of the incompressible Navier–Stokes equations with non-flat initial surface in the critical Besov space. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02823-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00208-024-02823-x