Abstract
We consider a linearized fluid-structure interaction problem, namely the flow of an incompressible viscous fluid in the half space \({\mathbb {R}}^n_+\) such that on the lower boundary a function h satisfying an undamped Kirchhoff-type plate equation is coupled to the flow field. Originally, h describes the height of an underlying nonlinear free surface problem. Since the plate equation contains no damping term, this article uses \(L^2\)-theory yielding the existence of strong solutions on finite time intervals in the framework of homogeneous Bessel potential spaces. The proof is based on \(L^2\)-Fourier analysis and also deals with inhomogeneous boundary data of the velocity field on the “free boundary” \(x_n=0\).
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1 Fluid-structure interaction
1.1 Introduction
We consider the following linear, coupled system
where u denotes the velocity of an incompressible, viscous fluid in \({\mathbb {R}}^n_+\) and p its pressure coupled with a boundary function h satisfying an undamped Kirchhoff-type plate equation. This problem is a linearised version of the following more abstract system of equations
in a time-depending domain \(\Omega _t = \{x=(x_1,...,x_n)\in {\mathbb {R}}^n : x_n > h(x',t)\}\), \(t \in (0,T)\), with lower free boundary, \(\Gamma _t = \{x=(x',x_n): x_n=h(x',t)\). Here, \(D(u)=\frac{1}{2}(\nabla u+(\nabla u)^\top )\) is the deformation tensor, and
denotes the force exerted by the incompressible fluid in \(\Omega _t\) onto the free boundary \(\Gamma _t\) with exterior normal \(n_t\) pointing downward. Moreover, at \(t=0\),
Additionally, a body force \(g_2\) exerts a force from the outside onto the boundary. Hence, the boundary is moving and its motion is tracked by the scalar function h describing the height of the boundary. On the boundary the fluid velocity, u, coincides with the velocity of the boundary, i.e., we assume the no-slip condition for its tangential part whereas its normal component coincides with \(\partial _t h\) (kinematic boundary condition); furthermore, for mathematical generality, we include an additional force \(g_1\) satisfying the compatibility condition \(g_1|_{t=0}=0\), which also models the phenomenon of leaking. In the domain \(\Omega _t\), the motion of the fluid is described by the Navier-Stokes equations with constant viscosity \(\mu >0\).
The linearised version with fixed boundary is obtained after transforming the equations to a problem in \({\mathbb {R}}^n_+\) and ignoring all nonlinear terms appearing in this way as well as the convective term \((u \cdot \nabla )u\). The first part of the term F vanishes due to \({\text {div}}u=0\) and \(u'=(u_1,\ldots ,u_{n-1})=0\) on \(\partial {\mathbb {R}}^n_+\) so that also \(\partial u_n/\partial x_n=0\) on \(\partial {\mathbb {R}}^n_+\), and only the pressure term remains, see (1.1)\(_4\). For more details on this transformation see [8, Chapter 2], [6] and [11].
There is a lot of literature concerning fluids interacting with an elastic plate. Often an additional damping term like \(-\partial _t \Delta h\) is introduced. This term allows the equation to have maximal \(L^p-\)regularity as considered by Denk and Saal, see [8], and [5, Chapter 4] in a space-time periodic setting. Weak solutions in \(L^2\)-spaces are constructed by Chambolle et al. [6] when adding a viscous damping term of either the form \(\Delta ^2\partial _t h\) or \(-\Delta \partial _{tt}h\) in the plate equation. Similar results have been shown by Grandmont [11] in the case of a three-dimensional cavity with one part of the boundary being elastic and the other one being rigid; weak solutions are shown to exist until a possible time where intersections of the two boundary parts occur. However, the focus in [11] is on the behavior of solutions in the limit of a vanishing damping term and on a uniform positive lower bound of the time of existence.
Fluid structure interaction problems with a classical nonlinear von Kármán shallow shell allowing for both transversal and lateral displacements are considered by Chueshov and Ryzhkova [7]. Lengeler and Růžička [13] discussed the case of a linearly elastic Koiter shell instead of a flat plate and hence replaced the operator \(\Delta ^2\) by an operator better suited for non-flat boundaries.
In the case of a bounded domain and no damping we still have the existence of a contraction semigroup, see Badra and Takahashi [1, Proposition 3.4]. In [4] Casanova, Grandmont and Hillairet construct weak solutions in a 2D periodic layer-type domain. For local-in-time strong solutions in this latter setting we refer to Beiraõ da Veiga [3]. Moreover, strong solutions are found by Badra and Takahashi [2] using a non-analytic semigroup of Gevrey class. Finally, in [10] the present authors construct weak solutions to the fluid-structure interaction problem of a viscous fluid coupled with a damped elastic plate under the nonlinear Coulomb boundary friction condition.
However, in our case of an unbounded domain even the existence of an \(L^2\)-semigroup is doubtful, which is why we will make more basic considerations to solve the equation in the case of non-vanishing initial data. The methods used here admit a solution in the case \(p=2\) only since the corresponding multipliers are bounded but they are not Fourier multipliers for \(p \ne 2\). Also note that although we show the existence of solutions on any finite time interval, we must exclude the case \(T=\infty \) since the Fourier transform of a solution of the undamped plate equation, \(\partial _t^2 h+\Delta ^2 h =0\), is given by terms involving cosine and sine functions which are not \(L^2-\)integrable on \((0, \infty )\). Adding a damping term as in [8], this issue is solved and allows the use of solution spaces with exponential weights. Additionally, in [8] the damping term also guarantees that the solution belongs to an inhomogeneous Sobolev space, which is not possible in the present undamped case.
This work is structured as follows: First we introduce the relevant solution spaces. Secondly, we show existence of solutions in the case of vanishing initial data using partial Fourier transforms. Finally, we reconstruct the initial data. Here we do not use abstract semigroup theory, as usually done, but the specific form of the undamped plate equation.
1.2 Solution spaces
One drawback of working with an undamped plate equation is that - unlike in [8] - we no longer obtain solutions \(u \in L^2(0,T;H^2({\mathbb {R}}^n_+))\) where \(H^2({\mathbb {R}}^n_+))\) denotes the usual inhomogeneous Sobolev space of order 2 over \(L^2\); rather, we have to work in homogeneous spaces, i.e., \(u \in L^2(0,T; {\dot{H}}^2({\mathbb {R}}^n_+))\). Since there are non-equivalent definitions of homogeneous Sobolev spaces and Bessel potential spaces, we choose a notion that fits well with our method of choice to obtain solutions, namely via the partial Fourier transform.
Let \({\mathcal {F}}\) and \({\mathcal {F}}^{-1}\) denote the Fourier transform and its inverse, respectively. Elements of \({\mathbb {R}}^n\) are often written in the form \(x=(x',x_n)\) with \(x'\in {\mathbb {R}}^{n-1}\), \(x_n\in {\mathbb {R}}\), and the phase variable \(\xi \) as \((\xi ',\xi _n)\). Similarly, vector fields \(u=(u_1,\ldots ,u_n)\) are splitted into \(u'=(u_1,\ldots ,u_{n-1})\) and \(u_n\); in particular, we write \(0'=(0,\ldots ,0)\in {\mathbb {R}}^{n-1}\).
Definition 1.1
Given \(s \in {\mathbb {R}}\) we define the homogeneous Sobolev space \({\dot{H}}^s({\mathbb {R}}^n)\) as
equipped with the norm
Here \(Z'({\mathbb {R}}^n):={\mathcal {S}}'({\mathbb {R}}^n)/P({\mathbb {R}}^n)\) where \(P({\mathbb {R}}^n)\) denotes the space of all polynomials on \({\mathbb {R}}^n\) and \({\mathcal {S}}'({\mathbb {R}}^n)\) denotes the set of Schwartz distributions. For the domain \({\mathbb {R}}^n_+\) we define the homogeneous Sobolev space \({\dot{H}}^s({\mathbb {R}}^n_+)\) via restriction:
We have the following useful properties that we will frequently use.
Proposition 1.2
Let \(s\in {\mathbb {R}}\) and let \(\Lambda =(-\Delta )^{1/2}\) denote the operator defined by \(\Lambda u={\mathcal {F}}^{-1} (|\xi |{\mathcal {F}} u)\).
-
i)
\({\dot{H}}^0 ({\mathbb {R}}^n)=L^2({\mathbb {R}}^n).\)
-
ii)
The mapping \(\Lambda ^\omega {\mathcal {F}}: {\dot{H}}^s({\mathbb {R}}^n) \rightarrow {\dot{H}}^{s-\omega }({\mathbb {R}}^n) \) is an isomorphism and maps \(Z'({\mathbb {R}}^n)\) onto itself for all \(\omega \in {\mathbb {R}}\).
-
iii)
If \(s>\frac{1}{2}\) then there is a bounded linear mapping \({\text {tr}}: {\dot{H}}^s({\mathbb {R}}^n_+) \rightarrow {\dot{H}}^{s-\frac{1}{2}}(\partial {{\mathbb {R}}^n_+})\cong {\dot{H}}^{s-\frac{1}{2}}({{\mathbb {R}}^{n-1}})\), the trace operator, such that \({\text {tr}}(\phi )(x')=\phi (x',0)\) for all \(\phi \in {\mathcal {S}}({{\mathbb {R}}^n})\) with \(0 \notin {\text {supp}} {\mathcal {F}}(\phi )\).
For a proof of iii) see Theorem 2.1 in [12]. For i) and ii) as well as for more information about homogeneous spaces see [14, Chapter 5].
Concerning Bochner spaces with respect to time over a Banach space X we write \(b \in {\dot{H}}^k(0,T;X)\) meaning that \(\partial ^k_t b \in L^2(0,T;X)\) for \(k \in {\mathbb {N}}\), \(T \in (0,\infty ]\). In most cases, X will be a homogeneous Bessel potential space \(\dot{H}^s(\Omega )\) where \(\Omega ={\mathbb {R}}^n, {\mathbb {R}}^n_+\) or \(\Gamma :={\mathbb {R}}^{n-1}=\partial {\mathbb {R}}^n_+\). Since the Fourier transform is used only on \({\mathbb {R}}^{n-1}\), it will, for the sake of simplicity, be denoted by \({\mathcal {F}}\) rather than \({\mathcal {F}}'\), its inverse by \({\mathcal {F}}^{-1}\) rather than \(({\mathcal {F}}')^{-1}\). However, the phase variable still is \(\xi '.\) For the one-dimensional Fourier transform with respect to time the phase variable will be called \(\tau \).
1.3 Main results
Now we can formulate the main result on existence of solutions. Recall that \(\Gamma ={\mathbb {R}}^{n-1}=\partial {\mathbb {R}}^n_+\).
Theorem 1.3
Let \(T\in (0,\infty )\) and
be given. Then the system
admits a solution
Furthermore, we have the estimate
Note that \(T\in (0,\infty )\) is arbitrary, but that C in (1.3) is independent of T.
2 Proofs
2.1 The case of vanishing initial data
First we assume \(f=0\) in (1.1)\(_1\) as well as homogeneous initial data in (1.1)\(_5\), i.e., \(u(0)=u_0=0\). Since \(g_1(0)=0\) we extend the functions \(g_1 \in L^2(0,\infty ;{\dot{H}}^{3/2}(\Gamma )) \cap {\dot{H}}^1(0,\infty ;{\dot{H}}^{-1/2}(\Gamma )) \) and \(g_2 \in L^2(0, \infty ; {\dot{H}}^{1/2}(\Gamma ))\) in time by zero to corresponding functions defined on \({\mathbb {R}}\). Then we apply the Fourier transform with respect to time and \(x' \in {\mathbb {R}}^{n-1}\) to get for \(v={{\hat{u}}}\) and \({{\hat{h}}}\) the system
From (2.1)\(_{1,2,3}\) we deduce \(\Delta p=0\) or, in other words,
which is solved in \({\mathcal {S}}'({\mathbb {R}}^{n-1})\) by
with a function \(\gamma =\gamma (\xi ', \tau ) \in {\mathbb {C}}\). Now we can solve (2.1)\(_1\) for \(v'\); the generic solution is given by
Indeed,
Concerning \(\alpha '\) (2.1)\(_4\) implies that
So for now we have
Using this representation we consider (2.1)\(_3\) to determine \(\partial _n v_n\):
This implies
We exploit this representation to solve (2.1)\(_2\) and get that
In particular, \( v_n|_{x_n=0}= \frac{\gamma A}{i \tau }- \frac{\gamma }{i \tau B}A^2.\) Now we determine two identities for \({\hat{h}}\) by using (2.1)\(_4\) and (2.1)\(_5\):
Combining these identities we solve for the still unknown term \(\gamma \) and see that
We define
and use the identity
where \(\frac{- \tau ^2 B}{(B-A) A} = -\frac{ \tau ^2}{A} +\mu i \tau (A+B)\). Now we conclude that
Summarizing (2.2), (2.4), (2.5), (2.6), (2.8) and (2.9) we obtain a solution \({{\hat{p}}}, v={{\hat{u}}}, {{\hat{h}}}\) in Fourier space by
Lemma 2.1
The functions
are uniformly bounded with respect to \((A, \tau ) \in (0,\infty ) \times {\mathbb {R}}\).
Proof
We note that \(B=\text {Re}B + \frac{i\tau }{2 \mu \text {Re} B}\) with \(\text {Re}B>0\) and decompose \({\mathcal {N}}(A, \tau )^{-1}\) into its real and imaginary part:
First we prove that \( \tau A{\mathcal {N}}(A, \tau )\) is bounded:
Next we consider \(\frac{ \tau ^2}{A}{\mathcal {N}}(A, \tau )\). If \(| \tau | \le A^2\), we use the imaginary part of \( {\mathcal {N}}(A, \tau )^{-1}\) to get as above the estimate
However, if \(| \tau |\ge A^2\), we use the real part of \({\mathcal {N}}(A, \tau )^{-1}\) and are led to the estimate
The function \( \tau ^{3/2}{\mathcal {N}}(A, \tau )\) is also bounded due to the previous considerations and Young’s inequality \(| \tau ^{3/2}| = \big |\tfrac{ \tau }{\sqrt{A}}\sqrt{ \tau A}\big |\le \frac{ \tau ^2}{2 A}+\frac{| \tau | A}{2}\).
Due to \(|B|\le \mu ^{-1/2}{\sqrt{| \tau |}}+A\) we deduce the boundedness of \( \tau B {\mathcal {N}}(A, \tau )\) from the previous cases. Finally, the boundedness of \((A^4-\tau ^{2}){\mathcal {N}}(A, \tau )\) follows from the identity
see (2.7), and the previous cases. \(\square \)
To simplify the analysis of the multiplier functions, we reduce the question of their boundedness to a problem in \({\mathbb {R}}^{n-1}\) rather than in \({\mathbb {R}}^n_+\) by the following lemma.
Lemma 2.2
-
i)
Let \(f: {\mathbb {R}}\times {\mathbb {R}}^{n-1} \rightarrow {\mathbb {C}}\) be measurable, \(c>0\) and \(\chi \in \{cA,cB \}\), see (2.3). Then we have the equivalence
$$\begin{aligned} f( \tau , \xi ')e^{-\chi x_n} \in L^2( {\mathbb {R}}\times {\mathbb {R}}^n_+) \Longleftrightarrow \frac{1}{\sqrt{\mathrm{Re}\chi }}f( \tau , \xi ') \in L^2( {\mathbb {R}}\times {\mathbb {R}}^{n-1}) \end{aligned}$$with equivalent norms.
-
ii)
There exists a constant \(C>0\) such that
$$\begin{aligned} A \left| \frac{e^{-Bx_n}-e^{-Ax_n}}{B-A} \right| \le Ce^{-\frac{1}{2} A x_n} \end{aligned}$$for all \(x_n,A>0\) and \( \tau \in {\mathbb {R}}\backslash \{ 0\}\).
Proof
i) The claim follows from the calculation
where we used Tonelli’s theorem in the first identity.
ii) We calculate
In the first inequality we used that \(\text {Re}((B-A)x_n)>0\). \(\square \)
Proposition 2.3
Let \(g_1 \in L^2(0,\infty ;{\dot{H}}^{3/2}(\Gamma )) \cap {\dot{H}}^1(0,\infty ;{\dot{H}}^{-1/2}(\Gamma ))\) with \(g_1(0)=0\) and \(g_2 \in L^2(0, \infty ; {\dot{H}}^{1/2}(\Gamma ))\) be given. Then the system
together with initial conditions \(u(0)=0\), \(h(0)=\partial _t h(0)=0\) has a solution
Furthermore, we have the estimate
The analogous statement holds if the time interval is replaced with (0, T).
Proof
First consider the case \(g_1=0\) in which the solution (2.10) simplifies to
where by Lemma 2.1
is a bounded multiplier function. By the previous two lemmata, we now show the corresponding estimates.
(i) \(u' \in L^2(0,\infty ; {\dot{H}}^2({{\mathbb {R}}^n_+}))\): Concerning tangential derivatives \(\partial _k \partial _i u_j\), \(i,j,k \in \{ 1,...,n-1\}\), we estimate as follows:
For the normal derivatives \(\partial _n^2 u'\) we have:
The second summand can be treated as above. Hence we only have to consider the first one:
For mixed derivatives \(\partial _k \partial _n u_j\), \(j,k \in \{ 1,...,n-1\}\) we have that
The first summand can be treated as in (2.13), the second one as in (2.12).
(ii) \(u_n \in L^2(0,\infty ;{\dot{H}}^2({\mathbb {R}}^n_+))\): Since \(\text {div}\, u=0\) and thus \(\partial _k \partial _n u_n = -\sum ^{n-1}_{j=1} \partial _k \partial _j u_j \in L^2((0,\infty ) \times {\mathbb {R}}^n_+)\) as shown in (i), it is left to consider \( \partial _k \partial _i u_n\) for \(i,k \ne n\). Moreover, the first summand of \(v_n\) can be treated in the same way as \(v'\), see (2.11). Therefore, we only have to discuss the second one:
(iii) \(p \in L^2(0,\infty ; {\dot{H}}^1({\mathbb {R}}^n_+))\): Since \(\partial _n {\hat{p}}= -A {\hat{p}}\), it suffices to consider the tangential derivatives \(\nabla ' {p}\) where, since \({\mathcal {M}}\) is bounded, we get that
(iv) \(h \in {\dot{H}}^1(0,\infty ;{\dot{H}}^{3/2}(\Gamma )) \cap {\dot{H}}^2(0,\infty ;{\dot{H}}^{-1/2}(\Gamma ))\): This follows immediately from \({\hat{h}}=\frac{{\mathcal {N}}}{\sqrt{A}}({\hat{g}}_2 \sqrt{A})\) and the boundedness of \( \tau A {\mathcal {N}}\) and \(\frac{ \tau ^2}{ A} {\mathcal {N}}\).
(v) \(u \in {\dot{H}}^1(0,\infty ;L^2({\mathbb {R}}^n_+))\): The estimate follows from \( \partial _t u=\Delta u-\nabla p \in {L}^2((0,\infty )\times {\mathbb {R}}^n_+)\).
If on the other hand \(g_2=0\), but
then we make use of the boundedness of \({\mathcal {M}}(A, \tau ):= {\mathcal {N}}(A, \tau ) (A^4-\tau ^2)\), see Lemma 2.1. Consider the set of solutions with \(g_2=0\), see (2.10):
Although the following estimates are similar to the previous ones, we present some details for the convenience of the reader:
(i) \(u' \in L^2(0,\infty , {\dot{H}}^2({{\mathbb {R}}^n_+}))\): For tangential derivatives \(\partial _k \partial _i u_j\), \(1\le i,j,k\le n-1\), we use the interpolation inequality
and that \(|B|\le \big |\frac{ \tau }{\mu }\big |^{1/2} + |A|\), and calculate as follows:
For the normal derivatives \(\partial _n^2 u'\) we compute:
The second summand can be treated as in (2.15). Hence we only have to consider the first one:
For mixed derivatives \( \partial _k \partial _n u_j \), \(k= 1,...,n-1\), there holds
The first summand can be treated as in (2.16), the second one as in (2.15).
(ii) \(u_n \in L^2(0,\infty ;{\dot{H}}^2({\mathbb {R}}^n_+))\): Since \(\partial _k \partial _n u_n = - \sum ^{n-1}_{j=1} \partial _k \partial _j u_j \in L^2((0,\infty ) \times {\mathbb {R}}^n_+)\), it is only left to estimate the term \( \partial _k \partial _i u_n\) for \(i,k \ne n\). The first summand of \(v_n\) can be treated in the same way as \(v_j\); thus we only have to consider the second one:
(iii) \(p \in L^2(0,\infty ; {\dot{H}}^1({\mathbb {R}}^n_+))\). For the tangential derivative \(\nabla ' {p}\) we have
for the last step we also use (2.15). Since \(\partial _n {\hat{p}}= -A {\hat{p}}\), we can show \(\partial _n {p} \in L^2(0,\infty ; {L}^2({\mathbb {R}}^n_+))\) as above.
(iv) \(h \in {\dot{H}}^1(0,\infty ;{\dot{H}}^{3/2}(\Gamma )) \cap {\dot{H}}^2(0,\infty ;{\dot{H}}^{-1/2}(\Gamma ))\). This follows immediately from the boundedness of \({\mathcal {M}}(A, \tau )-1\).
(v) \(u \in {\dot{H}}^1(0,\infty ;L^2({\mathbb {R}}^n_+))\): Here we refer to \( \partial _t u=\Delta u-\nabla p \in {L}^2((0,\infty )\times {\mathbb {R}}^n_+)\).
If \(g_1 \in L^2(0,T;{\dot{H}}^{3/2}(\Gamma )) \cap {\dot{H}}^1(0,T;{\dot{H}}^{-1/2}(\Gamma ))\) and \(g_2 \in L^2(0,T;{\dot{H}}^{1/2}(\Gamma ))\) we extend \(g_2\) by zero to an element in \(L^2(0,\infty ;{\dot{H}}^{1/2}(\Gamma ))\) and, since \(g_1(0)=0\), we extend \(g_1\) to an element in \(L^2(0,\infty ;{\dot{H}}^{3/2}(\Gamma )) \cap {\dot{H}}^1(0,\infty ;{\dot{H}}^{-1/2}(\Gamma ))\) by \(g_1 \circ \varphi \) where \(\varphi \in C^0([0,\infty ))\) has compact support in [0, 2T] and
Then we apply the above case and restrict the result to the finite time interval. \(\square \)
2.2 The case of non-vanishing initial data
Next we want to recover non-vanishing initial data starting with \(h_0 \in {\dot{H}}^{7/2}(\Gamma )\). Consider the classical plate equation
Applying the partial Fourier transform \({\mathcal {F}}\) with respect to the spatial variable \(x' \in \Gamma ={\mathbb {R}}^{n-1}\) we get a solution
Now the general idea is to replace h from Proposition 2.3 with \(h+\eta _0\) to satisfy the initial condition and to leave the plate equation unchanged. However this replacement alters the equation \(u_n|_{x_n=0}=\partial _t h\) which is why we need the following proposition on the Stokes system with inhomogeneous Dirichlet boundary data in the nth component.
Proposition 2.4
Let \(0<T<\infty \), \(h_0 \in {\dot{H}}^{7/2}(\Gamma )\), and let \({\eta }_0={\mathcal {F}}^{-1}(\cos (A^2 t) {\mathcal {F}}{h}_0 )\) be the solution of the plate equation (2.17). Then the Stokes system
has a solution
Furthermore, there holds the estimate
Proof
Choose \(\varphi \in C^1_c([0, \infty ))\) such that \(\varphi =1\) on [0, T], \(\varphi (t)=0\) for \(t>2T\) and \(t|\varphi '(t)|\le c\) on \({\mathbb {R}}\) with \(c>0\) independent of T. Obviously, we can replace \(\partial _t {\eta }_0(t,\xi )\) with \(\varphi (t)\,\partial _t {\eta }_0(t,\xi ) \) in (2.19)\(_3\). We want to ensure that
For the first property we have
Concerning the space \({\dot{H}}^1(0,\infty ; {\dot{H}}^{-1/2}(\Gamma ))\) we use the estimate \(|\sin s|\le |s|\), \(s\in {\mathbb {R}}\), to get that
Now it is easily seen that r satisfies (2.21).
Since \(\partial _t {\eta }_0(0,x')\,\varphi (0)=0 \), it is possible to extend \(r=\partial _t {\eta }_0(t,x')\, \varphi \) by zero to a function in \(L^2({\mathbb {R}};{\dot{H}}^{3/2}(\Gamma )) \cap {\dot{H}}^1({\mathbb {R}}; {\dot{H}}^{-1/2}(\Gamma ))\) still satisfying an estimate similar to (2.21)\(_2\). Then we apply in (2.19) the Fourier transform with respect to time and the first \(n-1\) spatial variables to obtain that
The following calculations resemble those at the beginning of Sect. 2.1, see (2.3)-(2.5). Since \({\text {div}}w_0=0\) we get from (2.22) that \(\Delta p_0=0\) is solved by
for a function \(\gamma =\gamma (\xi ',\tau )\). Using this identity and (2.22)\(_1\) we deduce
Now using (2.22)\(_3\) we see that
Next, from (2.22)\(_2\) we conclude that
cf. (2.5). Finally, (2.22)\(_4\) implies \({\hat{r}} = \gamma \left( \frac{A}{i \tau }-\frac{A^2}{i \tau B} \right) = \gamma \frac{A}{i \tau } \frac{B-A}{B}. \) Hence
In summary, we obtain a solution
Now we can prove a priori estimates of this solution in \(L^2\):
(i) \(w_0' \in L^2(0,\infty ;{\dot{H}}^2({\mathbb {R}}^{n}_+))\): Concerning tangential derivatives \(\partial _k \partial _i (w_0)_j\), \(i,j,k \in \{1,...,n-1 \}\) there holds, since \(|B|\le c| \tau |^{1/2} + A\),
For mixed derivatives \(\partial _i \partial _n (w_0)_j\) we see that
Finally, the normal derivative \(\partial ^2_n (w_0)_j\) satisfies the estimate
since \(A\le |B|\) and \(|B|^2\le c| \tau | + A^2\).
(ii) \((w_0)_n \in L^2(0,\infty ;{\dot{H}}^2({\mathbb {R}}^{n}_+))\): Since \(\partial _j \partial _n (w_0)_n = -\sum ^{n-1}_{k=1} \partial _k \partial _j (w_0)_k\), \(j \in \{1,...,n-1 \}\), we have to consider only \(\partial _k \partial _i (w_0)_n\) for \(i,k \in \{1,...,n-1 \}\). The first summand in (2.23)\(_4\) can be treated as \(({\hat{w}}_0)_j\) above; so we inspect the second one:
(iii) \({p}_0 \in L^2(0,\infty ;{\dot{H}}^1({\mathbb {R}}^{n}_+))\): Since \(\partial _n {\hat{p}}_0 = -A {\hat{p}}_0\), it suffices to consider the tangential gradient \(\nabla ' p_0\). Here we get that
(iv) \(w_0 \in {\dot{H}}^1(0,\infty ; L^2({\mathbb {R}}^n_+))\): This is immediately implied by the identity \(\partial _t w_0= \mu \Delta w_0 -\nabla p_0 \in L^2((0,\infty )\times {\mathbb {R}}^n_+)\).
Restricting ourselves to the time interval (0, T), recalling the interpolation inequality (2.14) and also (2.21) we deduce the proposition. \(\square \)
Now we reconstruct the initial data \(u_0 \in {\dot{H}}^2({\mathbb {R}}^n_+)\) with \({\text {div}}u_0=0\) and \({u_0'}|_{x_n=0}=0\), as well as the right-hand side \(f \in L^2((0,T) \times {\mathbb {R}}^n_+)\).
Proposition 2.5
Let \(T\in (0,\infty )\), \(f\in L^2((0,T)\times {\mathbb {R}}^n_+)\), and let \(u_0 \in {\dot{H}}^2\left( {\mathbb {R}}^n_+ \right) \) satisfy \({\text {div}}u_0=0\), \({u_0'}|_{\partial {\mathbb {R}}^n_+}=0\). Then the Stokes system
admits a solution
Furthermore, we have the estimate
Proof
To get rid of the terms involving \(u_0\) define
which satisfies (2.24)\(_{2,3,4}\). Moreover, since, in \(L^2((0,T)\times {\mathbb {R}}^n_+)\),
\(w_2\) can be estimated as z in (2.25) (with \(f=0\), \(p_1=0\)).
Next we consider the Stokes system
Since the right-hand side lies in \(L^2((0,T)\times {\mathbb {R}}^n_+)\), there exists a strong solution \(w_1 \in L^2(0,T;H^2({\mathbb {R}}^n_+))\cap H^1(0,T;L^2({\mathbb {R}}^n_+))\) and \(p_1 \in L^2(0,T; {\dot{H}}^1({\mathbb {R}}^n_+))\) with the corresponding maximal regularity estimate (see e.g. [9, Corollary 3.6]) with a constant \(C>0\) independent of T.
So we deduce that \(z:=w_1+w_2\) and \(p_1\) satisfy (2.24), (2.25). \(\square \)
Corollary 2.6
Let \(T\in (0,\infty )\), \(f\in L^2((0,T)\times {\mathbb {R}}^n_+)\), \(h_0 \in {\dot{H}}^{7/2}(\Gamma )\), \(u_0 \in {\dot{H}}^2({\mathbb {R}}^n_+)\) with \({\text {div}} u_0=0\) and \(u_0'|_{\partial {\mathbb {R}}^n_+}=0\). Then the Stokes system
together with the boundary condition
on \(\Gamma \times (0,T)\), has a solution
Furthermore, we have the corresponding estimate as in (2.25), including the term \(CT^{1/2} \Vert h_0 \Vert _{{\dot{H}}^{7/2}(\Gamma )}\).
Proof
Let \((w_0,p_0)\) be a solution of (2.19) and \((z,p_1)\) be a solution of (2.24). Then define
and \(q:=p_0+p_1 \in L^2(0,T;{\dot{H}}^1({\mathbb {R}}^n_+))\) to get the desired solution. \(\square \)
2.3 Proof of Theorem 1.3
Let
be a solution of (2.26)-(2.27). Due to \(q|_{\partial {\mathbb {R}}^{n}_+} \in L^2(0,T;{\dot{H}}^{1/2}(\Gamma ))\) Proposition 2.3 implies the existence of
such that
satisfying the initial conditions
To finish the proof define
where \(h \in {\dot{H}}^1(0,T;{\dot{H}}^{3/2}(\Gamma )) \cap {\dot{H}}^2(0,T;{\dot{H}}^{-1/2}(\Gamma )),\) to get a solution to (1.2). \(\square \)
Remark 2.7
Despite \(h \in {\dot{H}}^2(0,T;{\dot{H}}^{-1/2}(\Gamma ))\) we did not show that h is very regular with respect to space, let alone \(h(t) \in {\dot{H}}^{7/2}(\Gamma )\) for almost all \(t \in (0,T)\). This is due to the fact that while \( \tau ^{3/2}{\mathcal {N}}(A,\tau )\) is bounded, see Lemma 2.1, the multiplier function \( A^3 {\mathcal {N}}(A, \tau )\) is not uniformly bounded in \(A,\tau \).
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Farwig, R., Schmidt, A. An \(L^2\) approach to viscous flow in the half space with free elastic surface. J Elliptic Parabol Equ 7, 601–621 (2021). https://doi.org/10.1007/s41808-021-00111-2
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DOI: https://doi.org/10.1007/s41808-021-00111-2