Abstract
In this paper, we consider the 3-D motion of viscous gas with the vacuum free boundary. We use the conormal derivative to establish local well-posedness of this system. One of important advantages in the paper is that we do not need any strong compatibility conditions on the initial data in terms of the acceleration.
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1 Introduction
1.1 Formulation in Eulerian coordinates
In the paper, we consider a 3-D viscous compressible fluid in a moving domain \(\Omega (t)\) with an upper free surface \(\Gamma (t)\) and a fixed bottom \(\Gamma _b\). This model can be expressed by the 3-D compressible Navier–Stokes equations(CNS)
where \(\mathcal {V}(\Gamma (t))\) denotes the normal velocity of the free surface \(\Gamma (t)\), and \(n= n(t)\) is the exterior unit normal vector of \(\Gamma (t)\), the vector-field u denotes the Eulerian velocity field, \(\rho \) is the density of the fluid, and \(p=p(\rho )\) denotes the pressure function. The stress tensor \(\mathbb {S}(u)\) is defined by \(\mathbb {S}(u)=\mu \mathbb {D}(u)+\lambda (\nabla \cdot u)\mathbb {I}\), where the strain tensor \(\mathbb {D}(u)=\nabla u+\nabla u^{T}\) and dynamic viscosity \(\mu \) and bulk viscosity \(\nu \) are constants which satisfy the following relationship
The deviatoric (trace-free) part of the strain tensor \(\mathbb {D}(u)\) is then \(\mathbb {D}^0(u)=\mathbb {D}(u)-\frac{2}{3}{\mathrm{div}}\, u\, \mathbb {I}\). The viscous stress tensor in fluid is then given by \(\mathbb {S}(u)=\mu \,\mathbb {D}^0(u)+(\lambda +\frac{2}{3}\mu ) (\nabla \cdot u)\, \mathbb {I}\). Moreover, the pressure obeys the \(\gamma \)-law: \(p(\rho )=K\,\rho ^{\gamma }\), where K is an entropy constant and \(\gamma >1\) is the adiabatic gas exponent.
Equation (1.1)\(_1\) is the conservation of mass; Eq. (1.1)\(_2\) means the momentum conserved; the boundary condition (1.1)\(_3\) states that the pressure (and hence the density function) vanishes along the moving boundary \(\Gamma (t)\), which indicates that the vacuum state appears on the boundary \(\Gamma (t)\); the kinematic boundary condition (1.1)\(_4\) states that the vacuum boundary \(\Gamma (t)\) is moving with speed equal to the normal component of the fluid velocity; (1.1)\(_5\) means the fluid satisfies the kinetic boundary condition on the free boundary, (1.1)\(_6\) denotes the fluid is no-slip, no-penetrated on the fixed bottom boundary, and (1.1)\(_{7}\) are the initial conditions for the density, velocity, and domain.
In the paper, we assume the bottom \(\Gamma _{b}=\{y_3=b(y_h)\}\), and the moving domain \(\Omega (t)\) is horizontal periodic by setting \(\mathbb {T}^2_{y_h}\) with \(y_h:=(y_1, y_2)^T\) for \(\mathbb {T}=\mathbb {R}/\mathbb {Z}\).
1.2 Known results
Whether or not the appearance of vacuum state is related to the regularity of the solution to the compressible Navier–Stokes equations. Even if there is no vacuum in initial data, it cannot guarantee that vacuum state will be not generated in finite time in high-dimensional system. Whence initial data is close to a non-vacuum equilibrium in some functional space, Matsumura and Nishida [35, 36] proved global well-posedness of strong solutions to the 3-D CNS. Moreover, for the one dimensional case, Hoff and Smoller [17] proved that if the vacuum is not included at the beginning, no vacuum will occur in the future. Hoff and Serre [16] showed some physical weak solution does not have to depend continuously on their initial data when vacuum occurs.
When the initial density may vanish in open sets or on the (part of) boundary of the domain, the flow density may contain a vacuum, the equation of velocity becomes a strong degenerate hyperbolic-parabolic system and the degeneracy is one of major difficulties in study of regularity and the solution’s behavior, which is completely different from the non-vacuum case. For the existence of solutions for arbitrary data (the far field density is vacuum, that is, \(\rho (t, x)\rightarrow 0\) as \(x\rightarrow \infty \)), the major breakthrough is due to Lions [27] (also see [8, 14, 22]), where he obtains global existence of weak solutions, defined as solutions with finite energy with suitable \(\gamma \). Recently, Li and Xin [26] and Vasseur and Yu [39] independently studied global existence of weak solutions of CNS whence the viscosities depend on the density and satisfy the Bresch and Desjardins relation [1]. Yet little is known on the structure of such weak solutions except for the case that some additional assumptions are added (see [15] for example). Indeed, the works of Xin etc. [24, 40] showed that the homogeneous Sobolev space is as crucial as studying the well-posedness for the Cauchy problem of compressible Navier–Stokes equations in the presence of a vacuum at far fields even locally in time. Adding some compatible condition on initial data, Cho and Kim [3] develop local well-posedness for strong solutions. Moreover, if initial energy is small, Huang et al. [18] showed the global existence of classical solutions but with large oscillations to CNS.
Physically, the vacuum problem appears extensively in the fundamental free boundary hydrodynamical setting: for instance, the evolving boundary of a viscous gaseous star, formation of shock waves, vortex sheets, as well as phase transitions.
For free boundary problem of the multi-dimensional Navier–Stokes equations with non-vacuum state, there are many results concerning its local and global strong solutions, one may refer to [43, 44] and references therein.
But when the vacuum (in particular, the physical vacuum [28]) appears, the system becomes much harder. To understand the difficulty of the vacuum, we introduce the sound speed \(c:=\sqrt{p'(\rho )} (=\sqrt{K\gamma } \rho ^{\frac{\gamma -1}{2}}\) for polytropic gases) of the gas or fluid to describe the behavior of the smoothness of the density connecting to vacuum boundary. A vacuum boundary \(\Gamma (t)\) is called physical vacuum if there holds
near the boundary \(\Gamma (t)\), where n is the outward unit normal to the free surface. The physical vacuum condition (1.3) implies the pressure (or the enthalpy \(c^2\)) accelerates the boundary in the normal direction. Thus, the initial physical vacuum condition (1.3) is equivalent to the requirement that
which means that \(\rho ^{\gamma -1}_0(x)\sim dist(x,\Gamma (0))\), in other words, the initial sound speed \(c_0\) is only \(C^{\frac{1}{2}}\)-Hölder continuous near the interface \(\Gamma (0)\).
Due to lack of sufficient smoothness of the enthalpy \(c^2\) at the vacuum boundary, a rigorous understanding of the existence of physical vacuum states in compressible fluid dynamics has been a challenging problem, especially in multi-dimensional cases.
Recently, the local well-posedness theory for compressible Euler system with physical vacuum singularity was established in [4, 20, 21], and also global existence of smooth solutions for the physical vacuum free boundary problem of the 3-D spherically symmetric compressible Euler equations with damping was showed in [32]. And more recently, Hadzic and Jang [13] proved global nonlinear stability of the affine solutions to the compressible Euler system with physical vacuum, and Guo et al. [9] constructed an infinite dimensional family of collapsing solutions to the Euler-Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface, with continuous mass absorption at the origin.
The study of vacuum is important in understanding viscous surface flows [30]. Very little is rigorously known about well-posedness theories available about free boundary problems of CNS with physical vacuum boundary. For 1-D problem, global regularity for weak solutions to the vacuum free boundary problem of CNS was obtained in [30], which is further generalized by Zeng [45] which established the strong solutions. For the multidimensional case, regularity results related to spherically symmetric motions. Guo et al. [11] obtain a global weak solution to the problem with spherically symmetric motions and a jump density connects to vacuum. Later Liu [29] gives the existence of global solutions with small energy in spherically symmetric motions with the density connected to vacuum continuously or discontinuously. Anyway, almost all the well-posedness results require additional strongly singular compatibility conditions on initial data in terms of the acceleration for gaining more regularities of the velocity. Some related works can refer to [2, 6, 7, 12, 19, 25, 28, 30, 31, 37, 41, 42] and references therein.
The purpose of this paper is to establish the local well-posedness of the 3-D compressible Navier–Stokes equations (1.1) with physical vacuum boundary condition without any compatibility conditions, more precisely, we do not need any initial condition on the material derivative \(D_t u\) or its derivatives. For simplicity, we set \(\gamma =2\) and \(K=1\) in this paper.
As mentioned above, the main difficulty in obtaining regularity for the vacuum free boundary problem (1.1) lies in the degeneracy of the system near vacuum boundaries. In order to solve the system (1.1), the first idea is that we use Lagrangian coordinates to transform it to a system with fixed domain. One of advantage of Lagrangian coordinates is that the density \(\rho \) is solved directly by initial data and we only focus on the equation of velocity with coefficients related to Lagrangian coordinates.
The second and also key idea in our paper is that we use the conormal derivatives to obtain the high-order regularity. Because the density vanishes on the boundary, we can not close the energy estimates if we directly take normal derivatives to the system. So another choose is to take time derivatives in [4, 21] solving the compressible Euler equations with the physical vacuum, where high-order enough time-derivative estimates as long as spatial-derivative estimates allow us to close the energy estimates and then get the local-in-time existence of the strong solution of the Euler system. This high-order energy estimate in it is reasonable since the pressure term may cancel the singularity near the vacuum boundary when consider compatibility conditions on initial data in terms of the acceleration and its derivatives. However, this method may not work for the Navier–Stokes system (1.1) with constant viscosity coefficients. In fact, a strong singular compatibility conditions on initial data in terms of the acceleration and its derivatives will appear in it when we consider the high-order energy estimate, which is mainly due to the non-degenerate of the viscosity, but it seems very hard to find such kind of initial data satisfying these compatibility conditions. In order to get rid of this difficulty, our strategy is that we use conormal Sobolev space introduced in [34] to get the tangential regularity. Based on that, we multiply \(\partial _t v\) on the both sides of equations of v to get the estimates of \(\rho ^{\frac{1}{2}}\partial _t v\) which implies the two-order derivative on the normal direction. Form this, together with high-order tangential derivatives estimates, we get the \(W^{1,\infty }\) estimates of v and its conormal derivatives, which in turn guarantees the propagation of conormal regularities of the velocity.
1.3 Derivation of the system in Lagrangian coordinates and main result
In this paper, we consider the case that the upper boundary does not touch the bottom which means that
Take \(\Omega =\{x \in \mathbb {T}^2\times \mathbb {R}|\quad 0<x_3<1\}\) as the domain of equilibrium. Let \(\eta (t,x)\) be the position of the gas particle x at time t so that
Here \(\eta _0\) is a diffeomorphism from \({\Omega }\) to the initial moving domain \(\Omega (0)\) which satisfies that \(\Gamma (0)=\eta _0(\{x_3=1\})\) and \(\Gamma _{b}=\eta _0(\{x_3=0\})\). It is easy to construct a invertible transform \(\eta _0\) which satisfies that
Due to (1.5), we introduce the displacement 
which satisfies the following ODE
We define the following Lagrangian quantities:
Then, the system (1.1) is reformulated in Lagrangian coordinates as follows
with boundary conditions
and initial data
One may readily check from the definition of J that
which together with the equation of f in (1.7) yields
Hence, we find
where \(\rho _0\) is a given initial density function. We are interested in the initial density \(\rho _0\) satisfying
with some given function \(\overline{\rho }(x)\) (\(x\in \Omega \)), for any \(k \in \mathbb {N}\) with \(\nabla _h=(\partial _1,\partial _2)\), where d(x) is the distance function to the boundary \(\{x_3=1\}\).
Thus, it follows from (1.10) that
which implies that
Remark 1.1
For any smooth subdomain \(\mathcal {O}\) of \(\Omega \), we know that \(\eta _0(\mathcal {O})\) is a subdomain of \(\Omega (0)\) if \(\eta _0\) is a diffeomorphism from \(\Omega \) to \(\Omega (0)\). Hence, by using change of variables, we get
Hence, the assumption (1.11) is equivalent to the mass conservation law
Multiplying the both side of equation v by J, we obtain the equivalent form of the system (1.7)–(1.9) as follows
Next, we give some useful equations which we often use in what follows. Since \(\mathcal {A}[D\eta ]=I,\) one obtains that
Differentiating the Jacobian determinant, we get
Moreover, the following Piola identity holds:
for any \(i=1,2,3.\)
1.4 Main results
Before we state our main results, we give some definitions of functional spaces. First, define the operators:
Using \(Z^m\) to denote \(Z_3^{m_2}Z_h^{m_1}=Z_3^{m_2}Z_1^{m_{11}}Z_2^{m_{12}}\) with \(m_1=(m_{11}, m_{12})\) and |m| to denote \(|m|=|m_1|+m_2=m_{11}+m_{12}+m_2.\) Moreover, we use \(Z_3^{m_2}\) to denote \(\overline{\rho }^{m_2}\partial _{3}^{m_2}.\) By (1.11)–(1.13), it is easy to see
We recall the following conormal Sobolev space introduced by Masmoudi and Rousset [34].
where \(\alpha \in \mathbb {R}\). In particular, when \(\alpha =0\), we the spaces \(X^N_{\alpha }\) and \(\dot{X}^N_{\alpha }\) will be denoted by \(X^N\) and \(\dot{X}^N\) respectively for simplicity.
For \(T>0\), we define the energy space \(E_T\) as
with the instantaneous energy \(\mathcal {E}(t)\) (in terms to the velocity v)
and the dissipation \(\mathcal {D}(t)\)
Given \(\kappa >0\), we also introduce the space \(F_{\kappa }\) in terms to the flow map \(\eta \) as follows:
equipped with the norm
Now, we are in the position to state our main results.
Theorem 1.2
Under the assumptions (1.11)–(1.13), assume that there exists a positive number \(\sigma _0\) such that
If the initial data \((v_0,\, \eta _0)\in (X^{12}_{\frac{1}{2}}\cap H^1(\Omega )) \times \mathcal {F}_{\kappa }(\Omega )\) for some constant \(\kappa \in (0, \frac{1}{16})\), then the system (1.18) is locally well-posed. More precisely, there exists a positive time \(T>0\) such that the system (1.18) has a unique solution \((v,\,\eta )\in \,C([0, T]; X^{12}_{\frac{1}{2}}\cap H^1(\Omega )) \times C([0, T]; \mathcal {F}_{\kappa }(\Omega ))\) depending continuously on initial data \((v_0,\, \eta _0)\in (X^{12}_{\frac{1}{2}}\cap H^1(\Omega )) \times \mathcal {F}_{\kappa }(\Omega )\), and there hold
where C depends on initial data.
Remark 1.3
The assumption (1.11)–(1.13) on \(\rho _0\) is reasonable. In fact, if \(\Omega (0)=\Omega :=\{x \in \mathbb {T}^2\times \mathbb {R}|\quad 0<x_3<1\}\) and \(\rho _0=dist(x,\partial \Omega )\sim x_3(1-x_3)\), then the assumptions (1.11)– (1.13) are automatically satisfied .
Remark 1.4
In this paper, we consider the case that \(\gamma =2\). But our method may still work for all the cases \(\gamma >1\).
Remark 1.5
For any \(t \in [0, T]\), since \(\sigma _0\le \sup _{(t, x)\in [0,T]\times \Omega }J(t, x)\le 4\sigma _0\), the flow-map \(\eta (t, x)\) defines a diffeomorphism from the equilibrium domain \(\Omega \) to the moving domain \(\Omega (t)\) with the boundary \(\Gamma (t)\). From this, together with the fact that \(\eta _0\) is a diffeomorphism from the equilibrium domain \(\Omega \) to the initial domain \(\Omega (0)\), we deduce a diffeomorphism from the initial domain \(\Omega (0)\) to the evolving domain \(\Omega (t)\) for any \(t \in [0, T]\). Denote the inverse of the flow map \(\eta (t, x)\) by \(\eta ^{-1}(t, y)\) for \(t \in [0, T]\) so that if \(y = \eta (t, x)\) for \(y \in \Omega (t)\) and \(t \in [0, T]\), then \(x = \eta ^{-1}(t, y) \in \Omega \).
For the strong solution \((\eta , v)\) obtained in Theorem 1.2, and for \(y \in \Omega (t)\) and \(t \in [0, T]\), we denote that
Then the triple \((\rho (t,\,y), u(t, y), \Omega (t))\) (\(t \in [0, T]\)) defines a strong solution to the free boundary problem (1.1). Furthermore, we obtain the following theorem.
Theorem 1.6
Under the assumptions in Theorem 1.2, the free boundary problem (1.1) is locally well-posed, and the triple \((\rho (t,\,y), u(t, y), \Omega (t))\) (\(t \in [0, T]\)) defined in Remark 1.5 and (1.26) is the unique strong solution to the free boundary problem (1.1) satisfying \(\eta -Id \in \,C([0, T],\, \mathcal {F}_{\kappa })\).
The rest of the paper is organized as follows. In Sect. 2, we derive some preliminary estimates. Some necessary a priori estimates are obtained in Sect. 3. Finally in Sect. 4, the proof of Theorem 1.2 is proved.
Let us complete this section with some notations that we use in this context.
Notations Let A, B be two operators, we denote \([A, B]=AB-BA,\) the commutator between A and B. For \(a\lesssim b\), we mean that there is a uniform constant C, which may be different on different lines, such that \(a\le Cb\) and \(C_0\) denotes a positive constant depending on the initial data only.
2 Preliminary estimates
In what follows, we denote by C a positive constant which may depend on initial data \((v_0,\eta _0)\) if we don’t make a special explanation in it. This notation is allowed to change from one inequality to the next.
We first introduce the following inequality which we heavily use throughout the paper.
Lemma 2.1
(Hardy inequality, [23]) For any \(\varepsilon >0,\) there holds that
With Hardy inequality in hand, we may get the following interpolation equalities.
Lemma 2.2
For any \(\kappa \in (0,\frac{1}{16})\), there hold that, for \(0\le \ell \le 6\),
and for \(0\le \ell \le 4\),
Proof
For \(0\le \ell \le 6\), thanks to the Sobolev embedding theorem and Lemma 2.1, we have
According to the fact \(|Z \bar{\rho }|\le C\bar{\rho }\), we deduce from integration by parts that
where we used that \(\frac{14}{22}+(-\frac{1}{2}+\kappa )\frac{4}{11}\le \frac{21}{44}<\frac{1}{2}\) with \(\kappa \in (0, \frac{1}{16})\).
While by using integration by parts again, one can see that
which follows from the fact \(|Z \bar{\rho }|\le C\bar{\rho }\) that
Next, we deal with the last term in the above inequality. In fact, we may get from integration by parts that
which implies
Hence, one has
Inserting (2.4–2.5) into (2.3) ensures that for \(0\le \ell \le 6\)
that is, the inequality (2.1) holds.
The second inequality (2.2) comes from the Sobolev embedding theorem and (2.1):
for \(0\le \ell \le 4\), which ends the proof of Lemma 2.2. \(\square \)
To deal with nonlinear term, we need the following product laws in the spaces \(X^{N}\).
Lemma 2.3
There hold true that
and
Proof
By the Leibnitz formula, one can see that
Now, we focus only on the proof of the most difficulty case: \(|m_1|+|m_2|=12\). The others can be treated by a similar way. In fact, we divide its proof into three cases.
-
Case 1. \(8\le |m_1|\le 12\). By Hölder’s inequality, we prove
$$\begin{aligned} \Vert Z^{m_1}g~ Z^{m_2}f\Vert _{L^2}\le & {} \Vert Z^{m_1}g \Vert _{L^2}\Vert Z^{m_2}f\Vert _{L^\infty } \le C\Vert Z^{m_1}g \Vert _{L^2}\Vert Z^{m_2}f \Vert _{L^\infty _{x_3}(H^2_h)}\\\le & {} C\Vert g\Vert _{X^{12}} \sum _{|\ell |\le 6}\Vert Z^\ell f\Vert _{L^\infty _{x_3}(L^2_h)}, \end{aligned}$$where we used \(|m_2|+2\le 6\).
-
Case 2. \(6\le |m_1|\le 7\). Thanks to the Sobolev embedding theorem and Hölder’s inequality, one can obtain that
$$\begin{aligned} \Vert Z^{m_1}g~ Z^{m_2}f\Vert _{L^2}\le & {} \Vert Z^{m_1}g \Vert _{L^2_{x_3}(L^\infty _{h})}\Vert Z^{m_2}f\Vert _{L^\infty _{x_3}(L^2_h)}\\\le & {} C\Vert g\Vert _{X^{12}} \sum _{|\ell |\le 6}\Vert Z^\ell f\Vert _{L^\infty _{x_3}(L^2_h)}. \end{aligned}$$ -
Case 3. \( 0\le |m_1|\le 5\). For this case, we only need to exchange the position of f and g and apply the same argument as in the above two cases to get that
$$\begin{aligned} \Vert Z^{m_1}g~ Z^{m_2}f\Vert _{L^2}\le & {} C\Vert f\Vert _{X^{12}} \sum _{|\ell |\le 6}\Vert Z^\ell g\Vert _{L^\infty _{x_3}(L^2_h)}. \end{aligned}$$
Collecting all the above cases together, we obtain
which follows (2.6).
Next, since we the highest order in (2.7) is 11, we may readily verify (2.7) by the same process above, which ends the proof of Lemma 2.3. \(\square \)
We introduce a new quantity \(\mathfrak { D}(v)(t)\) which controls \(\Vert \nabla v\Vert _{L^\infty }\) according to Lemma 2.2:
In what follows, \(\mathcal {P}(\cdot )\) stands for some polynomial function which coefficients may depend on initial data.
Lemma 2.4
Assume that
Then there hold that for any \(t \in [0, T]\)
and
where the constant C depends on \(\Vert \xi _0\Vert _{\mathcal {F}_{\kappa }}\) and \(\sigma _0\).
Proof
Before giving the proof of this lemma, we state some estimates as preliminary.
First, taking \(f=g=\nabla v\) in (2.6), we obtain
While by Lemma 2.2, one can prove that
which along with (2.11) ensures (2.9).
Now we are in the position to prove the estimates in terms of \(J\mathcal {A}\) and \(\mathcal {A}\). Notice that
and every entry in \(J\mathcal {A}\) is a linear combination of
Then, thanks to Lemmas 2.2–2.3, (2.12) and Minkowski’s inequality, one has
which proves the first inequality in (2.10).
Similarly, we deduce that
Recalling the definition of J: \(J=\det (\nabla \eta _0+\int _0^t\nabla vds)\), J is a linear combination of the terms
Hence, similar to the proof of the first inequality in (2.10) in terms of \(J\mathcal {A}\), we may obtain
Owing to the fact \(J\ge \sigma _0\) and the formula to the composition of two functions, we obtain
We put \(\Vert \cdot \Vert _{L^\infty _{x_3}(L^2_h)}\) on the highest order term \(Z^{k_j}J\) and put \(\Vert \cdot \Vert _{L^\infty }\) to other lower terms (not more than order 4) with similar process to (2.12). It follows from Lemma 2.2 and (2.15) that
Therefore, due to (2.13) and (2.16), we find
For the high order estimate, similar to the proof of (2.9), by using Lemma 2.2, we achieve
and then
While by virtue of (2.15), (2.18) and Lemma 2.3, we deduce that
and
which completes the proof of Lemma 2.4. \(\square \)
Based on the above lemma, we may get the following estimates.
Lemma 2.5
Under the assumptions in Lemma 2.4, there hold
Proof
We mainly utilize Lemmas 2.3, 2.4 to prove (2.19). So one may focus only on the proof of the first inequality in (2.19), and the proofs of the others are the same as it, whose details will be omitted here.
First, by the definition of \(J\mathcal {A}\), we split \(\sum _{0\le |j|\le 1}\Vert \nabla v ~Z^j(J\mathcal {A})\Vert _{X^{11}}\) into three parts:
For \(I_1\), we have
For \(I_2\), taking \(g=\nabla v\) and \(f=J\mathcal {A}\) in (2.7) in Lemma 2.3 to obtain that
Applying Lemma 2.2 and (2.14), (2.17) in Lemma 2.4 to get
Similarly, we have
Plugging the estimates (2.21)–(2.23) into (2.20), we prove
which ends our proof. \(\square \)
Next we recall a version of Korn’s inequality involving only the deviatoric part \(\mathbb {D}^0\).
Lemma 2.6
(Korn’s lemma, Theorem 1.1 in [5]) Let \(n \ge 3\) and U be a Lipschitz domain in \(\mathbb {R}^n\), then there exists a constant C, independent of f, such that
for all \(f \in H^1(U)\).
3 A priori estimates
In this section, we give a priori estimates of the system (1.18). The main result of the section is as follows:
Proposition 3.1
Assume \((\xi ,\,v)\) is a smooth solution of system (1.18) on \([0,\bar{T}]\) with initial data \((\xi _0,\,v_0) \in \mathcal {F}_{\kappa } \times (X^{12}_{\frac{1}{2}}\cap \,H^1)\) and \(0<2\sigma _0\le J_0\le 3\sigma _0\), and \(\overline{\rho }\) satisfies (1.11)–(1.13). Then, there exists a positive constant \(T\le \bar{T}\) which depends on the initial data such that
Here, we use the bootstrap argument to prove this proposition. Now, we define a T such that there holds that
Before, we give the proof of the proposition, we prove some useful lemmas.
Lemma 3.2
Under the assumption of Proposition 3.1, we have
Proof
It is a direct result from Lemma 2.2 and Lemma 2.4. \(\square \)
Lemma 3.3
Under the assumption of Proposition 3.1, the following holds
Proof
Thanks to Korn’s lemma (Lemma 2.6), we have
For any function f(s), by Lemma 2.1, we have
By scaling, we have
Then (1.12) gives that
Taking \(\varepsilon \) small enough and \(f:=v\), we combine with Lemma 2.6 to get that
For given \(m\in \mathbb {N}^3\): \(1\le |m| \le N\),
which follows from the fact \([\mathbb {D}^0,Z^m]v\sim Z^{m-1}\, \nabla \,v\) that
Therefore, by a standard inductive argument in terms of \(m=0,1,\ldots ,N\) and the definition of space \(X^{N}\), we prove (3.2). \(\square \)
Lemma 3.4
Let the initial flow map \(\eta _0=Id+\xi _0: \Omega \rightarrow \Omega (0)\) satisfy its Jacobian \(2\sigma _0\le \,J_0\le 3\sigma _0\) and \(\xi _0 \in \mathcal {F}_{\kappa }\), and its inverse map \(\eta _0^{-1}: \Omega (0)\rightarrow \Omega \), \(v(x)=\widetilde{u}(\eta _0(x))\) with \(x \in \Omega \) and \(\widetilde{u}(y)=v(\eta _0^{-1}(y))\) with \(y \in \Omega (0)\), then there is a positive constant \(C_1\ge 1\) such that
Proof
First, taking changes of variables \(y=\eta _0(x)\), we have
which along with the assumptions \(2\sigma _0\le \,J_0\le 3\sigma _0\), \(\xi _0 \in \mathcal {F}_{\kappa }\), and (2.2) implies
Similarly, one may readily check
Therefore, we get (3.4), and complete the proof of Lemma 3.4. \(\square \)
Lemma 3.5
Under the assumption of Proposition 3.1, if (3.1) holds, then we have
Moreover, if T is small enough such that \(T^{\frac{1}{2}}\mathcal {P}(\mathfrak {C})<\frac{c_0}{2}\), then we have
Proof
We first to prove the first result. According to the fact
and \(\mathcal {A}_0^{-1}=D\eta _0,\) combining Lemmas 2.2, 3.2 with (3.1) , we have
which imply that
On the other hand, we use (3.1), the coordinate transformation from \(\Omega \) to \(\Omega (0)\) and Lemmas 2.6, 3.4 to get that
where \(\widetilde{u}=v\circ \eta _0^{-1}\). Hence, according to (3.1) and (3.3), we obtain that
which combining with (3.7) gives rise to
which we complete the first result. For the second one, we deduce
here we used (3.1) in the last step and assumption \(\mu >0,~\lambda +\frac{2}{3}\mu \ge 0\). Combining with the first result, we finish this proof. \(\square \)
3.1 Zeroth-order estimate of v
Now, we are in a position to give a priori estimates. First, multiplying by v on the first equation of (1.18) and integrating over \(\Omega \), from the Piola identity (1.21) and boundary conditions, we get the basic energy estimate:
Proposition 3.6
Assume v is a smooth solution of system (1.18) on [0, T]. Then, we have
3.2 First-order estimate of v
Here, to get the higher regularity of the v. We multiply \(\partial _t v\) on the both sides of (1.18) to get that
Proposition 3.7
Assume that (3.1) holds and v is a smooth solution of system (1.18) on [0, T], then there holds that for \(t\in [0, T]\)
Proof
Taking \(L^2\) product with \(\partial _t v\) to the first equation of (1.18) to get that
Due to the Piola identity (1.21) and the boundary condition \(\mathrm {S}_{\mathcal {A}}(v)\cdot \mathcal {N}|_{x_3=1}=0\) and \(v|_{x_3=0}=0\), integration by parts yields
Since \(\mathbb {D}_\mathcal {A}(v)\) and \((\nabla _\mathcal {A}\cdot v)\mathbb {I}\) are symmetric, it implies that
which gives that
To estimate the last two terms of right hand of the above equation, we recall that formula (1.19)–(1.20), Lemmas 2.2 and 3.2 to get that
which implies that
For the pressure term, we notice it contains \(\overline{\rho }^2\). Thus, we have
which implies that for all \(t\in [0, T]\), we have
where we used Lemma 2.4. Thus, by Hölder’s inequality, we get
This ends the proof of Proposition 3.7. \(\square \)
3.3 High-order estimates of v
In this subsection, we use the conormal derivative to get the regularity of the horizontal direction. For this, we recall the conormal Sobolev space with a parameter \(\delta \) introduced by Masmoudi and Rousset [34].
where \(\delta \) is a small positive constant which will be determined later on and \(\alpha \in \mathbb {R}\). In particular, when \(\delta =1\), the spaces \(X^N_{\alpha , \delta }\) and \(\dot{X}^N_{\alpha , \delta }\) will be denoted by \(X^N_{\alpha }\) and \(\dot{X}^N_{\alpha }\) respectively for simplicity.
For \(T>0\), \(\delta >0\), and \(t \in [0, T]\), we define the modified instantaneous energy \(\mathcal {E}_{\delta }(t)\) (in terms to the velocity v)
and the modified dissipation \(\mathcal {D}_{\delta }(t)\)
In particular, if \(\delta =1\), then \(\mathcal {E}_{\delta }(t)\) and \(\mathcal {D}_{\delta }(t)\) become the usual instantaneous energy \(\mathcal {E}(t)\) and the dissipation \(\mathcal {D}(t)\) respectively.
Let’s now state our main results of this subsection:
Proposition 3.8
Assume that (3.1) holds and v is a smooth solution of system (1.18) on [0, T], then it holds that
where the positive constants \(c_0\) and \(C_0\) are independent of \(\delta \), and \(\mathcal {P}(\mathfrak {C})\) may depend on \(\delta \).
Proof
Acting \(Z^m\) on the first equation of (1.18) and taking \(L^2\) inner product with \(\delta ^{2|m|}Z^mv\), then summing \(\sum _{|m|=0}^{12}\), we obtain
with
Estimate of dissipation term. For the dissipation term, by using integration by parts, we split it into three parts:
Next, we deal with the commutators \(I_3\), \(I_4\) and \(I_5\) step by step.
-
\({ {Estimates\, of\, } I_3}\). Thanks to Lemma 3.5, one can see that for any \(m:|m|=0, 1,\ldots ,12\)
$$\begin{aligned}&\int _\Omega J\mathbb {S}_\mathcal {A}(Z^mv): \nabla _{\mathcal {A}}Z^m v \,dx\\&\quad =\int _{\Omega } \left( \frac{\mu }{4}|\mathbb {D}^0_{\mathcal {A}} Z^mv|^2+\frac{\lambda +\frac{2}{3}\mu }{2} |\nabla _\mathcal {A}\cdot Z^mv|^2\right) \,J\,dx\\&\quad \ge c_1\Vert \mathbb {D}^0_{\mathcal {A}} Z^mv\Vert _{L^2}^2 \ge c_1\bigg ((c_0- t^2 \mathcal {P}(\mathfrak {C}))\Vert \nabla (Z^mv)\Vert _{L^2}^2-C_0\Vert \overline{\rho }^\frac{1}{2} Z^mv\Vert _{L^2}^2\bigg ), \end{aligned}$$
which implies
For \(|m|\ge 1\), by a direct calculation, we have
which implies that
Plugging (3.11) into (3.9) shows
-
\({ {Estimates\, of\, } I_4}\). For \(|m|\ge 1,\) by a direct calculation, we have
$$\begin{aligned}{}[Z^m,\mathbb {D}_{\mathcal {A}}]v&= Z^m\Big (\mathcal {A}_{i}^k\partial _kv_j+\mathcal {A}_{j}^k\partial _kv_i \Big )-\Big (\mathcal {A}_{i}^k\partial _k(Z^mv_j)+\mathcal {A}_{j}^k\partial _k(Z^mv_i) \Big )\\&=\mathcal {A}_{i}^k[Z^m,\partial _k]v_j+\mathcal {A}_{j}^k[Z^m,\partial _k]v_i\\&\quad +\sum _{\begin{array}{c} |m_1|+|m_2|=|m|,\\ |m_1|\ge 1 \end{array}}(Z^{m_1} \mathcal {A}_{i}^kZ^{m_2}\partial _kv_j+Z^{m_1}\mathcal {A}_{j}^kZ^{m_2}\partial _kv_i)\\&=m\partial _k \overline{\rho }\mathcal {A}_{i}^3Z^{m-1}\partial _3v_j+m\partial _k \overline{\rho }\mathcal {A}_{j}^3Z^{m-1}\partial _3v_i\\&\quad +\sum _{\begin{array}{c} \begin{array}{c} |m_1|+|m_2|=|m|,\\ |m_1|\ge 1 \end{array} \end{array}}(Z^{m_1} \mathcal {A}_{i}^kZ^{m_2}\partial _kv_j+Z^{m_1}\mathcal {A}_{j}^kZ^{m_2}\partial _kv_i). \end{aligned}$$
By the same argument, we have
Combining the above two estimates, we have
-
\({ {Estimates\, of \,} I_5}\). A direct calculation gives that
$$\begin{aligned}{}[Z^m,\nabla _{J\mathcal {A}}]\cdot \mathbb {S}_{\mathcal {A}}v&=Z^m(J\mathcal {A}_i^k\partial _k(\mathbb {S}_{\mathcal {A}} v)^i)-\partial _k(J\mathcal {A}_i^k(Z^m(\mathbb {S}_{\mathcal {A}} v))^i)\nonumber \\&=\partial _k\Big (Z^m(J\mathcal {A}_i^k(\mathbb {S}_{\mathcal {A}} v)^i)-J\mathcal {A}_i^k(Z^m(\mathbb {S}_{\mathcal {A}} v))^i\Big )+[Z^m,\partial _k](J\mathcal {A}_i^k(\mathbb {S}_{\mathcal {A}} v)^i). \end{aligned}$$(3.12)
For the commutator term, we see
where we used (1.13). Then one has
which combining with Lemma 2.5 follows
Now, we deal with the first term of the right hand of (3.12). By using integration by parts, one has
Because of \(\mathbb {S}_{\mathcal {A}} (v) \mathcal {N}=0\) on the boundary \(\{x_3=1\}\), \(J\mathcal {A}_i^3e_3= \mathcal {N}\), and \(Z_h^m(\mathbb {S}_{\mathcal {A}} v \mathcal {N})=0\) on \(\{x_3=1\}\), the second term on the above equality plus the second term of \(I_5\) is zero:
Hence, all we left is to deal with the commutator
By the same arguments as \(I_4\) and using Lemma 2.2–2.5, we deduce that
Combining all the above estimates, we get that
So far, we obtain
Estimate of \(I_2\). Now, we deal with the pressure.
-
\({ {Estimates\, of\, } I_{22}}\). Since \(Z^m \overline{\rho }^2\sim \overline{\rho }^2\) for any m, we use (3.10) and Lemmas 2.3–2.4 to get
$$\begin{aligned} I_{22} \le \delta (C_0+t^{\frac{1}{2}}\mathcal {P}(\mathfrak {C}))\Vert \nabla v\Vert _{X^{11}_{0, \delta }}. \end{aligned}$$ -
\({ {Estimates\, of\, } I_{21}}\). Because of \(\overline{\rho }|_{x_3=1}=0,\) the boundary terms vanish when we integrate by parts. By the same argument as \(I_5\), it is easy to see \(I_{21}\) is bounded by
$$\begin{aligned} I_{21}\le (C_0+t^{\frac{1}{2}}\mathcal {P}(\mathfrak {C}))(\Vert \nabla v\Vert _{X^{12}_{0, \delta }}+\delta \Vert \nabla v\Vert _{X^{11}_{0, \delta }}). \end{aligned}$$
Combining the two estimates, we get
Estimate of \(I_1\). For \(m\ge 1,\) it holds that
where \(f_k\) are smooth functions which are defined by \(\overline{\rho }\). Thus
From the formula above, \(I_1\) can be regarded as lower term to \(I_2\) plus dissipation term with the highest order 11. Since \(k\le m-1,\) extra \(\delta \) is left. Thus, we have
Collecting all estimates together, we finally obtain
which implies the desired results. \(\square \)
3.4 Estimate for \(\mathfrak {D}(v)\)
To close the energy estimates, all we left is the estimate of \(\mathfrak {D}(v)\) which should be controlled by the energy.
Lemma 3.9
Assume that (3.1) holds. Then there exists \(0<T\le \bar{T}\) and \(\delta _0>0\) which depend on the initial data, \(\sigma _0\) and \(\mathfrak {C}\) such that for any \(t\in [0,T]\) and \(\delta \in (0,\delta _0)\), there holds that
Proof
Here we only need to control the term \(\Vert \overline{\rho }^{\kappa -\frac{1}{2}}\triangle v\Vert _{L^2}\). To do that, we go back to the equation of v. Since
which implies that
Owing to Lemma 2.4, we have
For \(I_2\), by Lemma 2.1, Lemma 2.4 and (3.5)–(3.6), we have
Similarly, by the fact that
and
combine (3.5) with Lemma 3.2 to get
Collecting all above estimates to obtain
Next, we give the relationship between \(\triangle v\) and \(\nabla _{J_0\mathcal {A}_0}\cdot \mathbb {S}_{\mathcal {A}_0}(v)\). It is easy to find that
By Lemma 2.1 and the interpolation inequality, we have
where we use Young inequality in the last step and \(\theta \in (0,1).\)
Taking \(\varepsilon \) small enough and using (3.1), (3.14), (3.15), we have
Combining (3.14) and (3.16), we obtain the desired results. \(\square \)
3.5 Proof of Proposition 3.1
Now, from Propositions 3.7 to 3.8, we obtain that
Now, we give the estimates of J. By the definition of J, we have
which implies that
Then by the Lemma 3.9 and standard bootstrap argument, we finish the proof of Proposition 3.1.
4 Local well-posedness
In this section, we will first give existence and uniqueness of strong solutions of system (1.18), which is motivated by the method in [10]. First, we give some definitions of functional spaces. Given \(T>0,\) let \(\widetilde{Y}_T\) and \({Y}_T\) are defined by
where \(\Vert v\Vert _{\widetilde{Y}_T}^2:=\sup _{t\in [0,T]}\Vert \overline{\rho }^{\frac{1}{2}}v\Vert _{L^2}^2+\Vert v\Vert _{L^2_T (H^1)}^2\) and \(\Vert v\Vert _{Y_T}^2:=\sup _{t\in [0,T]}(\Vert v\Vert _{X^{12}_{\frac{1}{2}}}^2+\Vert \nabla v\Vert _{L^2}^2 )+\Vert \overline{\rho }^{-\frac{1}{2}+\kappa }\triangle v\Vert _{L^2_T(L^2)}^2 +\Vert \nabla v\Vert _{L_T^2(X^{12})}^2+\Vert \overline{\rho }^{\frac{1}{2}}\partial _t v\Vert _{L_T^2(L^2)}^2\).
Now, we define map \(\Theta : Y_T\rightarrow Y_T\) as follows. For any given \(\widetilde{v}\in Y_T,\) \(v:=\Theta (\widetilde{v})\) is the solution of the following linear \(\mathcal {A}\)-equations:
4.1 Existence and uniqueness of the strong solution to (4.1).
Our aim in this subsection is to construct strong solutions to linear \(\mathcal {A}\)-equations (4.1).
Lemma 4.1
Assume that \(\overline{\rho }^{\frac{1}{2}} v_0,\nabla v_0\in L^2\) and \(\widetilde{v}\in Y_T,\) then there exists a positive time \(T_1 \in (0, T]\) such that the system (4.1) has a unique strong solution v with
Moreover, the solution satisfies the following estimate
Proof
We split the proof of the lemma into four steps.
Step 1: Galerkin approximation. We first use Galerkin method to construct approximate solutions of the system (4.1). Let \(\{w_k\}_{k=1}^\infty \) are orthonormal basis of \(H^1(\Omega )\) which satisfy boundary condition \(\mathrm {S}_{\widetilde{\mathcal {A}}}(w_k)~ \widetilde{\mathcal {N}}|_{x_3=1}=0\) and \(w_k|_{x_3=0}=0\) and set approximate solution with the form
which solves the linear system
in the sense of the distribution, where \(d^m_k(0)=\int _{\Omega } v_0 w_k\) for \(k=1,\ldots , m\).
Taking the test function \(\phi =w_\ell \), \(\ell =1,\ldots , m\), from the weak formula of the system (4.2), we obtain the following ordinary differential equations
Notice that the matrix \(\Big (\int _{\Omega }\overline{\rho }w_k w_\ell dx \Big )_{m\times m}\) is invertible for any \(m\ge 1\), and the coefficient \(\int _\Omega \mathbb {S}_{\widetilde{\mathcal {A}}}w_k:\nabla _{\widetilde{J}\widetilde{\mathcal {A}}}w_\ell dx\) (in front of \(d^m_k(t)\)) is continuous in terms of \(t\in [0,T]\) because of \(\widetilde{v}\in Y_T\), we know that (4.3) is a non-generate linear ODE system with continuous coefficients. Due to the classical theory of ODE, we find solutions \(d^m_k(t)\in C^1([0,T]), k=1,\ldots , m\), which means approximate solutions \(v^m(t,x)\) exist and belong to the space \(C^1([0,T],H^1(\Omega )).\)
Step 2: Uniform estimates for \(v^m\). Multiplying \(d^m_\ell (t)\) on the both sides of (4.3) and taking the summation in terms of \(\ell =1,\ldots , m\), one has
Then Lemmas 2.4 and 3.3 give that
for t small enough.
By Gronwall’s inequality, we know there exists \(T_1>0\) independent of m such that
For any test function \(\phi \in C([0, T], H^1)\) with \(\phi |_{x_3=0}=0\) and \(\Vert \phi \Vert _{L^2_TH^1}\le 1,\) owing to the weak formula of the system (4.2), we deduce from (4.5) that
which follows from the dual argument that
Multiplying \(d^m_\ell (t)'\) on the both sides of (4.3) and taking the summation in terms of \(\ell =1,\ldots , m\), we have
Similar estimate in Proposition 3.7 implies that
Since \(\widetilde{v}\in Y_T\), we infer that
and
As a result, we get
Integrating time from 0 to \(T_1\) and using \(\widetilde{v}\in Y_T\) and Lemma 3.9, we obtain
Taking \(T_1\) small enough such that the second term on the right hand side absorbed by the left hand side, we obtain
Combining estimate (4.5), (4.6) and (4.7) together, there holds that
Step 3: Passing to the limit. Since
is uniformly bounded, up to the extraction of a subsequence, we know as \(m\rightarrow \infty \)
By lower semicontinuity and energy estimate (4.8), we use the fact \(\Vert v^m(0)-v_0\Vert _{L^2(\Omega )}\rightarrow 0\) as \(m\rightarrow \infty \) to infer that
and v is a weak solution to the linear \(\mathcal {A}\)-equations (4.1). Moreover, according to (4.10), we may obtain from Aubin–Lions’s lemma [38] that \(v \in C([0, T_1], X^0_{\frac{1}{2}} \cap \, H^1)\).
Step 4: The strong solution. Now, we prove the above weak solution v is a strong one. In fact, for a.e \(t\in [0,T],\) v(t) is a weak solution to the elliptic system in the sense of
for \(\phi \in H^1. \) Since \(\overline{\rho }^{-\frac{1}{2}}\Big (\nabla _{\widetilde{J}\widetilde{\mathcal {A}}}(\overline{\rho }^2\widetilde{J}^{-2})-\overline{\rho }\partial _t v\Big )\in L^2\) for a.e \(t\in [0,T],\) by elliptic regularity theory, we know this system admires a strong solution v solving (4.1) with \(\rho ^{-\frac{1}{2}+\kappa }\triangle v\in L^2([0,T],L^2)\). The uniqueness comes from energy estimates with zero initial data. \(\square \)
4.2 High regularity of v
In this subsection, we prove when \(\widetilde{v}\in Y_T,\) so does \(v:=\Theta (\widetilde{v}).\) It is mainly based on the priori estimates in Sect. 3.
Lemma 4.2
Assume that v is a strong solution obtained in Lemma 4.1 and \(\widetilde{v}\in Y_{T}\) with initial data \(v_0\in Y_0\), then we have \(v\in Y_{T_1}\) and satisfies
where the constant C depends on \(\Vert \widetilde{v}\Vert _{Y_T}\).
Proof
We take \(\widetilde{\mathcal {A}},\widetilde{J}\) instead of \(\mathcal {A}, J\) respectively in those estimates in Propositions 3.7 and 3.8. System (4.1) is a linear system due to \(\widetilde{\mathcal {A}},\widetilde{J}\) are regarded as known quantities, so for small \(T_1>0\), it is easy to arrive at the following estimate:
Passing to the limit, we get the desired results. \(\square \)
Remark 4.3
By Lemma 4.2, we know that \(\Theta : Y_{T_1}\rightarrow Y_{T_1} \) is well-defined.
4.3 Contraction
By Lemmas 4.1 and 4.2, we know that if \(\widetilde{v} \in Y_T\) with \(T>0\) sufficiently small , we can find a unique strong solution of equation (4.1) with regular \(v=\Theta (\widetilde{v})\in Y_T.\) In order to construct the solution to (1.18), we need to construct approximate solutions. The approximate solutions \(\{\xi ^{(n)},\,v^{(n)}\}_{n=1}^{\infty }\) we defined are iterated as follows:
with \(\{\xi ^{(1)},\,v^{(1)}\}\) be the solution of linear equation
where \(\mathcal {A}_0,J_0\) are given by \(\eta _0(x)= x+\xi _0(x)\) and \(\mathcal {N}_0=\partial _1\eta _0\times \partial _2\eta _0\) on \(\{x_3=1\}.\) Since (4.12) is a decouple linear system in terms of \(\xi ^{(n)}\) and \(v^{(n)}\), we need only to solve first \(v^{(n)}\) then \(\xi ^{(n)}\) according to the first equation in (4.12). Notice that (4.13) is linear, the assumption on initial data \( \Vert v_0\Vert _{Y_0}^2:=\Vert v_0\Vert _{X^{12}_{\frac{1}{2}}}^2+\Vert \nabla v_0\Vert _{L^2}^2 \le \frac{M}{2C_0}\) guarantees that \(v^{(1)}\in Y_T\) with bound \(\Vert v^{(1)}\Vert _{Y_T}^2\le M.\) By Lemma 4.2, we obtain \(\{v^{(n)}\}_{n=1}^{\infty }\subset Y_T\) for any \(n\ge 1.\)
Next, our goal in this subsection is to prove sequence \(\{v^{(n)}\}_{n=1}^{\infty }\) is contracted under norm \(\widetilde{Y}_T\).
First of all, we deduce \(\sigma (v^{(n)})\triangleq v^{(n+1)}-v^{(n)}\) satisfies the following equation
Lemma 4.4
Assume that \(\{v^{(n)}\}_{n=1}^{\infty }\) be the solutions of Eq. (4.12) with bound \(\Vert v^{(n)}\Vert _{Y_T}^2\le M\) for each \(n\ge 1.\) It holds that
Moreover, taking T small enough, the sequence \(v^{(n)}\) is a Cauchy sequence in the space \(\widetilde{Y}_T\).
Proof
Taking \(L^2\) inner product between (4.14) and \(\sigma (v^{(n)})\), we obtain
Estimate of dissipation term. Since
and
we get by using integration by parts that
Under the assumption \(\Vert v^{(n)}\Vert _{Y_T}^2\le M\), we have
where we use \(|J^{(n)}|\ge \sigma _0\) and Lemma 3.5 for \(\mathcal {A}^{(n)}.\)
For \(I_1\), owing to
then
Applying Holder inequality and Lemma 3.2 to \(\mathcal {A}^{(n)}\), one has
Similarly, we have
Combining all above estimates, we obtain
Estimate of pressure term. Integrating by parts and using \(\overline{\rho }|_{x_3=1}=0, ~\sigma (v^{(n)})|_{x_3=0}=0\), we prove that
Collecting all above estimates together, we finally obtain
Integrating (4.15) in \(t \in [0, T]\) and taking T small enough, we have
and then
By now, we get that when T takes small enough, then we get
which completes this Lemma. \(\square \)
4.4 Proof of Theorem 1.2.
From Lemma 4.4, we know \(\{v^{(n)}\}_{n=1}^{\infty }\) is Cauchy sequence in the space \(\widetilde{Y}_T.\) So as \(n\rightarrow \infty ,\)
Due to Lemma 4.2 that \(\Vert v^{(n)}\Vert _{Y_T}^2\le M\) uniformly in \(n\ge 1,\) sequence \(\{v^{(n)}\}_{n=1}^{\infty }\) have weakly convergent subsequence. Along with strong convergence (4.17), we infer that as \(n\rightarrow 0\)
So the function v satisfies equation (1.18) in weak sense. On the other hand, lower semicontinuity gives bound \(\Vert v\Vert _{Y_T}^2\le 2M\), and then (1.25) holds. As a result, thanks to Aubin–Lions’s lemma [38], we get that \((v,\,\eta )\in \,C([0, T]; X^{12}_{\frac{1}{2}}\cap H^1(\Omega )) \times C([0, T]; \mathcal {F}_{\kappa }(\Omega ))\) by using a standard procedure (cf. the proof of Theorem 3.5 in [33]), which is a strong solution to (1.18). The uniqueness comes from \(L^2\) energy estimates with zero initial data. More precise, let \((\xi _1,\,v_1)\) and \((\xi _2,\,v_2)\) are solutions to (1.18) with same initial data. The same process in Lemma 4.4 deduce that
which implies \(v_1=v_2\) and then \(\xi _1=\xi _2\) on the time interval [0, T]. Furthermore, applying (4.16) to the system (1.18), we may readily prove that the solution \((v,\,\eta )\in \,C([0, T]; X^{12}_{\frac{1}{2}}\cap H^1(\Omega )) \times C([0, T]; \mathcal {F}_{\kappa }(\Omega ))\) depends continuously on the initial data \((v_0,\,\eta _0)\in \,(X^{12}_{\frac{1}{2}}\cap H^1(\Omega )) \times \mathcal {F}_{\kappa }(\Omega )\). This finishes the proof of Theorem 1.2. \(\square \)
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Acknowledgements
G. Gui is partially supported by the National Natural Science Foundation of China under Grants 11571279 and 11931013. C. Wang is partially supported by NSF of China under Grant 11701016. Y. Wang is partially supported by China Postdoctoral Science Foundation 8206200009.
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Gui, G., Wang, C. & Wang, Y. Local well-posedness of the vacuum free boundary of 3-D compressible Navier–Stokes equations. Calc. Var. 58, 166 (2019). https://doi.org/10.1007/s00526-019-1608-y
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DOI: https://doi.org/10.1007/s00526-019-1608-y