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Regularity for compressible isentropic Navier-Stokes equations with cylinder symmetry
Journal of Inequalities and Applications volume 2016, Article number: 117 (2016)
Abstract
This paper is concerned with the regularity of the global solutions in \(H^{4}\) to the compressible isentropic Navier-Stokes equations with the cylinder symmetry in \(R^{3}\). Such a circular coaxial cylinder symmetric domain is an unbounded domain, but we assume that the corresponding solution depends only on one radial variable, r in \(G=\{r\in R^{+},0< a\leq r\leq b<+\infty\}\), in which the related domain G is a bounded domain. Some new ideas and more delicate estimates are introduced to prove these results.
1 Introduction
The compressible isentropic Navier-Stokes equations with density-dependent viscosity coefficients can be written for \(t>0\) as
where \(\rho({\mathbf{x}},t)\), \({\mathbf{U}}({\mathbf{x}},t)\) and \(P(\rho)=\rho^{\gamma}\) (\(\gamma>1\)) stand for the fluid density, velocity, and pressure, respectively, and
is the strain tensor and \(\mu(\rho)\), \(\lambda(\rho)\) are the Lamé viscosity coefficients satisfying
In this paper we establish the regularity of global solutions to the compressible Navier-Stokes equations with cylinder symmetry in \(R^{3}\). We will pay attention to the flows between two circular coaxial cylinders. We assume that the corresponding solutions depend only on the radial variable \(r\in G=\{r|0< a\leq r\leq b<+\infty\}\) and the time variable \(t\in[0,T]\). For simplicity, we will take \(\mu(\rho )=\rho^{\alpha}\) and \(\lambda(\rho)=\rho\mu'(\rho)-\mu(\rho)=(\alpha-1)\rho ^{\alpha}\) with \(\alpha>{1/2}\) and \(D({\mathbf{U}})=\nabla{\mathbf {U}}\). Then system (1.1)-(1.2) reduce to the following form:
We consider the initial boundary value problem (1.5)-(1.8) subject to the following initial and boundary conditions:
First we find it convenient to transfer problems (1.5)-(1.10) into that in Lagrangian coordinates and present the desired results. It is well known that Eulerian coordinates \((r,t)\) are connected to the Lagrangian coordinates \((\xi,t)\) by the following relation:
where \(\tilde{u}(\xi,t)=u(r(\xi,t),t)\) and
It follows from (1.5) and the boundary condition (1.10) that
Thus,
and G is transformed to \(\Omega=(0,1)\) with
Moreover, we have
For a function \(\phi(r,t)\), if we write \(\tilde{\phi}(\xi,t)=\phi (r(\xi,t),t)\), by virtue of (1.11), (1.16), and the chain rule, we have
In the following, without danger of confusion we denote \((\tilde{\rho},\tilde{u},\tilde{v},\tilde{w})\) still by \((\rho,u, v,w)\) and \((\xi,t)\) by \((x,t)\). Therefore, (1.5)-(1.8) in the Eulerian coordinates can be written in the Lagrangian coordinates in the new variables \((x,t)\) as follows (see also [1]):
subject to the following initial and boundary conditions:
where \(r(x,t)\) is determined by
The compressible Navier-Stokes system has been noticed academically by physicists and mathematicians for a relatively long time. We are interested in the case that the viscosity is density-dependent. Now let us first recall the related results in this direction. In the one-dimensional case, for the initial boundary value problems in a bounded domain, there have been many works (see, e.g., [2–11]) on the existence, uniqueness, and asymptotic behavior of weak solutions, based on the initial finite mass and the flow density being connected with the infinite vacuum either continuously or by a jump discontinuity. For the one-dimensional Cauchy problem, see, e.g., [12] and references therein for the results on global existence based on the density-dependent viscosity.
In two or three dimensions, the global existence and large time behavior of solution to compressible Navier-Stokes equations (1.1)-(1.2) with constant viscosity and density-dependent viscosity have been investigated for initial boundary value problems, we refer the reader to [1, 13–27] and references therein. Among them, Bresch and Desjardins [13, 14] obtained the global existence of 2D shallow water equations. For the spherically symmetric problem to a three-dimensional compressible isentropic Navier-Stokes problem, Guo et al. [19] analyzed the structure of the solution; Huang et al. [28] studied the global well-posedness of the classical solutions with large oscillations and vacuum; Lian et al. [22] obtained the global existence of spherically symmetric solution for the exterior problem and the initial boundary value problem; Zhang and Fang [27] investigated the global existence and uniqueness of the weak solution without a solid core. For the cylindrically symmetric problem to the three-dimensional compressible Navier-Stokes equations, when the viscosity coefficients are both constants, the uniqueness of the weak solutions was proved in [17, 18], the global existence of isentropic compressible cylindrically symmetric solution was established in [29]; this result was later generalized to the nonisentropic case in [21]. Recently, Cui and Yao [15] proved the asymptotic behavior of a compressible pth power Newtonian fluid with cylinder symmetry; Qin [24] established the exponential stability in \(H^{1}\) and \(H^{2}\) for an ideal fluid. Later on, Qin and Jiang [25] proved the global existence and the exponential stability in \(H^{4}\). Jiang and Zhang [21] established a boundary layer effect and the convergence rate as the shear viscosity μ goes to zero. When the viscosity coefficient \(\mu(\rho)\) is density-dependent and \(\lambda(\rho)\) is a positive constant, the global existence was obtained in [26]. When viscosity coefficients μ and λ are density-dependent, Liu and Lian [1] established the global existence and asymptotic behavior of cylindrically symmetric solutions, however, there is no result on the regularity for this system.
It is noticed that the above analysis concerns the existence of solution in \(H^{1}[0,1]\), the regularity in \(H^{4}[0,1]\) has never been investigated for the three-dimensional isentropic compressible Navier-Stokes equations. Therefore, we continue the work by Liu and Lian [1] and study the regularity of the solutions in \(H^{4}\). In order to obtain a higher regularity of global strong solutions, there are many complicated estimates on higher derivations of the solution involved; this is our difficulty. To overcome this difficulty, we shall use some proper embedding theorems, and the interpolation techniques as well as many delicate estimates.
The notation in this paper will be as follows: \(L^{\bar{p}}\), \(1\leq\bar{p}\leq+\infty\), \(W^{m,\bar{p}}\), \(m\in N\), \(H^{1}=W^{1,2}\), \(H_{0}^{1}=W_{0}^{1,2}\) denote the usual (Sobolev) spaces on \([0, 1]\). In addition, \(\Vert \cdot \Vert _{B}\) denotes the norm in the space B; we also put \(\Vert \cdot \Vert = \Vert \cdot \Vert _{L^{2}}\). Subscripts t and x denote the (partial) derivatives with respect to t and x, respectively. We use \(C_{i}\) (\(i=1,2,4\)) to denote the generic positive constant depending only on the \(H^{i}\) norm of the initial data \((\rho_{0},u_{0},v_{0},w_{0})\) and the variable t.
Before stating the main result, we assume the initial data
with \(\rho_{0}>0\) and \(\bar{\rho}=\frac{1}{b-a}\int_{a}^{b}\rho_{0}r\,dr\), and we define
Now we are in a position to state our main results.
Theorem 1.1
Let \(\gamma>1\), \(\alpha>1/2\). Assume that the initial data satisfies (1.26) and \(H_{0}(H_{0}+H_{1})< a^{2}\alpha^{2}(2\alpha -1)^{-2}\bar{\rho}^{\gamma+2\alpha-1}\). Then there exists a unique generalized global solution \((\rho(t),u(t),v(t),w(t))\in(H^{4}[0,1])^{4}\) to the problem (1.19)-(1.24) verifying that, for \(T>0\),
Corollary 1.1
Under the assumptions of Theorem 1.1, (1.27)-(1.28) implies \((\rho(t),u(t), v(t),w(t))\) is the classical solution verifying, for any \(t>0\),
The rest of the paper is arranged as follows. Section 2 is concerned with the proof of the regularity for a cylindrically symmetric solution to the compressible Navier-Stokes problem in detail.
2 Proof of Theorem 1.1
We will complete the proof of Theorem 1.1 and assume that the assumptions in Theorem 1.1 are valid. We begin with the following lemma.
Lemma 2.1
Under the assumptions in Theorem 1.1, there exist positive constants \(\rho_{*}>0\) and \(\rho^{*}>0\) with \(\rho_{*}<\bar{\rho}<\rho^{*}\) so that the unique global solution \((\rho(t),u(t),v(t),w(t))\) to problem (1.19)-(1.24) exists and satisfies, for any \(T>0\),
where ρ̄ is the same as in Theorem 1.1.
Proof
Estimates (2.1)-(2.4) were obtained in Ref. [1], the proof is complete. □
Lemma 2.2
Under the assumptions in Theorem 1.1, the following estimate holds for any \(T>0\):
Proof
Differentiating (1.20) with respect to x, exploiting (1.19), we have
which gives
with
Multiplying (2.7) by \(\rho^{\alpha-1}\rho_{xx}\), integrating the result over \([0,1]\), we deduce
which, by Young’s inequality and the interpolation inequality, implies
Integrating (2.8) with respect to t over \([0,T]\), using initial condition (1.26) and Lemma 2.1, we derive
which, by virtue of Gronwall’s inequality, gives (2.5). The proof is complete. □
Lemma 2.3
Under the assumptions in Theorem 1.1, the following estimates hold for any \(T>0\):
Proof
We infer from (1.20)-(1.22) and Lemmas 2.1-2.2 that
Differentiating (1.20)-(1.22) with respect to x, respectively, and exploiting Lemmas 2.1-2.2, we have
or
Differentiating (1.20)-(1.22) with respect to x twice, respectively, using Lemmas 2.1-2.2, we get
or
Differentiating (1.20)-(1.22) with respect to t, respectively, we deduce
Now differentiating (1.20) with respect to t twice, multiplying the resulting equation by \((\frac{u}{r})_{tt}\) in \(L^{2}[0,1]\), and using integration by parts and (1.19), we conclude
Employing Lemmas 2.1-2.2, (2.27)-(2.28), (1.19), and the interpolation inequality, we get, for any small \(\varepsilon\in(0,1)\),
Integrating (2.30) with respect to t, applying Lemmas 2.1-2.2, initial condition (1.26), and (2.31)-(2.33), we obtain (2.9).
Differentiating (1.21) with respect to t twice, multiplying the resulting equation by \((\frac{v}{r})_{tt}\) in \(L^{2}[0,1]\), and using integration by parts, we have
where
Using the interpolation inequality, Lemmas 2.1-2.2 and (2.27)-(2.28), we obtain for \(\epsilon\in(0,1)\),
Integrating (2.34) with respect to t, using Lemmas 2.1-2.2 and (2.35)-(2.36), and picking ϵ small enough, we conclude (2.10).
Similarly, differentiating (1.22) with respect to t twice, multiplying the resulting equation by \(w_{tt}\) in \(L^{2}[0,1]\), and using integration by parts, we deduce
Integrating (2.37) with respect to t, using Lemmas 2.1-2.2, we derive the estimate (2.11). The proof is complete. □
Lemma 2.4
Under the assumptions in Theorem 1.1, the following estimates hold for any \(T>0\) and \(\varepsilon\in(0,1)\):
Proof
Differentiating (1.20) with respect to t and x, then multiplying the result by \((\frac{u}{r})_{tx}\) in \(L^{2}[0,1]\), and integrating by parts, we deduce that
where
Now employing Lemmas 2.1-2.2 and the interpolation inequality, using Young’s inequality several times, we have, for \(\varepsilon\in(0,1)\),
On the other hand, we differentiate (1.20)-(1.22) with respect to x and t, and use Lemmas 2.1-2.2 and (2.12)-(2.29) to conclude
Integrating (2.41) with respect to t, using (2.42)-(2.45) and Lemmas 2.1-2.2, we obtain (2.38).
Analogously, differentiating (1.21) with respect to t and x, then multiplying the resultant by \((\frac{v}{r})_{tx}\) in \(L^{2}[0,1]\), and integrating by parts, we deduce that
where
Now we apply the interpolation inequality and Young’s inequality to estimate \(E_{0}(x,t)\), \(E_{1}(t)\), \(E_{2}(t)\) for any \(\varepsilon>0\),
Inserting (2.46) into (2.49), integrating (2.48) with respect to t, and using (2.49)-(2.51), we can get (2.39).
Differentiating (1.22) with respect to x and t, multiplying the resulting equation by \(w_{tx}\) in \(L^{2}[0,1]\), integrating by parts, we have
Employing Lemmas 2.1-2.2, the interpolation inequality and (2.47), we infer for any \(\varepsilon\in(0,1)\) that
Integrating (2.52) with respect to t, picking ε small enough, using Lemmas 2.1-2.3 and (2.53)-(2.54), we derive the estimate (2.40). The proof is complete. □
Lemma 2.5
Under the assumptions in Theorem 1.1, the following estimates hold for any \(T>0\):
Proof
Adding (2.38) and (2.39), using (2.9)-(2.10) and picking ε small enough, we easily obtain (2.55)-(2.56).
Differentiating (2.7) with respect to x, we have
where
An easy calculation with the interpolation inequality, Lemmas 2.1-2.2, and (2.55)-(2.56) gives
which, along with Lemmas 2.1-2.2 and (2.55)-(2.56), implies
Multiplying (2.58) by \(\rho^{\alpha-1}\rho_{xxx}\) in \(L^{2}[0,1]\), we deduce
which implies
Integrating (2.61) with respect to t, using (2.59), we conclude
which, by virtue of Gronwall’s inequality and (2.60), gives
By (2.18)-(2.20), (2.55)-(2.56), and Lemmas 2.1-2.4, we conclude
By virtue of (2.24)-(2.26), (2.55)-(2.56), (2.62), and Lemmas 2.1-2.4, we can get
which, along with (2.62)-(2.63), gives (2.57). The proof is complete. □
Lemma 2.6
Under the assumptions in Theorem 1.1, the following estimates hold for any \(T>0\):
Proof
Differentiating (1.20)-(1.22) with respect to t, respectively, we deduce
By virtue of Lemmas 2.1-2.5 and estimates (2.67)-(2.69), we conclude (2.64).
Differentiating (2.58) with respect to x, we have
where
and
Using the interpolation inequality, and the embedding theorem, Lemmas 2.1-2.5, we can deduce that
Inserting (2.45) into (2.71), and integrating (2.71) with respect to t over \([0,T]\), using Lemmas 2.1-2.5, we have
Multiplying (2.70) by \(\rho^{\alpha-1}\rho_{xxxx}\) in \(L^{2}[0,1]\), we can get
which implies
Integrating (2.74) with respect to t over \([0,T]\), using (2.72), we conclude
which, by virtue of Gronwall’s inequality, gives
Thus, we can obtain (2.65) by virtue of (2.75)-(2.76).
By (2.24)-(2.26), (2.64)-(2.65), (2.45)-(2.47), and Lemmas 2.1-2.5, we deduce that
On the other hand, we differentiate (1.20)-(1.22) with respect to x three times, use Lemmas 2.1-2.5 and (2.64)-(2.65) to conclude, for any \(t\in[0,T]\),
Thus we conclude from (2.77)-(2.80), (2.64)-(2.65), and Lemmas 2.1-2.5 that
which, combined with (2.77), implies (2.66). The proof is complete. □
Proof of Theorem 1.1.
Applying Lemmas 2.1-2.6, we readily get estimate (1.27)-(1.30) and complete the proof of Theorem 1.1. □
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Acknowledgements
This research was supported in part by NSFC (No. 11501199) and the Natural Science Foundation of Henan Province (No. 14B110037).
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Huang, L., Lian, R. Regularity for compressible isentropic Navier-Stokes equations with cylinder symmetry. J Inequal Appl 2016, 117 (2016). https://doi.org/10.1186/s13660-016-1055-7
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DOI: https://doi.org/10.1186/s13660-016-1055-7