JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2021/63392036339203Research ArticleAnalytical Solution to 1D Compressible Navier-Stokes Equationshttps://orcid.org/0000-0002-8027-3572DouChangshenghttps://orcid.org/0000-0003-1727-3358ZhaoZishuFeichtingerHans G.School of StatisticsCapital University of Economics and BusinessBeijing 100070Chinacueb.edu.cn2021275202120211242021195202127520212021Copyright © 2021 Changsheng Dou and Zishu Zhao.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

There exist complex behavior of the solution to the 1D compressible Navier-Stokes equations in half space. We find an interesting phenomenon on the solution to 1D compressible isentropic Navier-Stokes equations with constant viscosity coefficient on x,t0,+×R+, that is, the solutions to the initial boundary value problem to 1D compressible Navier-Stokes equations in half space can be transformed to the solution to the Riccati differential equation under some suitable conditions.

Top Young Talents of Beijing Gaochuang Project and CUEB’s Fund ProjectBJNSF1182007National Natural Science Foundation of China11671273CUEBQNTD202109
1. Introduction

We consider the 1D compressible Navier-Stokes equations in half space in the following: (1)ρt+ρux=0,x,t0,+×R+,(2)ρut+uux+px=μuxx,x,t0,+×R+,where ρx,t,ux,t stand for the density and velocity of compressible flow. μ is the constant viscosity coefficient. p=pρ means the pressure of the flow. We assume the initial data: (3)ρ,ux,0=ρ0,u0x,and the boundary condition: (4)u0,t=gt,

And let (5)ρ0x>0andρ0xC20,+,gtC10,+,pC10,+.

There is huge literature on the studies of the global existence and large time behavior of solutions to the 1D compressible Navier-Stokes equations. As the viscosity μ is a positive constant and the initial density away from vacuum, Kanel  addressed the problems for sufficiently smooth data, and Serre [2, 3] and Hoff  considered the problems for discontinuous initial data. As viscosity μ depends on density and has a positive constant lower bound,  gave the global well-posedness and large time behavior of solutions to the system without initial vacuum. However, when the initial data admits the presence of vacuum, many papers are related to compressible fluid dynamics . When we get the well-posedness of solutions to the compressible Navier-Stokes equations, the existence of vacuum is a major difficulty. Ding et al.  got the global existence of classical solutions to 1D compressible Navier-Stokes equations in bounded domains, provided that μC20,+ satisfies 0<μ¯μρC1+Pρ. Ye  obtained the global classical large solutions to the Cauchy problem (1) and (2) with the restriction μρ=1+ρβ; 0β<γ. Zhang and Zhu  derived the global existence of classical solution to the initial boundary value problem for the one-dimensional Navier-Stokes equations for viscous compressible and heat-conducting fluids in a bounded domain with the Robin boundary condition on temperature. Li et al.  derive the uniform upper bound of density and the global well-posedness of strong (classical) large solutions to the Cauchy problem with the external force. For two-dimensional case, global well-posedness of classical solutions to the Cauchy problem or periodic domain problem of compressible Navier-Stokes equations with vacuum was obtained in  when the first and second viscosity coefficients are μ and λρ, respectively. Li and Xin  derived the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density, provided the smooth initial data are of small total energy and the viscosity coefficients are two constants.

In this paper, we find an interesting phenomenon on the solution to 1D compressible Navier-Stokes equations (1) and (2) with constant viscosity coefficient, that is, the solutions to the problem (1) and (2) in half space can be transformed to the solution to the Riccati differential equation under some suitable conditions. Before stating the main results, we first denote (6)Ut=1+gt,A1x,t=μρ0x+tρ02x+t2ρ0x+t2ρ03x+t,B1x,t=ρ0x+tρ02x+t,C1x,t=pρ0x+tρ0x+t,Ax,t=ρ0tρ0t+A1x,t,Bx,t=ρ0tB1x,t,Cx,t=C1x,tρ0t.

Theorem 1.

The function ρ,ut,x=ρ0x+t,1+gtρ0t/ρ0x+t1 is the solution to compressible Navier-Stokes equations (1) and (2) with the initial data (3) and boundary condition (4), if and only if gt=Ut1 and Ut satisfies the Riccati differential equation: (7)Ut+QtUt+PtU2t=Rt,where

(8)ρ0x>0andρ0xC20,+,gtC10,+,pC10,+,

There exist three functions Qt, Pt, and Rt which only depend on t such that (9)Qt=ρ0tρt+A1x,t,Pt=ρ0tB1x,t,Rt=C1x,tρ0t,for x,t0,+×0,+.

Theorem 2.

If (10)2Q+Q22QPP2PP+3PP2+4Rt>0,then we have the global existence of (7).

Remark 3.

In Theorem 1, we do not know whether the solution UT of Ricatti equation exist globally, because the existence of general solution to Riccati equation is an open problem. If we add the condition of Pt, Qt, and Rt, such as (25) and (26), the global existence for (7) can be obtained, which can be seen in Theorem 2.

Remark 4.

In Theorem 1, the initial value ρ0x,u0x can be bounded in Sobolev space or not bounded in Sobolev space, which is determined by the given specific function for the initial value.

Bounded case. Furthermore, if assumed that (11)ρ0xL10,+C,ρ00=0,and gt=0, then we have the initial energy ρxu02xL10,+. In fact, (12)0ρ0xu02xdx=0ρ0x1+g0ρ00ρ0x12xdx=0ρ0xdxC.

As a result of basic energy estimate, we easily get (13)0ρx,tux,t2dxC.

Unbounded case. We can see the boundary condition of velocity gt0.

2. The Proof of Main ResultProof of Theorem 1.

If we have the analytical function: (14)ρt,x=ρ0x+t,we will get (15)ux,t=1+u0,texp0xρ0y+tρ0y+tdy1=1+gtρ0tρ0x,t1,though the equation (16)ρ0x+t+uρ0x+t+ρ0x+tux=0,and the boundary condition (4).

So, the derivatives of ux,t are (17)utx,t=gtρ0tρ0x+t+1+gtρ0tρ0x+tρ02x+t,(18)uxx,t=1+gtρ0tρ0x+tρ02x+t,(19)uxxx,t=1+gtρ0tρ02x+tρ0x+t2ρ0tρ0x+tρ0x+t2ρ03x+t,

Substituting (18) and (19) into the moment equation, we obtain (20)ρ0x+tgtρ0tρ0x+t+1+gtρ0ρ0x+tρ0tρ0x+tρ02x+t1+gtρ0tρ0x+t11+gtρ0tρ0x+tρ02x+t+pρ0x+tρ0x+t=μ1+gtρ0tρ0x+tρ0x+t2ρ0tρ0x+t2ρ03x+t,i.e., (21)gt+1+gtρ0tρ0t+μρ0x+tρ02x+t2ρ0x+t2ρ03x+t1+gt2ρ0tρ0x+tρ0x+tρ0x+t=pρ0x+tρ0tρ0x+t.

If Ut=1+gt, then Ut satisfies (22)Ut+Utρ0tρ0t+μρ0x+tρ02x+t2ρ0x+t2ρ03x+tU2ρ0tρ0x+tρ0x+tρ0x+t=pρ0x+tρ0tρ0x+t.and ρ0x+t/ρ02x+t2ρ0x+t2/ρ03x+t,ρ0x+t/ρ02x+t and pρ0x+tρ0x+t do not depend on the spatial variable x.

Therefore, Ut is the solution to the Riccati differential equation (7) with the conditions (8) and (9).

If gt=Ut1, Ut satisfies (7), and the conditions (8) and (9) hold; it is easy to get (23)ρx+tt+ρx+t1+gtρ0tρ0x+t1x=0,ρx+t1+gtρ0tρ0x+t1t+1+gtρ0tρ0x+t11+gtρ0tρ0x+t1x+ppx+tρx+t=μ1+gtρ0tρ0x+txx.

So, ρ,ut,x=ρ0x+t,1+gtρ0t/ρ0x+t1 is the solution to compressible Navier-Stokes equations (1) and (2) with the initial data (3) and boundary condition (4).

Since Riccati put forward the Riccati equation in the seventeenth century, there has been no general solution for it for more than 300 years. Although there are many special solutions, none of them can fundamentally solve this equation. Here, we give the global existence for Riccati equation (7) under some condition of Pt, Qt, and Rt, motivated by the results in the reference [26, 27].

Proof of Theorem 2.

By taking Wt=PtUt, equation (7) becomes (24)Wt=W2tftW+Rt,where ft=QtPt/Pt. Due to ρ0C20,+ and (10), we have (25)ftC10,+,RtC10,+,(26)12ft+14f2t+Rt>0.

With (25) and (26), we can obtain the global existence of the Riccati equation (7), according to [26, 27].

3. Example

In this section, we give some examples. First of all, it is easy to check that.

Example 1.

Suppose the initial data ρ0;u0 are both constants and the pressure pρ=ργforanyγ>0, we can get that the solution to compressible Navier-Stokes equations (1) and (2) satisfy the result of Theorem 1.

Specially, we can deduce the following interesting example if some nonphysical condition is given.

Example 2.

Assume pρ=ρ1, and suppose that (27)ρ0x=1x+1,u0x=2c0+12c01x+22c01c012.

Then, we can get the nontrivial analytical solution to the compressible Navier-Stokes equations (1) and (2) (28)ρx,t=1x+t+1,ux,t=2c0e2t+12c0e2t1x+t+11.

Moreover, we can get the particle path of compressible flow (29)xt=x0+12c01et2c0e2t1t1,where x0 stands for the initial position of the particle.

Proof.

From (27), we have the initial data (30)ρx,0=ρ0x=1x+1,ux,0=u0x=2c0+12c01x+22c01,and the compatibility condition (31)g0=u0,tt=0=u0,0=u00=22c01.

By the initial data, we have (32)A1x,t=μρ0x+tρ02x+t2ρ0x+t2ρ03x+t=0,B1x,t=ρ0x+tρ02x+t=1,C1x,t=pρ0x+tρ0x+t=1.

Consequently, (33)Qt=11+t,Pt11+t,Rt=1+t.

Due to the variable substitution (34)Ut=1+gt,we obtain (35)Ut11+tUt+11+tU2t=1+t.

Let (36)Vt=Utt+1,then we have that Vt satisfies (37)Vt=11+t2Vt11+tV2t.

The above equation (37) divided by 1/V2, we get (38)1Vt=211+t1Vt+11+t1Vt2.

By the method of constant variation and the compatibility condition g0=u00=2/2c01, we arrive at (39)Vt=21+t2c0e2t1.

Combining the variation substitutions (34) and (36), we get (40)gt=21+t2c0e2t1+t.

From the result of Theorem 1, we finally obtain (28).

Due to the particle path satisfies xt=ux,t, we have, from (28), (41)xt=2c0e2t+12c0e2t1x+t+11.

Direct calculation gives (42)xt=x0+1expt+02t12c0es1dst1,that is the function (29).☐

Remark 5.

In the theorem 1, we can get p>1,(43)ρ0xLp0,+=01x+1pdx1,ρ0x,tLp0,+=01x+t+1pdx11+tp1/p.

However, ρ0xL10,+ is not bounded, and it is hard to get the boundedness of ρ0u02L10,+ and ρu2L10,+ for the given initial data condition (27).

If c0=1/2, then the initial data of velocity u0x=1. Then, the solution of compressible Navier-Stokes equation (1) and (2) with the pressure pρ=ρ1 can be expressed as (44)ρx,t=1x+t+1,ux,t=e2t1e2t+1x+t+11.

And the particle path of compressible flow is (45)xt=x0+12ete2t+1t1.

Remark 6.

The functions ρx,t=1/x+t+1,ux,t=2c0e2t+1/2c0e2t1x+t+11 are also the solution to the following Euler equations: (46)ρt+ρux=0x,t0,+×R+,ρut+uux+px=0,x,t0,+×R+,with the assumption of the pressure and initial data. (47)pρ=ρ1,ρ0x=1x+1,u0x=2c0+12c01x+22c01c012,for the second order derivative of ux,t with the spatial variable x is 0.

Data Availability

All references in this paper can be found on the Web of Science; this manuscript mainly gets some interesting theorems which need serious proof, and there is no data analysis in this paper.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.

Acknowledgments

This work was supported by Special Fund for Fundamental Scientific Research of the Beijing Colleges in CUEB (Grant no. QNTD202109), NSFC (Grant no. 11671273), BJNSF (Grant no. 1182007), and Top Young Talents of Beijing Gaochuang Project and CUEB’s Fund Project for reserved discipline leader.

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