# Analytical Solution to 1D Compressible Navier-Stokes Equations

Authors: Changsheng Dou; Zishu Zhao
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

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 onx,t∈0,+∞×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.


---

## Body

## 1. Introduction

We consider the 1D compressible Navier-Stokes equations in half space in the following:(1)ρt+ρux=0,x,t∈0,+∞×R+,(2)ρut+uux+px=μuxx,x,t∈0,+∞×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>0andρ0x∈C20,+∞,gt∈C10,+∞,p⋅∈C10,+∞.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 viscosityμ is a positive constant and the initial density away from vacuum, Kanel [1] addressed the problems for sufficiently smooth data, and Serre [2, 3] and Hoff [4] considered the problems for discontinuous initial data. As viscosity μ depends on density and has a positive constant lower bound, [5–8] 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 [9–17]. 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. [18] got the global existence of classical solutions to 1D compressible Navier-Stokes equations in bounded domains, provided that μ∈C20,+∞ satisfies 0<μ¯≤μρ≤C1+Pρ. Ye [19] obtained the global classical large solutions to the Cauchy problem (1) and (2) with the restriction μρ=1+ρβ; 0≤β<γ. Zhang and Zhu [20] 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. [21] 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 [22–24] when the first and second viscosity coefficients are μ and λρ, respectively. Li and Xin [25] 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=μρ0″x+tρ02x+t−2ρ0′x+t2ρ03x+t,B1x,t=ρ0′x+tρ02x+t,C1x,t=p′ρ0x+tρ0′x+t,Ax,t=ρ0′tρ0t+A1x,t,Bx,t=−ρ0tB1x,t,Cx,t=−C1x,tρ0t.Theorem 1.
The functionρ,ut,x=ρ0x+t,1+gtρ0t/ρ0x+t−1 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=Ut−1 and Ut satisfies the Riccati differential equation:
(7)U′t+QtUt+PtU2t=Rt,where
(1)
(8)ρ0x>0andρ0x∈C20,+∞,gt∈C10,+∞,p⋅∈C10,+∞,(2)
There exist three functionsQt, Pt, and Rt which only depend on t such that
(9)Qt=ρ0′tρt+A1x,t,Pt=−ρ0tB1x,t,Rt=−C1x,tρ0t,for x,t∈0,+∞×0,+∞.Theorem 2.
If(10)2Q′+Q2−2QP′P−2P″P+3P′P2+4Rt>0,then we have the global existence of (7).Remark 3.
In Theorem1, 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 Theorem1, 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.
(1)
Bounded case. Furthermore, if assumed that(11)ρ0xL10,+∞≤C,ρ00=0,and gt=0, then we have the initial energy ρxu02x∈L10,+∞. In fact,
(12)∫0∞ρ0xu02xdx=∫0∞ρ0x1+g0ρ00ρ0x−12xdx=∫0∞ρ0xdx≤C.
As a result of basic energy estimate, we easily get(13)∫0∞ρx,tux,t2dx≤C.(2)
Unbounded case. We can see the boundary condition of velocitygt≠0.


## 2. The Proof of Main Result

Proof of Theorem 1.
If we have the analytical function:(14)ρt,x=ρ0x+t,we will get
(15)ux,t=1+u0,texp−∫0xρ0′y+tρ0y+tdy−1=1+gtρ0tρ0x,t−1,though the equation
(16)ρ0′x+t+uρ0′x+t+ρ0x+tux=0,and the boundary condition (4).
So, the derivatives ofux,t are
(17)utx,t=g′tρ0tρ0x+t+1+gtρ0′tρ0x+tρ02x+t,(18)uxx,t=−1+gtρ0tρ0′x+tρ02x+t,(19)uxxx,t=−1+gtρ0tρ02x+tρ0″x+t−2ρ0tρ0x+tρ0′x+t2ρ03x+t,
Substituting (18) and (19) into the moment equation, we obtain
(20)ρ0x+tg′tρ0tρ0x+t+1+gtρ0′ρ0x+t−ρ0tρ0′x+tρ02x+t−1+gtρ0tρ0x+t−11+gtρ0tρ0′x+tρ02x+t+p′ρ0x+tρ0′x+t=−μ1+gtρ0tρ0x+tρ0″x+t−2ρ0tρ0′x+t2ρ03x+t,i.e.,
(21)g′t+1+gtρ0′tρ0t+μρ0″x+tρ02x+t−2ρ0′x+t2ρ03x+t−1+gt2ρ0tρ0x+tρ0′x+tρ0x+t=−p′ρ0x+tρ0tρ0′x+t.
IfUt=1+gt, then Ut satisfies
(22)U′t+Utρ0′tρ0t+μρ0″x+tρ02x+t−2ρ0′x+t2ρ03x+t−U2ρ0tρ0x+tρ0′x+tρ0x+t=−p′ρ0x+tρ0tρ0′x+t.and ρ0″x+t/ρ02x+t−2ρ0′x+t2/ρ03x+t,ρ0′x+t/ρ02x+t and p′ρ0x+tρ0′x+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).
Ifgt=Ut−1, Ut satisfies (7), and the conditions (8) and (9) hold; it is easy to get
(23)ρx+tt+ρx+t1+gtρ0tρ0x+t−1x=0,ρx+t1+gtρ0tρ0x+t−1t+1+gtρ0tρ0x+t−11+gtρ0tρ0x+t−1x+p′px+tρ′x+t=μ1+gtρ0tρ0x+txx.
So,ρ,ut,x=ρ0x+t,1+gtρ0t/ρ0x+t−1 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 takingWt=PtUt, equation (7) becomes
(24)W′t=−W2t−ftW+Rt,where ft=Qt−P′t/Pt. Due to ρ0∈C20,+∞ and (10), we have
(25)ft∈C10,+∞,Rt∈C10,+∞,(26)12f′t+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 dataρ0;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.
Assumepρ=−ρ−1, and suppose that
(27)ρ0x=1x+1,u0x=2c0+12c0−1x+22c0−1c0≠12.
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+12c0e2t−1x+t+1−1.
Moreover, we can get the particle path of compressible flow(29)xt=x0+12c0−1e−t2c0e2t−1−t−1,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+12c0−1x+22c0−1,and the compatibility condition
(31)g0=u0,tt=0=u0,0=u00=22c0−1.
By the initial data, we have(32)A1x,t=μρ0″x+tρ02x+t−2ρ0′x+t2ρ03x+t=0,B1x,t=ρ0′x+tρ02x+t=−1,C1x,t=p′ρ0x+tρ0′x+t=−1.
Consequently,(33)Qt=−11+t,Pt11+t,Rt=1+t.
Due to the variable substitution(34)Ut=1+gt,we obtain
(35)U′t−11+tUt+11+tU2t=1+t.
Let(36)Vt=Ut−t+1,then we have that Vt satisfies
(37)V′t=11+t−2Vt−11+tV2t.
The above equation (37) divided by −1/V2, we get
(38)1Vt′=2−11+t1Vt+11+t1Vt2.
By the method of constant variation and the compatibility conditiong0=u00=2/2c0−1, we arrive at
(39)Vt=21+t2c0e2t−1.
Combining the variation substitutions (34) and (36), we get
(40)gt=21+t2c0e2t−1+t.
From the result of Theorem1, we finally obtain (28).
Due to the particle path satisfiesx′t=ux,t, we have, from (28),
(41)x′t=2c0e2t+12c0e2t−1x+t+1−1.
Direct calculation gives(42)xt=x0+1expt+∫02t12c0es−1ds−t−1,that is the function (29).☐Remark 5.
In the theorem1, we can get ∀p>1,(43)ρ0xLp0,+∞=∫0∞1x+1pdx≤1,ρ0x,tLp0,+∞=∫0∞1x+t+1pdx≤11+tp−1/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).Ifc0=−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=−ρ−1 can be expressed as
(44)ρx,t=1x+t+1,ux,t=e2t−1e2t+1x+t+1−1.And the particle path of compressible flow is(45)xt=x0+12e−te2t+1−t−1.Remark 6.
The functionsρx,t=1/x+t+1,ux,t=2c0e2t+1/2c0e2t−1x+t+1−1 are also the solution to the following Euler equations:
(46)ρt+ρux=0x,t∈0,+∞×R+,ρut+uux+px=0,x,t∈0,+∞×R+,with the assumption of the pressure and initial data.
(47)pρ=−ρ−1,ρ0x=1x+1,u0x=2c0+12c0−1x+22c0−1c0≠12,for the second order derivative of ux,t with the spatial variable x is 0.



---

*Source: 6339203-2021-05-27.xml*