## Existence of Global Strong Solutions for Initial Value Problems of One-Dimensional Compressible Navier-Stokes Equations

Received: 2019-05-24 Abstract

To consider the initial problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity. By using the energy estimates, the lower and upper bounds for the density is derived, that is, nether vacuum states nor concentration states can occur. Finally, the approximate solution is constructed by transforming the viscous coefficient, the existence of global strong solution is obtained by using the local existence conclusion of the strong solution and combining a prior estimates of the density function and the velocity function.

Keywords： Navier-Stokes equation ; Compressible ; Global strong solution ; Density-dependent viscosity

Guo Shangxi. Existence of Global Strong Solutions for Initial Value Problems of One-Dimensional Compressible Navier-Stokes Equations. Acta Mathematica Scientia[J], 2021, 41(3): 642-651 doi:

## 1 引言

$\begin{eqnarray} \left\{\begin{array}{ll} \rho_{t}+\rho^{2}u_{x} = 0, (x, t)\in{{\Bbb R}} \times{\Bbb R_{+}}, \\ u_{t}+p(\rho)_{x} = (\rho\mu(\rho)u_{x})_{x}, \end{array}\right. \end{eqnarray}$

## 2 主要结论

$\begin{equation} \rho-\overline{\rho}\in{L^{\infty}(0, T;H^{1}({{\Bbb R}} ))}, \rho_{t}\in{L^{2}(0, T;L^{2}({{\Bbb R}} ))}, \end{equation}$

$\begin{equation} u-\overline{u}\in{L^{\infty}(0, T;H^{1}({{\Bbb R}} ))}\bigcap{L^{2}(0, T;H^{2}({{\Bbb R}} ))}, u_{t}\in{L^{2}(0, T;L^{2}({{\Bbb R}} ))}. \end{equation}$

$\begin{eqnarray} 0<C^{-1}(T)\leq\rho(x, t)\leq{C(T)}<\infty, \forall(x, t)\in{{\Bbb R}} \times(0, T). \end{eqnarray}$

证明过程参考文献.

证明过程参考文献.

根据注$2.1{\rm (H_{2})}$可知存在$\eta>0$使得

$\begin{eqnarray} \int_{{{\Bbb R}} }p(\rho|\overline{\rho}){\rm d}x\geq\int_{x_{0}-r(T)}^{x_{0}}p(\rho|\overline{\rho}){\rm d}x\geq\frac{C}{2}r(T), \end{eqnarray}$

证明过程参考文献.

利用Sobolev嵌入定理, 引理3.1–3.4和Cauchy不等式, 有

$\begin{eqnarray} &&||\rho^{2n(\gamma-\alpha)}u^{2n}||_{L^{\infty}([-M, M])}{}\\ &\leq&{C}\int_{-M}^{M}\rho^{2n(\gamma-\alpha)}u^{2n}{\rm d}x+C\int_{-M}^{M}|(\rho^{2n(\gamma-\alpha)}u^{2n})_{x}|{\rm d}x{}\\ &\leq&{C(T, n)}\int_{-M}^{M}u^{2n}{\rm d}x+C(n)\int_{-M}^{M}|\rho^{2n(\gamma-\alpha)-1}\rho_{x}u^{2n}|{\rm d}x{}\\ &&+C(n)\int_{-M}^{M}|\rho^{2n(\gamma-\alpha)}u^{2n-1}u_{x}|{\rm d}x{}\\ &\leq&{C(T, n)}\int_{-M}^{M}(u-\overline{u})^{2n}{\rm d}x+C(T, n)\int_{-M}^{M}(\overline{u})^{2n}{\rm d}x+C(T, n)\int_{-M}^{M}|(\rho^{\alpha})_{x}u^{2n}|{\rm d}x{}\\ &&+C(n)\int_{-M}^{M}\rho^{\alpha+1}u_{x}^{2}{\rm d}x+C(n)\int_{-M}^{M}\rho^{4n(\gamma-\alpha)-(1+\alpha)}u^{4n-2}{\rm d}x{}\\ &\leq&{C(T, n)}\int_{-M}^{M}(\rho^{\alpha})_{x}^{2}{\rm d}x+{C(n)}\int_{-M}^{M}\rho^{\alpha+1}u_{x}^{2}{\rm d}x+C(T, M, n){}\\ &\leq&{C(n)}\int_{-M}^{M}\rho^{\alpha+1}u_{x}^{2}{\rm d}x+C(T, M, n), \end{eqnarray}$

$(1.1)_{1}$式代入$(1.1)_{2}$式得到

$\begin{eqnarray} (\rho^{\alpha})_{xt} = -\alpha{u_{t}}-\alpha(\rho^{\gamma})_{x}, \end{eqnarray}$

$\begin{eqnarray} (\rho^{\alpha})_{x} = (\rho_{0}^{\alpha})_{x}+\alpha(u_{0}-u)-\alpha\int_{0}^{t}(\rho^{\gamma})_{x}{\rm d}s, \end{eqnarray}$

利用引理3.1和引理3.2可知$\int_{{{\Bbb R}} }(\rho^{\alpha})_{x}^{2}{\rm d}x\leq{C(T)}$, 再结合引理3.3和$\alpha\in(0, 1]$得到$\int_{{{\Bbb R}} }\rho_{x}^{2}{\rm d}x\leq{C(T)}$, 因此

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Hoff D .

Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial date

Transactions of the American Mathematical Society, 1987, 303 (1): 169- 181

Hoff D .

Global solutions of the equations of one-dimensional, compressible flow with large date and forces, and with differing and states

Zeitschrift für Angewandte Mathematik und Physik, 1998, 49 (5): 774- 785

Hoff D , Serre D .

The failure of continuous dependence on initial date for the Navier-Stokes equations of compressible flow

SIAM Journal on Applied Mathematics, 1991, 51 (4): 887- 898

Liu T P , Xin Z P , Yang T .

Vacuum states for compressible flow

Discrete and Continuous Dynamical Systems, 1998, 4 (1): 1- 32

Okada M .

Free boundary value problems for the equation of the one-dimensional motion of viscous gas

Japan Journal of Industrial and Applied Mathematics, 1989, 6 (1): 161- 177

Okada M , Makino T .

Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity

Annali dell'Universita di Ferrara, 2002, 48 (1): 1- 20

Jiang S , Xin Z P , Zhang P .

Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

Methods and Applications of Analysis, 2005, 12 (3): 239- 252

Vong S W , Yang T , Zhu C J .

Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum(Ⅱ)

Journal of Differential Equations, 2003, 192 (2): 475- 501

Yang T , Yao Z A , Zhu C J .

Compressible Navier-Stokes equations with density-dependent viscosity and vacuum

Communications in Partial Differential Equations, 2001, 26 (5/6): 965- 981

Qin X L , Yao Z A , Zhao H X .

One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries

Communications on Pure and Applied Analysis, 2008, 7 (2): 373- 381

Yang T , Zhu C J .

Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum

Communications in Mathematical Physics, 2002, 230 (2): 329- 363

Yang T , Zhao H J .

A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity

Journal of Differential Equations, 2002, 184 (1): 163- 184

Li H L , Li J , Xin Z P .

Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations

Communications in Mathematical Physics, 2008, 281 (2): 401- 444

Mellet A , Vasseur A .

Existence and uniqueness of global strong solutions for one dimensional compressible Navier-Stokes equations

SIAM Journal on Mathematical Analysis, 2008, 39 (4): 1344- 1365

Liu S Q , Zhao J N .

Global strong solutions of the Cauchy problem for 1D compressible Navier-Stokes equations with density-dependent viscosity

Acta Mathematicae Applicatae Sinica, 2017, 33 (1): 25- 34

Solonnikov V A .

Solvability of the initial-boundary value problem for the equations of motion of a viscous compressible fluid

Zap Naucn Sem Leningrad Otdel Mat Inst Steklov(LOMI), 1976, 56, 128- 142

/

 〈 〉 