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

Guo Shangxi,

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

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

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

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

证明过程参考文献[15].

证明过程参考文献[15].

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[15].

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

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