一维可压缩Navier-Stokes方程初值问题强解的整体存在性

Abstract

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.

Key words: Navier-Stokes equation, Compressible, Global strong solution, Density-dependent viscosity

CLC Number:

O175.29

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

Trendmd

References 16

1	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
2	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 doi: 10.1007/PL00001488
3	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 doi: 10.1137/0151043
4	Liu T P , Xin Z P , Yang T . Vacuum states for compressible flow. Discrete and Continuous Dynamical Systems, 1998, 4 (1): 1- 32 doi: 10.3934/dcds.1998.4.1
5	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 doi: 10.1007/BF03167921
6	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 doi: 10.1007/BF02824736
7	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 doi: 10.4310/MAA.2005.v12.n3.a2
8	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 doi: 10.1016/S0022-0396(03)00060-3
9	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
10	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 doi: 10.3934/cpaa.2008.7.373
11	Yang T , Zhu C J . Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Communications in Mathematical Physics, 2002, 230 (2): 329- 363 doi: 10.1007/s00220-002-0703-6
12	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 doi: 10.1006/jdeq.2001.4140
13	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 doi: 10.1007/s00220-008-0495-4
14	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 doi: 10.1137/060658199
15	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 doi: 10.1007/s10255-016-0631-4
16	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

[1]	Mingyue Xu,Caidi Zhao,Caraballo Tomás. Degenerate Regularity of Trajectory Statistical Solutions for the 3D Incompressible Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2021, 41(2): 336-344.
[2]	Qing Chen. Optimal Time Decay Rate of the Highest Derivative of Solutions to the Compressible Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2021, 41(2): 345-356.
[3]	Jianhua Huang,Zaiyun Zhang,Yong Chen. The Moderate Deviation Principle for Stochastic 3D LANS-α Model Driven by Multiplicative Lévy Noise [J]. Acta mathematica scientia,Series A, 2019, 39(6): 1532-1544.
[4]	Na Wang,Shu Wang. The Boundary Layer for MHD Equations in a Plane-Parallel Channel [J]. Acta mathematica scientia,Series A, 2019, 39(4): 738-760.
[5]	Zhaoyang Shang. Blow-Up Criterion for Incompressible Magnetohydrodynamics Equations in Besov Space [J]. Acta mathematica scientia,Series A, 2019, 39(1): 67-80.
[6]	Zhihao Ge,Yuanyuan Ge. Multigrid Uzawa-Type Mixed Finite Element Methods for Nearly Incompressible Linear Elasticity Problem [J]. Acta mathematica scientia,Series A, 2018, 38(5): 873-882.
[7]	Wang Junli, Zhang Xingwei, Liu Jian. Instability of Plane Couette-Poiseuille Flow of Compressible Navier-Stokes Equation [J]. Acta mathematica scientia,Series A, 2018, 38(2): 322-333.
[8]	Chen Pengfei. The Inviscid and Non-Resistive Limit for 3D Nonhomogeneous Incompressible MHD Equations with a Slip Boundary Condition [J]. Acta mathematica scientia,Series A, 2018, 38(1): 83-95.
[9]	Wang Heyuan, Cui Jin. The Analysis of Global Stability and Numerical Simulation of Chaos Behaviors of Rotating Flow [J]. Acta mathematica scientia,Series A, 2017, 37(4): 783-792.
[10]	Qiu Hua, Zhang Yingshu, Fang Shaomei. The Global Existence Result of a Leray-α-Oldroyd Model to the Incompressible Viscoelastic Flow [J]. Acta mathematica scientia,Series A, 2017, 37(1): 102-112.
[11]	Bian Dongfen, Tang Tong. Blow-Up of Smooth Solutions to the Compressible MHD Equations [J]. Acta mathematica scientia,Series A, 2016, 36(4): 715-721.
[12]	Yang Xinguang, Zhao Mingxia, Hou Wei. Upper Semi-Continuity of Pullback Attractors for the 2D Non-Autonomous Navier-Stokes Equations with Weak Damping [J]. Acta mathematica scientia,Series A, 2016, 36(4): 722-739.
[13]	Wu Jihui, Wang Shu, Lu Lei. Global Regularity of the 3D Axially Symmetric Incompressible MHD Equations with Anisotropic Data [J]. Acta mathematica scientia,Series A, 2015, 35(5): 855-866.
[14]	ZHU Xu-Sheng, LI Fang-E. The Global Solutions of the Compressible Euler Equations With Nonlinear Damping in 3-Dimensions [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1111-1122.
[15]	ZHANG Zu-Jin. An Improved Regularity Criterion for the 3D Navier-Stokes Equations in Terms of Two Entries of the Velocity Gradient [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1327-1335.

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

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 16

Related Articles 15

Metrics

Comments

Recommended 10