Navier-Stokes equations with degenerate viscosity, vacuum and gravitational force

Mathematical Methods in the Applied Sciences

Volumn 30, Issue 3, 2007, Pages 347-374

  Duan, Renjun ^a   Yang, Tong ^a   Zhu, Changjiang ^b  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^b CENTRAL CHINA NORMAL UNIVERSITY   (China)

Author keywords

A priori estimates; Global existence; Navier Stokes equations; Vacuum

Indexed keywords

GRAVITATION; PARAMETER ESTIMATION; PROBABILITY DENSITY FUNCTION; VISCOSITY;

A PRIORI ESTIMATES; GLOBAL EXISTENCE;

NAVIER STOKES EQUATIONS;

EID: 33846490299     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.790     Document Type: Article

References (27)

1. Vong SW, Yang T, Zhu CJ. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum (II). Journal of Differential Equations 2003; 192:475-501.
2. Yang T, Yao ZA, Zhu CJ. Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Communications in Partial Differential Equations 2001; 26:965-981.
3. Yang T, Zhao HJ. A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Journal of Differential Equations 2002; 184:163-184.
4. Yang T, Zhu CJ. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Communications in Mathematical Physics 2002; 230:329-363.
5. Hoff D, Serre D. The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM Journal on Applied Mathematics 1991; 51:887-898.
6. Okada M. Free boundary value problems for the equation of one-dimensional motion of viscous gas. Japan Journal of Applied Mathematics 1989; 6:161-177.
7. Okada M, Makino T. Free boundary problem for the equations of spherically symmetrical motion of viscous gas. Japan Journal of Industrial and Applied Mathematics 1993; 10:219-235.
8. Xin Z, Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Communications in Pure and Applied Mathematics 1998; 51:229-240.
9. Luo T, Xin Z, Yang T. Interface behavior of compressible Navier-Stokes equations with vacuum. SIAM Journal on Mathematical Analysis 2000; 31:1175-1191.
10. Straškraba I, Zlotnik AA. Global properties of solutions to ID-viscous compressible baratropic fluid equations with density dependent viscosity. Zeitschrift für Angewandte Matematik und Physik 2003; 54:593-607.
11. Straškraba I. Global analysis of 1-D Navier-Stokes equations with density dependent viscosity. In Navier-Stokes Equations and Related Nonlinear Problems, Amann H et al. (eds). VSP Utrecht: TEV Vilnius, 1997; 371-390.
12. Zlonik AA. Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations. Differential Equations 2000; 36:701-716.
13. Grad H. Asymptotic theory of the Boltzmann equation II. In Rarefied Gas Dynamics, Laurmann J (ed.), vol. 1. Academic Press: New York, 1963; 26-59.
14. Liu T-P, Xin Z, Yang T. Vacuum states of compressible flow. Discrete and Continuous Dynamical Systems 1998; 4:1-32.
15. Okada M, Matušu-Nečasová Š, Makino T. Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity. Annals of University of Ferrara Sez, VII-Sc. Mat. 2002; 48:1-20.
16. Jiang S, Xin ZP, Zhang P. Global weak solutions to ID compressible isentropic Navier-Stokes equations with density-dependent viscosity. Methods and Applications of Analysis 2005; 12:239-252.
17. Fang DY, Zhang T. A note on compressible Navier-Stokes equations with vacuum state in one dimension. Nonlinear Analysis (TMA) 2004; 58:719-731.
18. Hoff D. Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Archive for Rational Mechanics and Analysis 1995; 132:1-14.
19. Jiang S. Global smooth solutions of the equations of a viscous, heat-conducting, one-dimensional gas with density-dependent viscosity. Mathematische Nachrichten 1998; 190:169-183.
20. Kawashima S, Nishida T. The initial-value problems for the equations of viscous compressible and perfect compressible fluids. RIMS, Kokyuroku 428, Kyoto University, Nonlinear Functional Analysis, June 1981; 34-59.
21. Lions PL. Mathematical Topics in Fluid Mechanics, vol. 1 and 2. Clarendon Press: Oxford, 1998.
22. Serre D. Sur l'equation mondimensionnelle d'un fluide visqueux, compressible et conducteur de chaleur. Comptes Rendus de l' Academie des Sciences 1986; 303:703-706.
23. Chen GQ, Hoff D, Trivisa K. Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data. Communications in Partial Differential Equations 2000; 25:2233-2257.
24. Chen GQ, Kratka M. Global solutions to the Navier-Stokes equations for compressible heat-conducting flow with symmetry and free boundary. Communications in Partial Differential Equations 2002; 27:907-943.
25. Yang T. Some recent results on compressible flow with vacuum. Taiwanese Journal of Mathematics 2000; 4:33-44.
26. Hoff D, Smoller J. Non-formation of vacuum states for compressible Navier-Stokes equations. Communications in Mathematical Physics 2001; 216:255-276.
27. Nishida T. Equations of motion of compressible viscous fluids. Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations 1986; 97-128.