Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas
Authors:
Song Jiang and Ping Zhang
Journal:
Quart. Appl. Math. 61 (2003), 435-449
MSC:
Primary 76N10; Secondary 35L65, 35Q30
DOI:
https://doi.org/10.1090/qam/1999830
MathSciNet review:
MR1999830
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Abstract: We prove the existence of global weak solutions to the Navier-Stokes equations for a one-dimensional viscous polytropic ideal gas. We require only that the initial density is in ${L^\infty } \cap L_{loc}^2$ with positive infimum, the initial velocity is in $L_{loc}^2$, and the initial temperature is in $L_{loc}^1$ with positive infimum. The initial density and the initial velocity may have differing constant states at $x = \pm \infty$. In particular, piecewise constant data with arbitrary large jump discontinuities are included. Our results show that neither vacuum states nor concentration states can form and the temperature remains positive in finite time.
- S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids, Studies in Mathematics and its Applications, vol. 22, North-Holland Publishing Co., Amsterdam, 1990. Translated from the Russian. MR 1035212
- A. A. Amosov and A. A. Zlotnik, Generalized solutions “in the large” of the equations of the one-dimensional motion of a viscous heat-conducting gas, Dokl. Akad. Nauk SSSR 301 (1988), no. 1, 11–15 (Russian); English transl., Soviet Math. Dokl. 38 (1989), no. 1, 1–5. MR 953594
- A. A. Amosov and A. A. Zlotnik, Solvability “in the large” of a system of equations for the one-dimensional motion of an inhomogeneous viscous heat-conducting gas, Mat. Zametki 52 (1992), no. 2, 3–16, 155 (Russian, with Russian summary); English transl., Math. Notes 52 (1992), no. 1-2, 753–763 (1993). MR 1187869, DOI https://doi.org/10.1007/BF01236769
- Gui-Qiang Chen, David Hoff, and Konstantina Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2233–2257. MR 1789926, DOI https://doi.org/10.1080/03605300008821583
- Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481
H. Fujita-Yashima, M. Padula, and A. Novotný, Equation monodimensionnelle d’un gaz visqueux et calorifère avec des conditions initiales moins restrictives, Ricerche di Matematica XLII, 199–248 (1993)
- David Hoff, Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data, J. Differential Equations 95 (1992), no. 1, 33–74. MR 1142276, DOI https://doi.org/10.1016/0022-0396%2892%2990042-L
- David Hoff, Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states, Z. Angew. Math. Phys. 49 (1998), no. 5, 774–785. MR 1652200, DOI https://doi.org/10.1007/s000330050120
- D. A. Iskenderova and Sh. S. Smagulov, The Cauchy problem for equations of a viscous heat-conducting gas with degenerate density, Zh. Vychisl. Mat. i Mat. Fiz. 33 (1993), no. 8, 1251–1259 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 33 (1993), no. 8, 1109–1117. MR 1234035
- Song Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Comm. Math. Phys. 178 (1996), no. 2, 339–374. MR 1389908
- Song Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys. 200 (1999), no. 1, 181–193. MR 1671920, DOI https://doi.org/10.1007/s002200050526
- Ja. I. Kanel′, The Cauchy problem for equations of gas dynamics with viscosity, Sibirsk. Mat. Zh. 20 (1979), no. 2, 293–306, 463 (Russian). MR 530493
- A. V. Kazhikhov, On the Cauchy problem for the equations of a viscous gas, Sibirsk. Mat. Zh. 23 (1982), no. 1, 60–64, 220 (Russian). MR 651877
- A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, Prikl. Mat. Meh. 41 (1977), no. 2, 282–291 (Russian); English transl., J. Appl. Math. Mech. 41 (1977), no. 2, 273–282. MR 0468593, DOI https://doi.org/10.1016/0021-8928%2877%2990011-9
- Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. Compressible models; Oxford Science Publications. MR 1637634
- Takeyuki Nagasawa, On the one-dimensional motion of the polytropic ideal gas nonfixed on the boundary, J. Differential Equations 65 (1986), no. 1, 49–67. MR 859472, DOI https://doi.org/10.1016/0022-0396%2886%2990041-0
- Takeyuki Nagasawa, On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with stress-free condition, Quart. Appl. Math. 46 (1988), no. 4, 665–679. MR 973382, DOI https://doi.org/10.1090/S0033-569X-1988-0973382-3
- Takeyuki Nagasawa, On the one-dimensional free boundary problem for the heat-conductive compressible viscous gas, Recent topics in nonlinear PDE, IV (Kyoto, 1988) North-Holland Math. Stud., vol. 160, North-Holland, Amsterdam, 1989, pp. 83–99. MR 1041376, DOI https://doi.org/10.1016/S0304-0208%2808%2970507-3
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Denis Serre, Sur l’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 14, 703–706 (French, with English summary). MR 870700
- A. A. Zlotnik and A. A. Amosov, On the stability of generalized solutions of equations of one-dimensional motion of a viscous heat-conducting gas, Sibirsk. Mat. Zh. 38 (1997), no. 4, 767–789, i (Russian, with Russian summary); English transl., Siberian Math. J. 38 (1997), no. 4, 663–684. MR 1474911, DOI https://doi.org/10.1007/BF02674573
S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, “Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,” North-Holland, Amsterdam, New York, 1990
A. A. Amosov and A. A. Zlotnik, Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas, Soviet Math. Dokl. 38, 1–5 (1989)
A. A. Amosov and A. A. Zlotnik, Solvability “in the large” of a system of equations of the one-dimensional motion of an inhomogeneous viscous heat-conducting gas, Math. Notes 52, 753–763 (1992)
G. Q. Chen, D. Hoff, and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data, Comm. Partial Differential Equations 25, 2233–2257 (2000)
L. C. Evans, “Weak Convergence Methods for Nonlinear Partial Differential Equations,” CBMS No. 74, AMS, 1990
H. Fujita-Yashima, M. Padula, and A. Novotný, Equation monodimensionnelle d’un gaz visqueux et calorifère avec des conditions initiales moins restrictives, Ricerche di Matematica XLII, 199–248 (1993)
D. Hoff, Global well-posedness of the Cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data, J. Diff. Eqs. 95, 33–74 (1992)
D. Hoff, Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states, Z. Angew. Math. Phys. 49, 774–785 (1998)
D. A. Iskenderova and Sh. S. Smagulov, The Cauchy problem for the equations of a viscous heat-conducting gas with degenerate density, Comput. Math. Phys. 33, 1109–1117 (1993)
S. Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Comm. Math. Phys. 178, 339–374 (1996)
S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys. 200, 181–193 (1999)
Y. I. Kanel, Cauchy problem for the equations of gas dynamics with viscosity, Siberian Math. J. 20, 208–218 (1979)
A. V. Kazhikhov, Cauchy problem for viscous gas equations, Siberian Math. J. 23, 44–49 (1982)
A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech. 41, 273–282 (1977)
P. L. Lions, “Mathematical topics in Fluid Mechanics, Vol. 2, Compressible Models,” Clarendon Press, Oxford, 1998
T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary, J. Diff. Eqs. 65, 49–67 (1986)
T. Nagasawa, On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with stress-free condition, Quart. Appl. Math. 46, 665–679 (1988)
T. Nagasawa, On the one-dimensional free boundary problem for the heat-conductive compressible viscous gas, In: Mimura, M., Nishida, T. (eds.), Recent Topics in Nonlinear PDE IV, Lecture Notes in Num. Appl. Anal. 10, pp. 83–99, Kinokuniya/North-Holland, Amsterdam, Tokyo, 1989
M. H. Protter and H. F. Weinberger, “Maximum Principles in Differential Equations,” Prentice Hall, Englewood Cliffs, New Jersey, 1967
D. Serre, Sur l’équation monodimensionnelle d’un fluids visqueux, compressible et conducteur de chaleur, C. R. Acad. Sci. Paris, Sér. I 303, 703–706 (1986)
A. A. Zlotnik and A. A. Amosov, On stability of generalized solutions to the equations of one-dimensional motion of a viscous heat-conducting gas, Siberian Math. J. 38, 663–684 (1997)
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