Existence of classical solutions of the equations of motion for compressible fluids of second grade
Author:
Š. Matušů-Nečasová
Journal:
Quart. Appl. Math. 58 (2000), 369-378
MSC:
Primary 76N10; Secondary 35Q35, 76A05
DOI:
https://doi.org/10.1090/qam/1753405
MathSciNet review:
MR1753405
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The existence of the classical solutions has been proved for the equations of motion for compressible fluids of second grade in a bounded domain.
- Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI https://doi.org/10.1002/cpa.3160170104
- H. Beirão da Veiga, Boundary-value problems for a class of first order partial differential equations in Sobolev spaces and applications to the Euler flow, Rend. Sem. Mat. Univ. Padova 79 (1988), 247–273. MR 964034
- Lamberto Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308–340 (Italian). MR 138894
- J. Ernest Dunn and Roger L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity $2$ and fluids of second grade, Arch. Rational Mech. Anal. 56 (1974), 191–252. MR 351249, DOI https://doi.org/10.1007/BF00280970
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- Vincenzo Coscia and Giovanni P. Galdi, Existence, uniqueness and stability of regular steady motions of a second-grade fluid, Internat. J. Non-Linear Mech. 29 (1994), no. 4, 493–506. MR 1287760, DOI https://doi.org/10.1016/0020-7462%2894%2990018-3
- Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205
J.-L. Lions and E. Magenes, Nonhomogeneous boundary value problems, Vol. I, Springer-Verlag, Berlin, Heidelberg, New York, 1972
J. Málek, J. Nečas, and M. Růžička, On the non-Newtonian incompressible fluids, Mathematical Methods and Models in Applied Sciences, Vol. 3, 35–63 (1993)
- J. Nečas, Theory of multipolar viscous fluids, The mathematics of finite elements and applications, VII (Uxbridge, 1990) Academic Press, London, 1991, pp. 233–244. MR 1132501
- Antonín Novotný, About steady transport equation. I. $L^p$-approach in domains with smooth boundaries, Comment. Math. Univ. Carolin. 37 (1996), no. 1, 43–89. MR 1396161
M. Padula, Recent contributions and open questions in the mathematical theory of steady motions of viscous compressible fluids, Preprint, 1996
K. R. Rajagopal, personal communications
- R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational Mech. Anal. 4 (1955), 323–425. MR 68413, DOI https://doi.org/10.1512/iumj.1955.4.54011
- Roger Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis 20 (1975), no. 1, 32–43. MR 0430568, DOI https://doi.org/10.1016/0022-1236%2875%2990052-x
- J. E. Dunn and K. R. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, Internat. J. Engrg. Sci. 33 (1995), no. 5, 689–729. MR 1321925, DOI https://doi.org/10.1016/0020-7225%2894%2900078-X
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 17, 35–92 (1964)
H. Beraio da Veiga, Boundary value problems for a class of first order partial differential equations in Sobolev spaces and applications to the Euler flow, Rend. Sem. Mat. Univ. Padova 79, 247–273 (1988)
L. Cattabriga, Su un problema el contorno relativo al sistema di equanioni di Stokes, Rend. Math. Sem. Univ. Padova 31, 308–340 (1961)
J. E. Dunn and R. L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal. 56, 191–252 (1974)
A. Friedman, Partial Differential Equations, Holt, Rinehard, and Winston, Inc., New York, 1969
G. P. Galdi and V. Coscia, Existence, uniqueness and stability of regular steady motions of a second grade fluid, Internat J. Nonlinear Mech. 29, 493–506 (1994)
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I, II, Springer-Verlag, New York, 1994
J.-L. Lions and E. Magenes, Nonhomogeneous boundary value problems, Vol. I, Springer-Verlag, Berlin, Heidelberg, New York, 1972
J. Málek, J. Nečas, and M. Růžička, On the non-Newtonian incompressible fluids, Mathematical Methods and Models in Applied Sciences, Vol. 3, 35–63 (1993)
J. Nečas, Theory of multipolar viscous fluids, The Mathematics of Finite Elements and Applications, (Ed. J. R. Whiteman), Academic Press, London, 1991, pp. 233–244
A. Novotný, About steady transport equations I — $L^{p}$-approach in domains with smooth boundaries, Comment. Math. Univ. Carolina 37, 43–89 (1996)
M. Padula, Recent contributions and open questions in the mathematical theory of steady motions of viscous compressible fluids, Preprint, 1996
K. R. Rajagopal, personal communications
R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational Mech. Anal. 4, 323–425 (1955)
R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20, 32–43 (1975)
J. E. Dunn and K. R. Rajagopal, Fluids of differential type: Critical review and thermodynamic analysis, Internat. J. Engrg. Sci. 33, 689–729 (1995)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
76N10,
35Q35,
76A05
Retrieve articles in all journals
with MSC:
76N10,
35Q35,
76A05
Additional Information
Article copyright:
© Copyright 2000
American Mathematical Society