In [18]–[19], P.L. Lions studied (among others) the compactness and regularity of weak solutions to steady compressible Navier-Stokes equations in the isentropic regime with arbitrary large external data, in particular, in bounded domains. Here we investigate the same problem, combining his ideas with the method of decomposition proposed by Padula and myself in [29]. We find the compactness of the incompressible part $u$ of the velocity field $v$ and we give a new proof of the compactness of the ``effective pressure'' ${\Cal P} = \rho ^\gamma - (2\mu _1 +\mu _2) \operatorname{div} v$. We derive some new estimates of these quantities in Hardy and Triebel-Lizorkin spaces.
@article{118834,
author = {Anton\'\i n Novotn\'y},
title = {Some remarks to the compactness of steady compressible isentropic Navier-Stokes equations via the decomposition method},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {37},
year = {1996},
pages = {305-342},
zbl = {0852.76077},
mrnumber = {1399004},
language = {en},
url = {http://dml.mathdoc.fr/item/118834}
}
Novotný, Antonín. Some remarks to the compactness of steady compressible isentropic Navier-Stokes equations via the decomposition method. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 305-342. http://gdmltest.u-ga.fr/item/118834/
Sobolev spaces, Academic Press, 1975. | MR 0450957
An $L^p$-theory for the $n$-dimensional stationary compressible Navier- Stokes equations and the incompressible limit for compressible fluids. The equilibrium solutions, Comm. Math. Phys. 109 (1987), 229-248. (1987) | MR 0880415
Solutions of some problems of vector analysis with the operators div and grad, Trudy Sem. S.L. Soboleva (1980), 5-41. (1980)
Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308-340. (1961) | MR 0138894 | Zbl 0116.18002
Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (1993), 247-286. (1993) | MR 1225511 | Zbl 0864.42009
Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511-547. (1989) | MR 1022305
Stationary solutions of the Navier-Stokes equations for a compressible viscous and heat-conductive fluid, preprint, Univ. Bonn, 1988.
Stationary solutions of the Navier-Stokes equations with slip boundary conditions, Comm. Part. Diff. Eqs. 14 (1989), 1579-1606. (1989) | MR 1026775
An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I, Springer, 1994. | MR 1284205 | Zbl 0949.35005
An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II, Springer, 1994. | MR 1284206 | Zbl 0949.35005
About steady compressible flows in 2D exterior domains, Pacif. Journal Math., in press.
Parameter dependent pseudodifferential boundary value problems in anisotropic $L^p$ Sobolev spaces with applications to Navier-Stokes problem, Acta. Math., in press.
Pseudodifferential boundary value problems in $L^p$-spaces, Comm. Part. Diff. Eq. 15 (1990), 289-340. (1990)
The stationary Navier-Stokes equations in $L^p$-spaces, Ph.D. Theses, Math. Inst. Copenhagen, 1993.
Function Spaces, Academia, Prague, 1977. | MR 0482102
Etudes de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1-82. (1933)
Sur le mouvement d'une liquide visqueux emplisant l'espace, Acta Math. 63 (1934), 193-248. (1934) | MR 1555394
Compacité des solutions des équations de Navier-Stokes compressibles isentropiques, C.R. Acad. Sci. Paris 317 (Serie I) (1993), 115-120. (1993) | MR 1228976 | Zbl 0781.76072
Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques, C.R. Acad. Sci. Paris 316 (Serie I) (1993), 1335-1340. (1993) | MR 1226126 | Zbl 0778.76086
Private communication, .
Quelques méthodes de resolution des problemes aux limites nonlinéaires, Mir, 1972 (in Russian), French original: Dunod, 1969.
Exterior stationary problems for the equations of motion of compressible viscous and heat-conductive fluids, Proc. EQUADIFF 89, eds. Dafermos C.M., Ladas G., Papanicolau G., Dekker publ., 1989, pp. 473-479. | MR 1021749 | Zbl 0679.76076
Exterior stationary problems for the equations of motion of compressible viscous and heat-conductive fluids, manuscript in Japanese. | Zbl 0679.76076
Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 (1983), 445-464. (1983) | MR 0713680 | Zbl 0543.76099
Steady flows of viscous compressible fluids - $L^2$-approach, Proc. of EQUAM 92, eds. Salvi, Straskraba (1993) Stab. Appl. Anal. Cont. Media 3.3 (1993), 181-199. (1993) | MR 1245633
Steady flows of viscous compressible fluids in exterior domains under small perturbations of great potential forces, Math. Model. Meth. Appl. Sci. 3.6 (1993), 725-757. (1993) | MR 1245633
Compactness of steady compressible isentropic Navier-Stokes equations via the decomposition method (the whole $\Bbb R^3$), Proc. Symposium Navier-Stokes equations. Theory, numerical analysis and applications, Oberwolfach 1994, eds. Heywood J., Masuda K., Rautmann R., Solonnikov V.A., in press.
On steady compressible Navier-Stokes equations in plane domains with corners, Math. Annalen 304.1 (1996), 121-150. (1996) | MR 1367886
$L^p$-approach to steady flows of viscous compressible fluids in exterior domains, preprint, Univ. Ferrara, 1992; Arch. Rat. Mech. Anal. 126 (1994), 243-297. (1994) | MR 1293786
Physically reasonable solutions to steady compressible Navier- Stokes equations in 3D-exterior domains II $(v_\infty = 0)$, J. Math. Kyoto Univ., in press.
Physically reasonable solutions to steady compressible Navier- Stokes equations in 3D-exterior domains I $(v_\infty \neq 0)$, preprint, Univ. Toulon, 1994. | MR 1293786
On the decay at infinity of steady flow of viscous gas in an exterior domain I $(v_\infty = 0)$, preprint, Univ. Toulon, 1994.
Existence and uniqueness of stationary solutions for viscous compressible heat-conductive fluid with large potential and small nonpotential external forces, Sib. Math. J. 34 (1991), 120-146. (1991) | MR 1255466
A remark on the well possedness of the problem of a steady flow of a viscous barotropic gas in a pipe, Comm. Part. Diff. Eq. 21.1-2 (1996), 23-35. (1996) | MR 1373763
$L^p$-approach for steady flows of viscous compressible heat conductive gas, $M^3AS$, in press.
About the incompressible limit of steady compressible Navier-Stokes equations in exterior domains, preprint, Univ. Toulon, 1995.
On the uniqueness of viscous compressible flows, Proc. IV. Symposium - Trends in Applications of Pure Mathematics to Mechanics, editor Brilla E., Pitman, 1981.
Existence and uniqueness for viscous steady compressible motions, Proc. Sem. Fis. Mat., Trieste, Dinamica dei fluidi e dei gaz ionizzati, 1982. | Zbl 0644.76086
Existence and uniqueness for viscous steady compressible motions, Arch. Rat. Mech. Anal. 77 (1987), 89-102. (1987) | MR 0860302 | Zbl 0644.76086
A representation formula for steady solutions of a compressible fluid moving at low speed, Transp. Th. Stat. Phys. 21 593-613. | MR 1194463
On the exterior steady problem for the equations of a viscous isothermal gas, Proc. EVEQ 92, Prague, eds. John O., Stará J., Comm. Math. Univ. Carolinae 34 (1993), 275-293. | MR 1241737 | Zbl 0778.76087
Existence of global solutions for 2-dimensional viscous compressible flow, J. Funct. Anal. 69 (1986), 1-20. (1986) | MR 0864756
Erratum, J. Funct. Anal. 76 (1988), 231. (1988) | MR 0923054 | Zbl 0641.76015
Erratum, J. Funct. Anal. 77 (1988), 232. (1988) | MR 0930400 | Zbl 0641.76015
Steady flows of viscous ideal gas in domains with noncompact boundaries: existence and asymptotic behaviour in a pipe, to appear.
On the free boundary problem for stationary compressible Navier-Stokes equations, Comm. Math. Phys. 129 (1990), 169-204. (1990) | MR 1046283
About the solvability of the initial boundary value problem for the viscous compressible fluid (in Russian), .
Free boundary value problem for a viscous compressible flow with the surface tension, An Int. Tribute, World Sci. Publ. Singapore (1991), pp. 1270-1303.
Harmonic Analysis, Princeton Univ. Press, 1993. | MR 1232192 | Zbl 1106.42300
Nonlinear equations and weak convergence, Proc. of 14th Conference on PDEs, Hřensko, 1989, pp. 103-146 (in Czech).
On the free boundary value problem for compressible viscous fluid motion, J. Math. Kyoto Univ. 21.4 (1981), 839-859. (1981) | MR 0637520 | Zbl 0499.76061
Two phase free boundary value problem for compressible viscous fluid motion, J. Math. Kyoto Univ. 24.2 (1984), 243-267. (1984) | MR 0751700
Navier Stokes Equations, Mir, 1981 (in Russian), English original North Holland, 1979. | MR 0603444 | Zbl 1157.35333
Theory of Functional Spaces, Birkhauser, 1983.
On the existence of stationary solutions to compressible Navier-Stokes equations, Ann. Inst. H. Poincaré 4 (1987), 99-113. (1987) | MR 0877992 | Zbl 0627.76080
Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Sc. Norm. Sup. Pisa 4 (1983), 607-647. (1983) | MR 0753158 | Zbl 0542.35062
Navier-Stokes equations for compressible fluids: global existence and qualitative properties of solutions in the general case, Comm. Math. Phys. 103 (1989), 259-296. (1989) | MR 0826865