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Lp-approach to steady flows of viscous compressible fluids in exterior domains


We investigate steady compressible flows in three-dimensional exterior domains for small data and for both zero and nonzero (but constant) velocity at infinity. We prove existence and uniqueness of solutions in L p-spaces, p>3, and study their regularity as well as their decay at infinity.

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  1. [A]

    Adams, R. A., 1976, Sobolev Spaces, Academic Press.

  2. [ADN]

    Agmon, S., Douglis A. & Nirenberg L., 1964, Estimates Near the Boundary of Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions II, Comm. Pure Appl. Math. 17, 35–92.

  3. [BV1]

    Beirão da Veiga, H., 1987, 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, 229–248.

  4. [BV2]

    Beirão da Veiga H., 1987, Existence Results in Sobolev Spaces for a Stationary Transport Equation, Ricerche di Matematica, volume in honour of Prof. C. Miranda.

  5. [BV3]

    Beirão da Veiga, H., 1988, 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.

  6. [B]

    Bers, L., 1954, Existence and Uniqueness of a Subsonic Flow Past a Given Profile, Comm. Pure Appl. Math. 7, 441–504.

  7. [Bo]

    Bogovskij, M. E., 1980, Solutions of Some Problems of Vector Analysis with the Operators div and grad (in Russian), Trudy Sem. S. L. Soboleva, 5–41.

  8. [BM]

    Borchers, & W. Miyakawa, T., 1990, Algebraic L 2Decay for Navier-Stokes Flows in Exterior Domains, Acta Math. 165, 189–227.

  9. [C]

    Cattabriga, L., 1961, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31, 308–340.

  10. [CF]

    Chang, I. D. & Finn R., 1961, On the Solutions of a Class of Equations Occuring in Continuum Mechanics with Applications to the Stokes Paradox, Arch. Rational Mech. Anal. 7, 388–441.

  11. [Fl]

    Farwig, R., 1988, Stationary Solutions of the Navier Stokes Equations for a Compressible Viscous and Heat Conductive Fluid, preprint, Univ. Bonn.

  12. [F2]

    Farwig, R., 1989, Stationary Solutions of the Navier-Stokes Equations with Slip Boundary Conditions, Comm. Part. Diff. Eqs. 14, 1579–1606.

  13. [Fi1]

    Fichera, G., 1956, Sulle equazioni differenziali lineari ellitico paraboliche del secondo ordine, Atti Acad. Naz. Lincei, Mem. Sc. Fis. Mat. Nat., Sez. I, 5, 1–30.

  14. [Fi2]

    Fichera, G., 1960, On a Unified Theory of Boundary Value Problem for Elliptic-Parabolic Equations of Second Order in Boundary Problems, Differential Equations, Univ. Wisconsin Press, 87–120.

  15. [F]

    Finn, R., 1965, On the Exterior Stationary Problem for the Navier-Stokes Equations and Associated Perturbation Problems, Arch. Rational Mech. Anal. 19, 363–406.

  16. [FG]

    Finn, R. & Gilbarg, D., 1957, Three-Dimensional Subsonic Flows and Asymptotic Estimates for Elliptic Partial Differential Equations, Acta Math. 98, 265–296.

  17. [Fr]

    Friedman, A., 1964, Partial Differential Equations of Parabolic Type, Prentice-Hall.

  18. [Ga]

    Gagliardo, E., 1957, Caratterizzazioni delle trace sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova 27, 284–305.

  19. [G1]

    Galdi, G. P., 1992, On the Asymptotic Properties of Leray's Solution to the Exterior Stationary Three-Dimensional Navier-Stokes Equations with Zero Velocity at Infinity, Degenerate Diffusion, Ni, W.-M., Peletier, L. A., & Vazquez, J. L., Eds., Springer-Verlag, 95–103.

  20. [G2]

    Galdi, G. P., 1991, On the Oseen Boundary Value Problem in Exterior Domains, Proc. of the Oberwolfach Meeting “The Navier-Stokes Equations: Theory and Numerical Methods”, Heywood, J. G., Masuda, K., Rautmann, R., & Solonnikov, V. A., Eds., Springer Lecture Notes in Mathematics 1530, 111–131.

  21. [G3]

    Galdi, G. P., 1994, An Introduction to the Mathematical Theory of the Navier Stokes Equations. Volume I: Linearized stationary problems, Springer Tracts in Natural Philosophy 38.

  22. [G4]

    Galdi, G. P., 1992, On the Energy Equation and on the Uniqueness for D-Solutions to Steady Navier-Stokes Equations in Exterior Domains, Mathematical Problems Related to the Navier-Stokes Equation, Galdi, G. P., Ed., Advances in Mathematics for Applied Science, 11, World Scientific, 34–78.

  23. [G5]

    Galdi, G. P., 1992, On the Asymptotic Structure of D-Solutions to Steady Navier-Stokes Equations in Exterior Domains, Mathematical Problems Related to the Navier-Stokes Equation, Galdi, G. P., Ed., Advances in Mathematics for Applied Sciences 11, World Scientific, 79–103.

  24. [GM]

    Galdi, G. P. & Maremonti, P., 1986, Monotonic Decreasing and Asymptotic Behavior of the Kinetic Energy for Weak Solutions of the Navier-Stokes Equations in Exterior Domains, Arch. Rational Mech. Anal. 94, 253–266.

  25. [GS]

    Galdi, G. P. & Simader, C. G., 1990, Existence, Uniqueness and L q-Estimates for the Stokes Problem in an Exterior Domain, Arch. Rational Mech. Anal. 112, 291–318.

  26. [H]

    Heywood, J., 1980, The Navier-Stokes Equations: on the Existence, Regularity and Decay of Solutions, Indiana Univ. Math. J. 29, 639–681.

  27. [KS]

    Kozono, H. & Sohr, H., 1991, New a Priori Estimates for the Stokes Equation in Exterior Domains, Indiana Univ. Math. J. 40, 1–27.

  28. [LP]

    Lax, P. D. & Phillips, R. S., 1958, Symmetric Positive Linear Differential Equations, Comm. Pure Appl. Math. 11, 333–418.

  29. [L]

    Leray, J., 1934, Sur le mouvement d'une liquide visqueux emplissant l'espace, Acta Math. 63, 193–248.

  30. [Mi]

    Mizohata, S., 1973, The Theory of Partial Differential Equations. Cambridge Univ. Press.

  31. [MS]

    Maremonti, P. & Solonnikov, V. A., 1986, Su una diseguaglianza per le soluzioni del problema di Stokes in domini esterni, preprint, Univ. Naples.

  32. [MN1]

    Matsumura, A. & Nishida, T., 1989, Exterior Stationary Problems for the Equations of Motion of Compressible Viscous and Heat-Conductive Fluids. Proc. EQUADIFF 89. Dafermos, C. M., Ladas, G. & Papanicolau, G., Eds. Dekker 473–479.

  33. [MN2]

    Matsumura,. A. & Nishida, T., 1989, Exterior Stationary Problems for the Equations of Motion of Compressible Viscous and Heat-Conductive Fluids, manuscript in Japanese.

  34. [MP]

    Matsumura, A. & Padula, M., 1992, Stability of Stationary Flow of Compressible Fluids Subject to Large External Potential Forces, Stab. Anal. Cont. Media 2, 183–202.

  35. [N]

    Nečas, J., 1967, Les methodes directes en theorie des equations elliptiques, Masson.

  36. [N1]

    Novotný, A., 1993, Existence and Uniqueness of Stationary Solutions for Viscous Compressible Heat-Conductive Fluid with Great Potential and Small Nonpotential Forces, Proc. EQUADIFF 91, Barcelona. C. Perello, Ed.

  37. [N2]

    Novotný, A., About the Steady Transport Equation IL p-Approach in Domains with Smooth Boundaries, to appear.

  38. [N3]

    Novotný, A., A Note about the Steady Compressible Flows in R 3, R + 3 — LpApproach, to appear.

  39. [NP1]

    Novotný, A. & Padula M., 1991, Existence and Uniqueness of Stationary Solutions for Viscous Compressible Heat-Conductive Fluid with Large Potential and Small Nonpotential External Forces, Sib. Math. J. 34, 120–146.

  40. [NP2]

    Novotný, A. & Padula M., On Physically Reasonable Solutions for Steady Compressible Navier-Stokes Equations in 3-D Exterior Domains, to appear.

  41. [NPe]

    Novotný, A. & Penel P., An L pApproach for Steady Flows of Viscous Compressible Heat Conductive Gas, to appear.

  42. [O]

    Oleinik, O. A., 1967, Linear Equations of Second Order with Nonnegative Characteristics Form, Amer. Math. Soc. Transl. 65, 167–199.

  43. [OR]

    Oleinik, O. A. & Radkevič E. V., 1973, Second Order Equations with Nonnegative Characteristic Form, Amer. Math. Soc. and Plenum Press.

  44. [P1]

    Padula, M., 1981, On the Uniqueness of Viscous Compressible Flows, Proc. IV. Symposium — Trends in Applications of Pure Mathematics to Mechanics. E. Brilla, Ed., Pitman.

  45. [P2]

    Padula, M., 1982, Existence and Uniqueness for Viscous Steady Compressible Motions, Proc. Sem. Fis. Mat., Trieste, Dinamica dei fluidi e dei gaz ionizzati.

  46. [P3]

    Padula, M., 1987, Existence and Uniqueness for Viscous Steady Compressible Motions, Arch. Rational Mech. Anal. 77, 89–102.

  47. [P4]

    Padula, M., 1993, A Representation Formula for Steady Solutions of a Compressible Fluid Moving at Low speed, Transp. Theory and Stat. Phys. 21, 593–613.

  48. [P5]

    Padula, M., 1993, On the Exterior Steady Problem for the Equations of a Viscous Isothermal Gas, Comm. Math. Univ. Carolinae 34, 275–293.

  49. [PP]

    Padula, M. & Pileckas, C, Steady Flows of a Viscous Ideal Gas in Domains with non Compact Boundaries: Existence and Asymptotic Behavior in a Pipe, to appear.

  50. [S]

    Serrin, J., 1959, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik., Flügge, S., Ed., Springer-Verlag.

  51. [Si]

    Simader, Ch., 1990, The Weak Dirichlet and Neumann Problem for the Laplacian in L qfor Bounded and Exterior Domains. Applications., Nonlinear Analysis, Function Spaces and Applications Vol. 4. Krbec, M., Kufner, A., Opic, B., & Rakosnik, J, Eds. Teubner, 180–223.

  52. [SiSo1]

    Simader, Ch. & Sohr, H., 1992, The Weak Dirichlet Problem for Δ in L qin Bounded and Exterior Domains, Stab. Anal. Cont. Media 2, 183–202.

  53. [SiSo2]

    Simader, ch. & Sohr, H., 1992, A New Approach to the Helmholtz & Decomposition and the Neumann Problem in Lq-Spaces for Bounded and Exterior Domains, Mathematical Problems Relating to the Navier-Stokes Equations, Galdi, G. P., Ed., World Scientific.

  54. [Sm]

    Smirnov, V. A., 1964, A Course of Higher Mathematics, V, Pergamon Press

  55. [SW]

    Stein, E. M. & Weiss, G., 1958, Fractional Integrals on n-Dimensional Euclidean Space, J. Math. Mech. 7, 503–514.

  56. [V1]

    Valli, A., 1987, On the Existence of Stationary Solutions to Compressible Navier-Stokes Equations, Ann. Inst. H. Poincaré 4, 99–113.

  57. [V2]

    Valli, A., 1983, Periodic and Stationary Solutions for Compressible Navier-Stokes Equations Via a Stability Method, Ann. Sc. Sup. Pisa 4, 607–647.

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Novotný, A., Padula, M. Lp-approach to steady flows of viscous compressible fluids in exterior domains. Arch. Rational Mech. Anal. 126, 243–297 (1994).

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