INTRODUCTION TO THE MATHEMATICAL THEORY OF COMPRESSIBLE FLOW (original) (raw)

2021, INTRODUCTION TO THE MATHEMATICAL THEORY OF COMPRESSIBLE FLOW

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This book provides a mathematical framework for understanding the flow of Newtonian compressible fluids in both steady and unsteady regimes, exploring fundamental equations like the Euler and Navier-Stokes equations. It aims to present recent advancements and nontrivial results in a format that is accessible to a broad audience, while also being self-contained for readers of varying expertise. Highlights include discussions on heuristic approaches to proofs and the omission of non-Newtonian fluid dynamics, focusing on isentropic and barotropic flows.

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References (264)

  1. I. Fonseca and W. Gangbo: Degree theory in analysis and applications
  2. P.-L. Lions: Mathematical topics in fluid mechanics, Vol. 1: Incompressible models
  3. J. E. Beasley (ed.): Advances in linear and integer programming
  4. L. W. Beineke and R. J. Wilson (eds): Graph connections: Relationships between graph theory and other areas of mathematics
  5. I. Anderson: Combinatorial designs and tournaments
  6. G. David and S. W. Semmes: Fractured fractals and broken dreams
  7. Oliver Pretzel: Codes and algebraic curves
  8. M. Karpinski and W. Rytter: Fast parallel algorithms for graph matching problems
  9. P.-L. Lions: Mathematical topics in fluid mechanics, Vol. 2: Compressible models
  10. W. T. Tutte: Graph theory as I have known it
  11. Andrea Braides and Anneliese Defranceschi: Homogenization of multiple integrals
  12. Thierry Cazenave and Alain Haraux: An introduction to semilinear evolution equations
  13. J. Y. Chemin: Perfect incompressible fluids
  14. Giuseppe Buttazzo, Mariano Giaquinta and Stefan Hildebrandt: One-dimensional variational problems: an introduction
  15. Alexander I. Bobenko and Ruedi Seiler: Discrete integrable geometry and physics
  16. Doina Cioranescu and Patrizia Donato: An introduction to homogenization
  17. E. J. Janse van Rensburg: The statistical mechanics of interacting walks, polygons, animals and vesicles
  18. S. Kuksin: Hamiltonian partial differential equations
  19. Alberto Bressan: Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem
  20. B. Perthame: Kinetic formulation of conservation laws
  21. A. Braides: Gamma-convergence for beginners
  22. Robert Leese and Stephen Hurley: Methods and algorithms for radio channel assignment
  23. Charles Semple and Mike Steel: Phylogenetics
  24. Luigi Ambrosio and Paolo Tilli: Topics on analysis in metric spaces 26. Eduard Feireisl: Dynamics of viscous compressible fluids
  25. Antonín Novotný and Ivan Straškraba: Introduction to mathematical theory of compressible flow
  26. 4.11.1 The method of contractive mapping A mapping F : X → X is called contractive if there exists a constant q ∈ [0, 1) such that REFERENCES
  27. Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York.
  28. Agmon, S., Douglis, A., and Nirenberg, L. (1959). Estimates near the boundary for solutions of elliptic partial differential equations satisfying general bound- ary conditions I. Commun. on Pure and Appl. Math., 12, 623-727.
  29. Aizicovici, S. and Feireisl, E. (2003). On the long-time behavior of compressible fluid flows subjected to highly oscillating external forces. Czechoslovak Math. J., 53(3), 757-767.
  30. Alinhac, S. (1993). Temps de vie des solutions régulières des équations com- pressibles axisymétriques en dimension deux. Inventiones Mathematicae, 111, 625-670.
  31. Alinhac, S. (1995). Blowup for Nonlinear Hyperbolic Equations. Birkhäuser, Boston.
  32. Amann, H. (1995). Linear and Quasilinear Parabolic Problems, Volume I, II. Birkhäuser, Basel.
  33. Aze, D. (1997). Elements d'analyse fonctionnelle convexe et variationnelle. Ellipses, Paris.
  34. Babenko, K.I. (1973). On stationary solutions of the problem of flow past a body of a viscous compressible fluid. Mat. Sbornik , 91(133)(1), 1-25.
  35. Barták, J., Herrmann, L., Lovicar, V., and Vejvoda, O. (1991). Partial Differ- ential Equations of Evolution. Ellis Horwood, New York.
  36. Bause, M., Heywood, J., Novotný, A., and Padula, M. (2001). An interactive scheme for steady compressible viscous flows modified to treat large potential forces. In Math. Fluid Mech. (ed. Birkhäuser), pp. 27-46. Birkhäuser, Basel.
  37. Bause, M., Heywood, J., and Novotný, A. (2003). On some approximation schemes for steady compressible viscous flow. J. Math. Fluid Mech., 5, 201- 230. Beirão da Veiga, H. (1987). An L p -theory for the n-dimensional, stationary, compressible Navier-Stokes equations, and incompressible limit for compress- ible fluids. The equilibrium solutions. Comm. Math. Phys., 109, 229-248.
  38. Berezansky, Y. M., Sheffel, Z. G., and Us, G. F. (1991). Functional Analysis I,II. Operator Theory Advances and Applications, Volume 85. Birkhäuser, Basel.
  39. Bergh, J. and Löfström (1976). Interpolation Spaces, An Introduction. Springer, Berlin.
  40. Bogovskii, M. E. (1980). Solution of some vector analysis problems connected with operators div and grad. Trudy Sem. S.L. Soboleva, 80(1), 5-40. In Russian.
  41. Bourdaud, G. (1988). Analyse fonctionnelle dans l'espace euclidien. L'Univer- sité Paris VII, Paris. REFERENCES
  42. Bressan, A. (2000). Hyperbolic Systems of Conservation Laws. The One- dimensional Cauchy Problem. Oxford Lecture Series. Oxford University Press, Oxford.
  43. Bressan, A. and Colombo, R. M. (1995). The semigroup generated by 2 × 2 conservation laws. Arch. Rat. Mech. Anal., 133, 1-75.
  44. Brezis, H. (1973). Opérateurs maximaux monotones et semi-groupes de con- tractions dans les espaces de Hilbert. North Holland, Amsterdam.
  45. Brezis, H. (1987). Analyse fonctionnelle, théorie et applications. Masson, Paris.
  46. Calderón, A. P. and Zygmund, A. (1956). On singular integrals. Amer. J. Math., 78, 289-309.
  47. Calderón, A. P. and Zygmund, A. (1957). Singular integral operators and dif- ferential equations. Amer. J. Math., 79, 901-921.
  48. Chang, I. D. and Finn, R. (1961). On the solutions of a class of equations in continuum mechanics with applications to Stokes paradox. Arch. Rat. Mech. Anal., 7, 388-441.
  49. Chemin, Y. (1990). Dynamique des gaz à masse totale finie. Asympt. Anal., 3, 215-220.
  50. Chen, G.-Q. (1997). Remarks on global solutions to compressible Euler equa- tions with spherical symmetry. Proc. Roy. Soc. Edinburgh A, 127, 243-259.
  51. Chen, G.-Q. and Dafermos, C. M. (1995). The vanishing viscosity method in the one-dimensional thermoelasticity. Trans. Amer. Math. Soc., 347, 531-541.
  52. Chen, G.-Q., Frid, H., and Li, Y. (2002). Uniqueness and stability of Riemann solutions with large oscillations in gas dynamics. Commun. Math. Phys., 228(2), 201-218.
  53. Chen, G.-Q. and Glimm, J. (1996). Global solution to the compressible Euler equations with geometrical structure. Commun. Math. Phys., 179, 153-193.
  54. Chen, G.-Q. and LeFloch, P. (2000). Compressible Euler equations with general pressure law. Arch. Rat. Mech. Anal., 153(3), 221-259.
  55. Chen, G.-Q. and Wang, D. (2001). The Cauchy problem for the Euler equa- tions for compressible fluids. In Handbook of Mathematical Fluid Dynamics, Volume 1, pp. 421-543. North-Holland, Amsterdam.
  56. Choe, H. J. and Kim, H. (2003). Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Diff. Equations, 190, 504-523.
  57. Chorin, A. J. and Marsden, J. E. (1979). A Mathematical Introduction to Fluid Mechanics. Springer, New York.
  58. Clark, D. C. (1971). The vorticity at infinity for solutions of the stationary Navier-Stokes equations in exterior domains. Indiana Univ. Math. J., 20, 633-654.
  59. Coifman, R., Lions, P.-L., Meyer, Y., and Semmes, S. (1993). Compensated compactness and hardy spaces. J. Math. Pure Appl., 72, 247-86.
  60. Coifman, R. and Meyer, Y. (1975). On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc., 212, 315-331.
  61. Coifman, R. R., Rochberg, R., and Weiss, G. (1976). Factorization theorems for Hardy spaces in several variables. Ann. Math., 103, 611-635.
  62. Courant, R. and Friedrichs, K. O. (1962). Supersonic Flow and Shock Waves. Interscience Publishers, New York.
  63. Dafermos, C. (2000). Hyperbolic Conservation Laws in Continuum Physics. Springer, Heidelberg.
  64. Danchin, R. (2001). Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch. Rat. Mech. Anal., 160, 1-39.
  65. Desjardins, B. (1997). Regularity of weak solutions of compressible isentropic Navier-Stokes equations. Commun. Partial Diff. Equations, 22, 977-1008.
  66. Desjardins, B. and Chi-Kun, L. (1999). A survey of the compressible Navier- Stokes equations. Taiwanese J. Math., 3(2), 123-37.
  67. Ding, X., Chen, G.-Q., and Lero, P. (1985). Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics I-II. Acta Mathematica Scientia Series, 5, 483-500, 501-540.
  68. DiPerna, R. J. (1983a). Convergence of approximate solutions to conservation laws. Arch. Rat. Mech. Anal., 82, 27-70.
  69. DiPerna, R. J. (1983b). Convergence of the viscosity method for isentropic gas dynamics. Commun. in Math. Phys., 91, 1-30.
  70. DiPerna, R. J. and Lions, P.-L. (1989). Ordinary differential equations, trans- port theory, and Sobolev spaces. Inventiones Mathematicae, 98, 511-547.
  71. Doktor, A. (1977). Global solution of mixed problem for a certain system of nonlinear conservation laws. Czechoslovak Math. J., 27, 69-95.
  72. Dunford, N. and Schwartz, J. T. (1958). Linear Operators, Volume I. Inter- science Publishers, New York, London.
  73. Dunford, N. and Schwartz, J. T. (1963). Linear Operators, Volume II. Inter- science Publishers, New York, London.
  74. Dutto, P. and Novotný, A. (2001). Physically reasonable solutions to steady compressible Navier-Stokes equations in 2d exterior domains with nonzero velocity at infinity. J. Math. Fluid Mech., 3, 99-138.
  75. Edwards, R. E. (1965). Functional Analysis, Theory and Applications. Holt, Rinehart and Winston, New York.
  76. Erban, R. (2001). On the static-limit solutions to the Navier-Stokes equations of compressible flow. J. Math. Fluid Mech., 3, 393-408.
  77. Evans, L. C. (1992). Measure Theory and Fine Properties of Functions.C R C Press, Boca Raton, FL.
  78. Evans, L. C. (1998). Partial Differential Equations in Mathematics, Volume 19. AMS, Providence.
  79. Farwig, R. (1989). Stationary solutions of the Navier-Stokes equations with slip boundary conditions. Commun. Partial Diff. Equations, 14, 1579-1606.
  80. Farwig, R. (1992). The stationary exterior 3-d problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces. Math. Zeitschrift, 221, 409-447.
  81. Feireisl, E. (2000). The dynamical systems approach to the Navier-Stokes equa- tions of compressible fluid. In Advances in Math. Fluid Mech. (ed. J. Málek, J. Nečas, and M. Rokyta). Springer, Berlin.
  82. Feireisl, E. (2001). On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable. Comment. Math. Univ. Carolinae, 42(1), 83-98.
  83. Feireisl, E. (2002). Compressible Navier-Stokes equations with a non-monotone pressure law. J. Diff. Equations, 184, 97-108.
  84. Feireisl, E. (2003a). Dynamics of Compressible Flow. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford.
  85. Feireisl, E. (2003b). On the motion of rigid bodies in a viscous compressible fluid. Arch. Rat. Mech. Anal., 167, 281-308.
  86. Feireisl, E. (2003c). Propagation of oscillations, complete trajectories and at- tractors for compressible flows. Nonlinear Differ. Equ. Appl., 10, 33-55.
  87. Feireisl, E. (2003d). Shape optimization in viscous compressible fluids. Appl. Math. and Optimization, 47, 59-78.
  88. Feireisl, E., Matušů-Nečasová, Š., Petzeltová, H., and Straškraba, I. (1999). On the motion of a viscous compressible fluid driven by a time-periodic external force. Arch. Rat. Mech. Anal., 149, 69-96.
  89. Feireisl, E., Novotný, A., and Petzeltová, H. (2001). On the existence of glob- ally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech., 3, 358-392.
  90. Feireisl, E., Novotný, A., and Petzeltová, H. (2002). On the domain dependence of solutions to the compressible Navier-Stokes equations of a barotropic fluid. Math. Meth. Appl. Sci., 25, 1045-1073.
  91. Feireisl, E. and Petzeltová, H. (1998). On the zero-velocity limit solutions to the Navier-Stokes equations of compressible flows. Manuscr. Math., 97, 109-116.
  92. Feireisl, E. and Petzeltová, H. (1999). Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow. Arch. Rat. Mech. Anal., 150, 77-96.
  93. Feireisl, E. and Petzeltová, H. (2000). On integrability up to the boundary of the weak solutions of the Navier-Stokes equations of compressible flows. Commun. Partial Diff. Equations., 25(3-4), 755-767.
  94. Feireisl, E. and Petzeltová, H. (2001a). Asymptotic compactness of global tra- jectories generated by the Navier-Stokes equations of compressible fluid. J. Diff. Equations., 173, 390-409.
  95. Feireisl, E. and Petzeltová, H. (2001b). Bounded absorbing sets for the Navier- Stokes equations of compressible fluid. Commun. Partial Diff. Equations, 26(7 &8 ) , 1133-1144.
  96. Feistauer, M. (1993). Mathematical Methods in Fluid Dynamics. Longman Scientific & Technical, Harlow.
  97. Feistauer, M., Felcman, J., and Straškraba, I. (2003). Mathematical and Com- putational Methods for Compressible Flow. Oxford University Press, Oxford.
  98. Filatov, A. N. and Sharova, L. V. (1976). Integral Inequalities in Theory of Nonlinear Oscillations. Izd. Nauka, Moscow. In Russian.
  99. Finn, R. (1959). Estimates in infinity for stationary solutions of the Navier- Stokes equations. Bull. Math. Soc. Sci. Math. Phys. R.P.R., 3(53), 387-418.
  100. Finn, R. (1965). On the exterior stationary problem for the Navier-Stokes equations and associated perturbation problems. Arch. Rat. Mech. Anal., 19, 363-406.
  101. Fonseca, I. and Gangbo, W. (1995). Degree Theory in Analysis and Applica- tions. Oxford Science Publications. Oxford University Press, Oxford.
  102. Frehse, J., Goj, S., and Málek, J. (2002). A Stokes-like system for mixtures. Rheinische Friedrich-Wilhelms-Universität Bonn, SFB 611, no. 7.
  103. Friedrichs, K. O. (1948). Nonlinear hyperbolic differential equations in two independent variables. Amer. J. Math., 70, 555-558.
  104. Friedrichs, K. O. (1954). Symmetric hyperbolic linear differential equations. Commun. on Pure and Appl. Math., 7, 345-392.
  105. Gajewski, H., Gröger, K., and Zacharias, K. (1974). Nichtlineare Operatorgle- ichungen und Operatordifferentialgleichungen. Academie Verlag, Berlin.
  106. Galdi, G. P. (1994a). An Introduction to the Mathematical Theory of Navier- Stokes Equations, Volume 1. Springer, Berlin.
  107. Galdi, G. P. (1994b). An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol.I, II, Volume 38, 39 of Springer Tracts in Nat- ural Philosophy. Springer.
  108. Glimm, J. (1965). Solutions in the large for nonlinear hyperbolic systems of equations. Commun. on Pure and Appl. Math., 18(4), 697-715.
  109. Godlewski, E. and Raviart, P. A. (1996). Numerical Approximation of Hyper- bolic Systems of Conservation Laws. Springer, Berlin. Godunov, S. K. and Romenskij, E. J. (1998). Elements of Continuum Mechanics and Conservation Laws. Nauchnaya kniga. Novosibirsk. In Russian.
  110. Grassin, M. and Serre, D. (1997). Existence de solutions globales et régulières aux équations d'Euler pour un gaz parfait isentropique. C.R. Acad. Sci. Paris, 325, 721-726.
  111. Hewitt, T. and Stromberg, K. (1975). Real and Abstract Analysis. Springer, New York.
  112. Heywood, J. G. (1976). On uniqueness questions in the theory of viscous flow. Acta Math., 136, 61-102.
  113. Heywood, J. G. and Padula, M. (1999). On the existence and uniqueness the- ory for steady compressible viscous flow. In Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel.
  114. Hoff, D. (1992a). Global well-posedness of the cauchy problem for nonisentropic gas dynamics with discontinuous initial data. J. Diff. Eqs., 95, 33-73.
  115. Hoff, D. (1992b). Spherically symmetric solutions of the navier-stokes equations for compressible, isothermal flow with large, discontinuous initial data. Ind. Univ. Math. J., 41, 1225-1302.
  116. Hoff, D. (1995a). Global solutions of the Navier-Stokes equations for multidi- mensional, compressible flow with discontinuous initial data. J. Diff. Equa- tions, 120, 215-254.
  117. Hoff, D. (1995b). Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rat. Mech. Anal., 132, 1-14. REFERENCES
  118. Hoff, D. (1997). Discontinuous solutions of the Navier-Stokes equations for multidimensional heat-conducting flows. Arch. Rational Mech. Anal., 139, 303-354.
  119. Hsiao, L. (1997). Quasilinear Hyperbolic Systems and Dissipative Mechanisms. World Scientific, Singapore.
  120. Jiang, S. (1998). Global solutions of the Cauchy problem for a viscous polytropic ideal gas. Ann. Scuola Norm. Sup. Pisa Cl Sci., XXVI(4), 47-74.
  121. Jiequan, L., Tong, Z., and Shuli, Y. (1998). The Two-Dimensional Riemann Problem in Gas Dynamics, Volume 98 of Pitman Monographs. Longman, Harlow.
  122. Jin, B. J. and Padula, M. (2003). Steady flows of compressible fluids in a rigid container with upper free boundary. To appear in Math. Annalen.
  123. John, F. (1974). Formation of singularities in one-dimensional wave propaga- tion. Commun. on Pure and Appl. Math., 27, 377-405.
  124. John, F. (1981). Blow-up for quasilinear wave equations in three space dimen- sions. Commun. on Pure and Appl. Math., 34, 29-51.
  125. Kato, T. (1972). Nonstationary flows of viscous and ideal fluids in IR 3 . J. Funct Anal., 9, 296-305.
  126. Kato, T. (1975). The Cauchy problem for quasilinear symmetric hyperbolic systems. Arch. Rat. Mech. Anal., 58, 181-205.
  127. Kobayashi, T. and Shibata, Y. (1998). On the Oseen equation in exterior domains. Math. Annalen, 310, 1-45.
  128. Kolmogorov, A. N. and Fomin, S. V. (1972). Elements of the Theory of Func- tions and Functional Analysis (3rd edn). Nauka, Moscow. Russian.
  129. Kračmar, S., Novotný, A., and Pokorný, M. (2001). Estimates of Oseen kernels in weighted L p -spaces. J. Math. Soc. Japan, 53(1), 59-111.
  130. Krejč´ı, P. (1996). Hysteresis, Convexity and Dissipation in Hyperbolic Equa- tions, Volume 8 of Mathematical Sciences and Applications. Gakkōtosho, Tokyo.
  131. Krejč´ı, P. and Straškraba, I. (1997). A uniqueness criterion for the Riemann problem. Hiroshima Math. J., 27(2), 307-346.
  132. Kruzhkov, S. (1970). First order quasilinear equations with several space vari- ables. Math. USSR Sb., 10, 217-243.
  133. Kufner, A., John, O., and Fuč´ık, S. (1977). Function Spaces. Academia, Prague.
  134. Kurzweil, J. (1986). Ordinary Differential Equations. Introduction to the Theory of Ordinary Differential Equations in the Real Domain, Volume 13 of Studies in Appl. Math. Elsevier, Amsterdam.
  135. Kweon, J. R. and Kellogg, R. B. (2003). Regularity of solutions to the Navier- Stokes system for compressible flows on a polygon. submitted SIAM J. Math. Anal.
  136. Ladyzhenskaya, O. A. and Solonnikov, V. A. (1976). On some problems of vector analysis and generalized formulations of boundary value problems for the Navier-Stokes equations. Zapiski Nauchn. Sem. LOMI , 59, 81-116. English Transl.: J. Sov. Math., 10,(2), 257-285 (1978).
  137. Ladyzhenskaya, O. A. and Solonnikov, V. A. (1977). On the solvability of boundary value problems for the Navier-Stokes equations in regions with non- compact boundaries. Vestnik Leningrad. Univ., 13(3), 39-47. English Transl.: Vestnik Leningrad. Univ. Math., 10, 271-280 (1982).
  138. Landau, L. D. and Lifschitz, E. M. (1959). Fluid Mechanics. Pergamon Press, Oxford.
  139. Lax, P. (1953). Nonlinear hyperbolic equations. Commun. on Pure and Appl. Math., 6, 231-258.
  140. Lax, P. D. (1954). Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. on Pure and Appl. Math., 7, 159-193.
  141. Li, T. (1994). Global Classical Solutions for Quasilinear Hyperbolic Systems. Wiley, Chichester.
  142. Li, T. and Yu, W. (1985). Boundary Value Problems for Quasilinear Hyperbolic Systems, Volume V. Duke Univ. Math. Ser., Durham NC, Duke University.
  143. Lieb, E. H. and Loss, M. (1997). Analysis, Graduate Studies in Mathematics, Volume 14. AMS, Providence.
  144. Lin, L.-W. (1987). On the vacuum state for the equations of isentropic gas dynamics. J. Math. Anal. Appl., 121, 406-425.
  145. Lions, J.-L. and Magenes, E. (1968). Problèmes aux limites non homogènes et applications. Dunod, Paris.
  146. Lions, P.-L. (1993a). Compacité des solutions des équations de Navier-Stokes compressibles isentropiques. C.R. Acad. Sci. Paris, 317, 115-120.
  147. Lions, P.-L. (1993b). Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques. C.R. Acad. Sci. Paris, 316, 1335- 1340.
  148. Lions, P.-L. (1998). Mathematical Topics in Fluid Mechanics, Volume 2 of Oxford Lecture Series in Mathematics and its Applications. Clarendon Press, Oxford.
  149. Lions, P.-L. (1999). Bornes sur la densité pour les équations de Navier-Stokes compressible isentropiques avec conditions aux limites de Dirichlet. C.R. Acad. Sci. Paris, 328, 659-662.
  150. Lions, P. L., Perthame, B., and Souganidis, P. E. (1996). Existence and stabil- ity of entropy solutions for the hyperbolic systems of isentropic gas dynam- ics in Eulerian and Lagrangian coordinates. Commun. on Pure and Appl. Math., 49(6), 599-638.
  151. Liu, T.-P. (1979). The development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations. J. Diff. Eqs., 33, 92-111.
  152. Lizorkin, P. I. (1952). (L p ,L q )-multipliers of Fourier integrals. Dokl. Akad. Nauk SSSR, 4, 808-811. Engl. transl.: Soviet Math. Dokl. 4 (1963), 1420- 1424.
  153. Lizorkin, P. I. (1969). About Fourier multipliers in L p,θ .InImbedding Theorems and Applications, Baku 1966, Moscow, pp. 137-142. Nauka.
  154. Lizorkin, P. I. and Nikol'skii, S.M. (1965). Classification of differentiable func- tions on the basis of spaces with dominating mixed derivative. Trudy Mat. Inst. Steklov , 77, 143-167.
  155. Ljusternik, L. A. and Sobolev, V. I. (1974). Elements of Functional Analysis. Hindustan Publ. Corp., Delphi-New York.
  156. Lovicar, V. and Straškraba, I. (1991). Remark on cavitation solutions of sta- tionary compressible Navier-Stokes equations in one dimension. Czechoslovak Mathematical Journal , 41(4), 653-662.
  157. Lu, Y. (2003). Hyperbolic Conservation Laws and the Compensated Compact- ness, Volume 128 of Monographs and Surveys in Pure and Appl. Math. Chap- man & Hall, New York.
  158. Lukeš, J. and Malý, J. (1995). Measure and Integral. Matfyzpress, Fac. of Math. and Phys.,Charles Univ., Prague.
  159. Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel.
  160. Majda, A. (1984). Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York.
  161. Makino, T., Kawashima, S., and Ukai, S. (1986). Sur la solution à support compact des équations d'Euler compressible. Japan J. Appl. Math., 3, 249- 257. Makino, T., Mizohata, K., and Ukai, S. (1992). The global weak solutions of the compressible Euler equations with spherical symmetry. Japan J. Indust. Appl. Math., 9, 431-449.
  162. Málek, J. and Nečas, J. (1996). A finite-dimensional attractor for the three dimensional flow of incompressible fluid. J. Diff. Equations, 127, 498-518.
  163. Málek, J., Nečas, J., Rokyta, M., and Růžička, M. (1996). Weak and Measure- Valued Solutions to Evolutionary PDE's. Chapman and Hall, London.
  164. Matsumura, A. and Nishida, T. (1979). The initial value problem for the equa- tions of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A Math. Sci., 55, 337-342.
  165. Matsumura, A. and Nishida, T. (1980). The initial value problem for the equa- tions of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ., 20, 67-104.
  166. Matsumura, A. and Nishida, T. (1989). Exterior stationary problems for the equations of motion of compressible heat-conductive fluids. In Proc. Equadiff 89 (ed. C. M. D. et al.), Amsterdam, pp. 473-479. Dekker.
  167. Matsumura, A. and Padula, M. (1992). Stability of stationary flows of com- pressible fluids subject to large external potential forces. Stab. and Appl. Anal. in Cont. Media, 2, 183-202.
  168. Matsumura, A. and Yamagata, N. (2001). Global weak solutions of the Navier- Stokes equations for multidimensional compressible flow subject to large ex- ternal forces. Osaka J. Math., 38, 399-418.
  169. Matušů, Š., Okada, M., and Makino, T. (1997). Free boundary problem for the equation of spherically symmetric motion of viscous gas (iii). Japan J. Industrial and Appl. Math., 14(2), 199-213.
  170. Mitrea, D. and Mitrea, M. (2001). Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds. AMS, Providence RI.
  171. Mizohata, K. (1994). Equivalence of Eulerian and Lagrangian weak solutions of the compressible Euler equations with spherical symmetry. Kodai Math. J., 17, 69-81.
  172. Mucha, P. B. and Zajaczkowski, W. (2002). Global existence of solutions of the Dirichlet problem for the compressible Navier-Stokes equations. ZAMM To appear.
  173. Murat, F. (1978). Compacité par compensation. Annali della Scuola Normale Superiore di Pisa, 5, 489-507.
  174. Navier, H. (1827). Mémoire sur les lois du mouvement des fluides. Mem. Acad. Sci. Inst. de France, 2(6), 389-440.
  175. Nazarov, S., Novotný, A., and Pileckas, K. (1996). On steady compress- ible Navier-Stokes equations in plane domains with corners. Math. Anna- len, 304(1), 121-150.
  176. Nečas, J. (1967). Les méthodes directes en théorie des équations elliptiques. Academia, Prague.
  177. Nirenberg, L. (1974). Topics in Nonlinear Functional Analysis. Courant Insti- tute of Mathematical Sciences, New York University, New York.
  178. Nitsche, J. A. (1981). On Korn's second inequality. RAIRO Anal. Numér., 15, 237-248.
  179. Novo, S. (2003). The compressible Navier-Stokes equations with influx-outflux boundary conditions. J. Math. Fluid Mech. To appear.
  180. Novo, S. and Novotný, A. (2002). On the existence of weak solutions to steady compressible Navier-Stokes equations when the density is not square inte- grable. J. Math. Kyoto Univ., 42(3), 531-550.
  181. Novo, S. and Novotný, A. (2003a). Existence of weak solutions to compressible barotropic Navier-Stokes equations in domains with several conical exits. Preprint ANLA, Université de Toulon et du Var.
  182. Novo, S. and Novotný, A. (2003b). Some notes to the existence of weak solutions to the steady compressible Navier-Stokes equations in bounded domains with Lipschitz boundary. Preprint ANLA, Université de Toulon et du Var.
  183. Novo, S., Novotný, A., and Pokorný, M. (2003). Steady compressible isentropic Navier-Stokes equations in domains with noncompact boundaries. Preprint, Université de Toulon et du Var.
  184. Novotný, A. (1996). Some remarks about the compactness of steady compress- ible Navier-Stokes equations via the decomposition method. Comment. Math. Univ. Carolinae, 37(2), 305-342.
  185. Novotný, A. (1998). On the steady transport equation. In Advanced Topics in Theoretical Fluid Mechanics (ed. J. Málek, J. Nečas, and M. Rokyta), Volume 392 of Pitman Res. Notes in Math. Series, pp. 118-46. Longman, Essex.
  186. Novotný, A. and Padula, M. (1994). L p approach to steady flows of viscous compressible fluids in exterior domains. Arch. Rat. Mech. Anal , 126, 243-297. REFERENCES
  187. Novotný, A. and Padula, M. (1997). Physically reasonable solutions for steady compressible Navier-Stokes equations in 3d exterior domains (v ∞ = 0). Math. Annalen, 308, 439-489.
  188. Novotný, A., Padula, M., and Penel, P. (1996). Remark on the well posedness of the problem of a steady flow of viscous barotropic gas in a pipe. Commun. Partial Diff. Equations, 21, 23-24.
  189. Novotný, A. and Pileckas, K. (1998). Steady compressible Navier-Stokes equa- tions with large potential forces via a method of decomposition. Math. Meth. Appl. Sci., 21, 665-684.
  190. Novotný, A. and Straškraba, I. (2000). Stabilization of solutions to compressible Navier-Stokes equations. J. Math. Kyoto Univ., 40(2), 217-245.
  191. Novotný, A. and Straškraba, I. (2001). Convergence to equilibria for com- pressible Navier-Stokes equations with large data. Ann. di Mat. Pura ed Appl., CLXXIX(IV), 263-287.
  192. Padula, M. (1981). Existence and continuous dependence for solutions to the equations of a one-dimensional model in gas dynamics. Meccanica, 17, 128- 135.
  193. Padula, M. (1982). Existence and uniqueness for viscous steady compressible motions. In Proc. Sem. Fis. Mat., Dinamica dei Fluidi e dei gaz ionizzati, Trieste.
  194. Padula, M. (1986). Existence of global solutions for 2-dimensional viscous com- pressible flow. J. Funct. Anal., 69, 1-20.
  195. Padula, M. (1987). Existence and uniqueness for viscous, steady compressible motions. Arch. Rat. Anal., 77, 89-102.
  196. Padula, M. (1988a). Erratum. J. Funct. Anal., 76, 231-231.
  197. Padula, M. (1988b). Erratum. J. Funct. Anal., 77, 232-232.
  198. Padula, M. (1997). Steady flows of barotropic viscous fluids. In Classical Problems in Mechanics, Volume 1, pp. 253-345. Dipartimento di Matemat- ica Seconda Università di Napoli, Caserta.
  199. Penel, P. and Straškraba, I. (1999). Global behaviour of compressible fluid with a free boundary and large data. In Appl. Nonlin. Anal. (ed. A. Sequeira), pp. 427-442. Kluwer Academic, New York.
  200. Penel, P. and Straškraba, I. (2003). Global behavior of one-dimensional com- pressible flows with vanishing viscosity near vacuums. J. of Concrete and Applicable Math., 1(4), 1-17.
  201. Pileckas, K. (1980). On unique solvability of boundary value problems for the Stokes system of equations in domains with noncompact boundaries. Trudy Mat. Inst. Steklov , 147, 115-123. English Transl.: Proc. Mat. Inst. Steklov, 147(2),117-126.
  202. Pileckas, K. (1996). Recent advances in the theory of Stokes and Navier-Stokes equations in domains with noncompact boundaries. In Nonlinear Partial Dif- ferential Equations and their Applications, Volume 304 of Pitman Research Notes in Mathematics, pp. 30-85. Longman, Harlow. Collège de France Seminar.
  203. Pileckas, K. and Solonnikov, V. A. (1977). Certain spaces of solenoidal vectors and solvability of the boundary value problem for the Navier-Stokes system of equations in domains with nonocompact boundaries. Zapiski Nauchn. Sem. LOMI , 73, 136-151. English Transl.: J. Sov. Math., 34(6), 2101-2111.
  204. Plotnikov, P. I. and Sokolowski, J. (2002). On compactness domain depen- dence and existence of steady state solutions to compressible Navier-Stokes equations. Preprint, Univ. H. Poincaré, Nancy.
  205. Poisson, S. D. (1829). Mémoire sur les équations générales de l'équilibre et du mouvement des corps solides élastiques et des fluides. J. Ecole Polytech- nique, 13, 1-174.
  206. Rajagopal, K. R. and Tao, L. (1995). Mechanics of Mixtures, Volume 35 of Series on Advances in Mathematics for Applied Sciences. World Scientific, London.
  207. Rajagopal, K. R. and Truesdell, C. (1999). An Introduction to the Mechanics of Fluids. Birkhäuser, Boston.
  208. Rammaha, M. A. (1989). Formation of singularities in compressible fluids in two space dimensions. Proc. Amer. Math. Soc., 107, 705-714.
  209. Rudin, W. (1974). Real and Complex Analysis. McGraw-Hill, London, New York.
  210. Salvi, R. and Straškraba, I. (1993). Global existence for viscous compressible fluids and their behaviour as t →∞ . J. Fac. Sci. Univ. Tokyo Section IA Mathematics, 40(1), 17-52.
  211. Serre, D. (1991). Variation de grande amplitude pour la densité d'un fluid viscqueux compressible. Physica D, 48, 113-128.
  212. Serre, D. (1996). Systèmes de lois de conservation. Diderot, Paris. English translation: Systems of Conservation Laws, volumes 1-2, Cambridge University Press, 1999.
  213. Serre, D. (1997). Solutions classiques globales des équations d'Euler pour un fluide parfait compressible. Annales de l'Institut Fourier, Grenoble, 47, 139- 153.
  214. Shearer, J. W. (1994). Global existence and compactness in L p for the quasi- linear wave equation. Commun. in Partial Diff. Equations, 19, 1829-1877.
  215. Sideris, T. C. (1985). Formation of singularities in three-dimensional compress- ible fluids. Commun. in Math. Phys., 101, 475-485.
  216. Sideris, T. C. (1991). The lifespan of smooth solutions to the three-dimensional Euler equations and the incompressible limit. Indiana Univ. Math. J., 40, 535-550.
  217. Sideris, T. C. (1997). Delayed singularity formation in 2D compressible flow. Amer. J. Math., 119, 371-422.
  218. Šilhavý, M. (1997). The Mechanics and Thermodynamics of Continuum Media. Springer, Berlin.
  219. Simader, C. G. (1990). The weak weak Dirichlet and Neumann problem for the Laplacian in L q for bounded and exterior domains. In Applica- tions of Nonlinear Analysis, Function Spaces and Applications (ed. M. Krbec, A. Kufner, B. Opic, and J. Rákosník), Volume 4, pp. 180-223. Teubner, Leipzig.
  220. Simader, C. G. and Sohr, H. (1992a). A new approach to the Helmholtz de- composition and the Neumann problem in L q -spaces for bounded and exterior domains. In Mathematical Problems Relating to the Navier-Stokes Equations (ed. G. P. Galdi), Volume 11 of Advances in Mathematics for Applied Sciences, pp. 1-34. World Scientific, London.
  221. Simader, C. G. and Sohr, H. (1992b). The weak Dirichlet problem for ∆ in L q in bounded and exterior domains. Stab. Anal. Cont. Media, 2, 183-202.
  222. Simader, C. G. and Sohr, H. (1996). The Dirichlet Problem for the Laplacian in Bounded and Ubounded Domains, Volume 360 of Pitman Res. Notes in Math. Longman, Essex.
  223. Simon, J. (1987). Compact sets in the space L p (0,T; B). Ann. Mat. Pura Appl., 146, 65-96.
  224. Slemrod, M. (1996). Resolution of spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit. Proceedings of the Royal Society of Edinburgh, 126A, 1309-1340.
  225. Smoller, J. (1983). Shock Waves and Reaction-Diffusion Equations. Springer, New York.
  226. Solonnikov, V. A. (1981). On the solvability of boundary and initial boundary value problems for the Navier-Stokes system in domains with noncompact boundaries. Pacific J. Math., 93(2), 443-458.
  227. Solonnikov, V. A. (1983). Stokes and Navier-Stokes equations in domains with noncompact boundaries. In Collège de France Seminar, 4, Volume 84 of Pit- man Res. Notes in Math., pp. 240-349. Longman, Essex.
  228. Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Func- tions. Princeton University Press.
  229. Stokes, G. G. (1845). On the theories of the integral friction of fluids in motion. Trans. Cambridge Phil. Soc., 8, 287-319.
  230. Straškraba, I. (1984). A note on critical times of 2 × 2 quasilinear hyperbolic systems. Aplikace matematiky, 29, 294-302.
  231. Straškraba, I. (1996). Asymptotic development of vacuums for 1-dimensional Navier-Stokes equations of compressible flow. Nonlinear World , 3, 519-535.
  232. Straškraba, I. (1997). Global analysis of 1-D Navier-Stokes equations with den- sity dependent viscosity. In Navier-Stokes Equations and Related Nonlinear Problems (ed. H. A. et al.), pp. 371-390. NSEC-6, VSP Utrecht, TEV Vilnius, Palanga, Lithuania.
  233. Straškraba, I. (1998). Large time behavior of solutions to compressible Navier- Stokes equations. In Navier-Stokes Equations: Theory and Numerical Methods (ed. R. Salvi), Volume 388, pp. 125-138. Longman, Edinburgh.
  234. Straškraba, I. and Valli, A. (1988). Asymptotic behaviour of the density for one-dimensional Navier-Stokes equations. Manuscr. Math., 62, 401-416.
  235. Straškraba, I. and Zlotnik, A. (2001). On global behaviour of solutions to equations of one-dimensional viscous compressible barotropic fluid with vari- able viscosity. Dokl. Russian Acad. Sci., 378(6), 737-742.
  236. Straškraba, I. and Zlotnik, A. (2002). On a decay rate for 1d-viscous compress- ible barotropic fluid equations. J. Evol. Equ., 2, 69-96.
  237. Straškraba, I. and Zlotnik, A. (2003a). Global behavior of 1d-viscous com- pressible barotropic fluid with a free boundary and large data. J. Math. Fluid Mech., 5, 119-143.
  238. Straškraba, I. and Zlotnik, A. (2003b). Global properties of solutions to 1d- viscous compressible barotropic fluid equations with density dependent vis- cosity. ZAMP , 54, 593-607.
  239. Tanabe, H. (1961). Evolution equations of parabolic type. Japan Acad. Proc., 37, 610-613.
  240. Tanaka, N. and Tani, A. (2003). Surface waves for a compressible fluid. J. Math. Fluid Mech., 5, 303-363.
  241. Tartar, L. (1975). Compensated compactness and applications to partial dif- ferential equations. In Nonlin. Anal. and Mech. (ed. L. Knopps), Res. Notes in Math. 39, Boston, pp. 136-211. Heriot-Watt Sympos.: Pitman.
  242. Taylor, A. E. (1967). Introduction to Functional Analysis. Wiley, New York.
  243. Temam, R. (1983). Problèmes mathématiques en plasticité. Gautiers-Villars, Paris.
  244. Torchinsky, A. (1986). Real-Variable Methods in Harmonic Analysis, Volume 123 of Pure and Applied Mathematics. Academic Press, New York.
  245. Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Opera- tors. North-Holland, Amsterdam.
  246. Triebel, H. (1995). Interpolation Theory, Function Spaces, Differential Opera- tors. Barth, Heidelberg.
  247. Vaigant, V. A. (1994). An example of the nonexistence with respect to time of global solutions of Navier-Stokes equations for compressible viscous barotropic fluid. Dokl. Acad. Nauk , 339(2), 155-156. Russian.
  248. Vaigant, V. A. and Kazhikhov, A. V. (1995). On the existence of global so- lutions to two-dimensional Navier-Stokes equations of compressible viscous fluid. Sibirskij Mat. Z., 36, 1108-1141. Russian.
  249. Valli, A. (1983). Periodic and stationary solutions for compressible Navier- Stokes equations via a stability method. Ann. Scuola Normale Sup. Pisa, 4(1), 607-646.
  250. Valli, A. (1987). On the existence of stationary solutions to compressible Navier-Stokes equations. Ann. Inst. H. Poincaré, Anal. Non-Lin., 4, 99-113.
  251. Valli, A. (1992). Mathematical results for compressible flows. In Mathematical Topics in Fluid Mechanics (ed. J. Rodrigues and A. Sequeira), Volume 274, pp. 193-229. Wiley, New York.
  252. Valli, A. and Zajaczkowski, W. M. (1986). Navier-Stokes equations for com- pressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. in Math. Phys., 103, 256-296.
  253. Visintin, A. (1984). Strong convergence results related to strong convexity. Commun. Partial Diff. Equations, 9(5), 439-466.
  254. Vladimirov, V. S. (1967). Equations of Mathematical Physics. Nauka, Moscow.
  255. Volpert, A. I. (1967). The space BV and quasilinear equations. Mat. USSR Sbornik , 2, 225-267.
  256. Wagner, D. H. (1987). Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions. J. Diff. Eqs., 68, 118-136.
  257. Whitham, G. B. (1974). Linear and Nonlinear Waves. Wiley, New York.
  258. Xin, Z. P. (1998). Blowup of smooth solutions to the compressible Navier- Stokes equation with compact density. Commun. on Pure and Appl. Math., LI, 229-240.
  259. Yi, Z. (1992). An L p theorem for compensated compactness. Proc. Royal Soc. Edinburgh, 122 A, 177-189.
  260. Yosida, K. (1974). Functional Analysis. Springer, New York.
  261. Zeidler, E. (1985-89). Nonlinear Functional Analysis and its Applications, Volumes 1-4. Springer, New York.
  262. Zhang, T. and Zheng, Y. (1997). Exact spiral solutions of two dimensional compressible Euler equations. Discrete and Continuous Dynamical Systems, 3, 117-133.
  263. Zhang, T. and Zheng, Y. (1998). Axisymmetric solutions of the Euler equations for square polytropic gases. Arch. Rat. Mech. Anal., 142, 253-279.
  264. Ziemer, W. P. (1989). Weakly Differentiable Functions. Springer, New York.