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Analysis of Viscous Fluid Flows: An Approach by Evolution Equations | SpringerLink

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Analysis of Viscous Fluid Flows: An Approach by Evolution Equations

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Mathematical Analysis of the Navier-Stokes Equations

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Abstract

This course of lectures discusses various aspects of viscous fluid flows ranging from boundary layers and fluid structure interaction problems over free boundary value problems and liquid crystal flow to the primitive equations of geophysical flows. We will be mainly interested in strong solutions to the underlying equations and choose as mathematical tool for our investigations the theory of evolution equations. The models considered are mainly represented by semi- or quasilinear parabolic equations and from a modern point of view it is hence natural to investigate the underlying equations by means of the maximal L p-regularity approach.

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References

  1. K. Abe, Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions. Acta Math. 211, 1–46 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Abe, Y. Giga, M. Hieber, Stokes resolvent estimates in spaces of bounded functions. Ann. Sci. Ec. Norm. Super. 48, 521–543 (2015)

    MathSciNet  MATH  Google Scholar 

  3. H. Abels, Bounded imaginary powers and H ∞-calculus for the Stokes operator in two-dimensional exterior domains. Math. Z. 251, 589–605 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound. 9, 31–65 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Abels, L. Dienig, Y. Terasawa, Existence of weak solutions for a diffusive interface models of non-Newtonian two-phase flows, Nonlinear Anal. Ser. B, 15, 149–157 (2014)

    Article  MATH  Google Scholar 

  6. H. Amann, Dynamic theory of quasilinear parabolic equations I: abstract evolution equations. Nonlinear Anal. 12, 895–919 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Amann, Stability of the rest state of a viscous incompressible fluid. Arch. Ratational Mech. Anal. 126, 231–242 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Amann, Linear and Quasilinear Parabolic Problems, vol. I (Birkhäuser, Basel, 1995)

    Google Scholar 

  9. H. Amann, Stability and bifurcation in viscous incompressible fluids. Zapiski Nauchn. Seminar. POMI 233, 9–29 (1996)

    MATH  Google Scholar 

  10. H. Amann, On the strong solvability of the Navier–Stokes equations. J. Math. Fluid Mech. 2, 16–98 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Amann, Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Necas Center for Mathematical Modeling Lecture Notes (Prague, 2009)

    Google Scholar 

  12. H. Amann, Linear and Quasilinear Parabolic Problems, vol. II (Birkhäuser, Basel, 2019)

    Google Scholar 

  13. W. Arendt, Ch. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, 2nd edn. (Birkhauser, Basel, 2011)

    Book  MATH  Google Scholar 

  14. H. Bahouri, J, Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 (Springer, Grundlehren, 2011)

    Google Scholar 

  15. J. T. Beale, Large-time regularity of viscous surface waves. Arch. Rational Mech. Anal. 84, 307–352 (1983–1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Benedek, A.P. Calderón, R. Panzone, Convolution operators on Banach space valued functions. Proc. Nat. Acad. Sci. USA 48, 356–365 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Berkson, T. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces. Studia Math. 112, 13–49 (1994)

    MathSciNet  MATH  Google Scholar 

  18. D. Bothe, J. Prüss, L p-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39, 379–421 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Boulakia, S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 777–813 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21, 163–168 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Bourgain, Vector-valued singular integrals and the H 1 − BMO duality, in Probabilty Theory and Harmonic Analysis (Marcel-Dekker, New York, 1986), pp. 1–19

    Google Scholar 

  22. J. Bourgain, Vector-valued Hausdorff-Young inequalities and applications, in Geometric Aspects of Functional Analysis (Springer, Berlin, 1988), pp. 239–249

    Google Scholar 

  23. D. Burkholder, Martingale transforms and the geometry of Banach spaces, in Probability in Banach Spaces, III (Springer, Berlin, 1981), pp. 35–50

    Google Scholar 

  24. D. Burkholder, A geometrical condition that implies the existence of certain singular integrals of Banach-space-valued functions, in ed. by W. Beckner, A.P. Calderón, R. Fefferman, P.W. Jones. Confernece Harm Anal., 1981, pp. 270–286, Wadsworth, 1983.

    Google Scholar 

  25. Ch. Cao, E. Titi, Global well–posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. Math. 166, 245–267 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Mathemtical Geophysics. Oxford Lecture Series in Mathematics and Its Applications, vol. 32 (Oxford University Press, Oxford, 2006)

    Google Scholar 

  27. P. Clément, S. Li, Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl. 3, 17–32 (1993–1994)

    MathSciNet  MATH  Google Scholar 

  28. P. Clément, B. de Pagter, F. Sukochev, H. Witvliet, Schauder decomposition and multiplier theorems. Studia Math. 138, 135–163 (2000)

    MathSciNet  MATH  Google Scholar 

  29. P. Clément, J. Prüss, An operator-valued transference principle and maximal regularity on vector-valued L p-sapces, in Evolution Equations, ed. by G. Lumer, L. Weis. Lectures Notes Pure Applied Mathematics, vol. 215 (Marcel Dekker, New York, 2001), pp. 67–87

    Google Scholar 

  30. R. Danchin, P. Mucha, A critical functional framework for the inhomogeneous Navier–Stokes equations in the half space. J. Funct. Anal. 256, 881–927 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Danchin, P. Mucha. Critical functional framework and maximal regularity in action on systems of incompressible flows. Memoirs Soc. Math. France 143, 151 (2015)

    MathSciNet  MATH  Google Scholar 

  32. G. Da Prato, P. Grisvard, Sommes d’opérateurs linéaires et équations différentielles opérationelles. J. Math. Pures Appl. 54, 305–387 (1975)

    MathSciNet  MATH  Google Scholar 

  33. F. De Anna, C. Liu, Non-isothermal general Ericksen-Leslie system: derivation, analysis and thermodynamic consistency. Arch. Rational Mech. Anal. 231, 637–717 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Denisova, A priori estimates for the solution of the linear nonstationary problem connected with the motion of a drop in a liquid medium, Proc. Stekhlov Inst. Math. 3, 1–24 (1991)

    Google Scholar 

  35. I. Denisova, Problem of the motion of two viscous incompressible fluids separated by a closed free interface, Acta Appl. Math. 37, 31–40 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. R. Denk, M. Hieber, J. Prüss, \(\mathcal R\)-Boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166, 144 (2003)

    Google Scholar 

  37. R. Denk, G. Dore, M. Hieber, J. Prüss, A. Venni, New thoughts on old ideas of R.T. Seeley. Math. Annalen, 166, 545–583 (2004)

    Google Scholar 

  38. R. Denk, M. Hieber, J. Prüss, Optimal L p-L q-estimates for parabolic boundary value problem with inhomogeneous data. Math. Z. 257, 193–224 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. R. Denk, M. Geissert, M. Hieber, J. Saal, O. Sawada, The spin-coating process: analysis of the free boundary value problem. Comm. Partial Differ. Equ. 36, 1145–1192 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. W. Desch, M. Hieber, J. Prüss, L p-theory of the Stokes equation in a half-space. J. Evol. Equ. 1, 115–142 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. B. Desjardins, M. Esteban, On weak solutions for fluid rigid structure interaction: compressible and incompressible models. Comm. Partial Differ. Equ. 25, 1399–1413 (2000)

    MathSciNet  MATH  Google Scholar 

  42. B. Desjardins, E. Dormy, E. Grenier, Stability of mixed Ekman-Hartmann boundary layers. Nonlinearity 12, 181–199 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. L. Diening, M. Ruzicka, Strong solutions for generalized Newtonian fluids. J. Math. Fluid Mech. 7, 413–450 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. G. Dore, A. Venni, On the closedness of the sum of two closed operators. Math. Z. 196, 189–201 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Duistermaat, J. Kolk, Distributions. Cornerstones Series (Birkhäuser, New York, 2011)

    Google Scholar 

  46. X. Duong, L. Yan, Bounded holomorphic functional calculus for non-divergence form differential operators. Diff. Int. Equ. 15, 709–730 (2002)

    MathSciNet  MATH  Google Scholar 

  47. V.W. Ekman, On the influence of the earth’s rotation on ocean currents. Arkiv Matem. Astr. Fysik, (Stockholm) 11, 1–52 (1905)

    Google Scholar 

  48. K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer, Berlin, 2000)

    MATH  Google Scholar 

  49. J. L. Ericksen, Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9, 371–378 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  50. L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence, 1998)

    MATH  Google Scholar 

  51. R. Farwig, H. Sohr, Generalized resolvent estimates for the Stokes operator in bounded and unbounded domains. J. Math. Soc. Japan 46, 607–643 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  52. E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167, 281–308 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. E. Feireisl, E. Rocca, G. Schimperna, On a non-isothermal model for nematic liquid crystals. Nonlinearity 24, 243–257 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. E. Feireisl, M. Frémond, E. Rocca, G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals. Arch. Ration. Mech. Anal. 205, 651–672 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. K. Furukawa, Y. Giga, M. Hieber, A. Hussein, T. Kashiwabara, M. Wrona, Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier–Stokes equations. Submitted.

    Google Scholar 

  56. G.P. Galdi, On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, in Handbook of Mathematical Fluid Dynamics, ed. by S. Friedlander, D. Serre, vol. I (Elsevier, North-Holland, 2002), pp. 653–791

    Google Scholar 

  57. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady State Problems, 2nd edn. (Springer, New York, 2011)

    Google Scholar 

  58. G.P. Galdi, J. Neustupa, Steady state Navier–Stokes flow around a moving body, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, ed. by Y. Giga, A. Novotny, vol. 1 (Springer, Beriln, 2018), pp. 341–418

    Google Scholar 

  59. M. Geissert, M. Hess, M. Hieber, C. Schwarz, K. Stavrakidis, Maximal L p–L q-estimates for the Stokes equation: a short proof of Solonnikov’s theorem. J. Math. Fluid Mech., 12, 47–60 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. M. Geissert, H. Heck, M. Hieber, O. Sawada, Weak Neumann implies Stokes. J. Reine Angew. Math. 669, 75–100 (2012)

    MathSciNet  MATH  Google Scholar 

  61. M. Geissert, K. Götze, M. Hieber, L p-theory fro strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids. Trans. Amer. Math. Soc. 365, 1393–1439 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  62. Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L p-spaces. Math. Z. 178, 297–329 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  63. Y. Giga, Domains of fractional powers of the Stokes operator in L r spaces. Arch. Ration. Mech. Anal. 89, 251–265 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  64. Y. Giga, H. Sohr, Abstract L p-estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  65. Y. Giga, M. Gries, M. Hieber, A. Hussein, T. Kashiwabara, Analyticity of solutions to the primitive equations. Math. Nachrichten, to appear.

    Google Scholar 

  66. Y. Giga, K. Inui, A. Mahalov, S. Matsui, J. Saal, Rotating Navier–Stokes equations in \(\mathbb R^3_+\) with initial data nondecreasing at infinity: the Ekman boundary layer problem. Arch. Ration. Mech. Anal. 186, 177–224 (2007)

    Google Scholar 

  67. Y. Giga, K. Inui, A. Mahalov, S. Matsui, J. Saal, Uniform global solvability of the rotating Navier–Stokes equations for nondecaying initial data. Indiana Univ. Math. J. 57, 2775–2791 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  68. Y. Giga, M. Gries, M. Hieber, A. Hussein, T. Kashiwabara, Bounded H ∞-calculus for the Hydrostatic Stokes operator on L p-spaces and applications. Proc. Amer. Math. Soc. 145, 3865–3876 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  69. L. Grafakos, Classical Fourier Analysis (Springer, Berlin, 2008)

    MATH  Google Scholar 

  70. G. Grubb, V.A. Solonnikov, Boundary value problems for the nonstationary Navier–Stokes equations treated by pseudo-differential methods. Math. Scand. 69, 217–290 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  71. M. Haase, The Functional Calculus for Sectorial Operators (Birkhäuser, Basel, 2006)

    Google Scholar 

  72. M. Hess, M. Hieber, A. Mahalov, J. Saal, Nonlinear stability of the Ekman boundary layers. Bull. London Math. Soc. 42, 691–706 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  73. M. Hieber, T. Kashiwabara, Global strong well–posedness of the three dimensional primitive equations in L p–spaces. Arch. Ration. Mech. Anal.221, 1077–1115 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  74. M. Hieber, M. Murata, The L p-approach to the fluid rigid body interaction problem for compressible fluids. Evol. Equ. Contr. Theory 4, 69–87 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  75. M. Hieber, J. Prüss, Functional calculi for linear operators in vector-valued L p-spaces via the transference principle. Adv. Diff. Equ. 3, 847–872 (1998)

    MathSciNet  MATH  Google Scholar 

  76. M. Hieber, J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress I: the incompressible isotropic case. Math. Ann. 369, 977–996 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  77. M. Hieber, J. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flow, in ed. by Y. Giga, A. Novotny. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, vol. 2 (Springer, Berlin, 2018), pp. 1057–1134

    Google Scholar 

  78. M. Hieber, J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress II: the compressible isotropic case. Arch. Ration. Mech. Anal. 233, 1441–1468 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  79. M. Hieber, J. Saal, The Stokes equation in the L p-setting: well-posedness and regularity properties. in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. ed. by Y. Giga, A. Novotny, vol. 1 (Springer, Berlin, 2018), pp. 117–206

    Google Scholar 

  80. M. Hieber, H. Saito, Strong solutions for two-phase free boundary problems for a class of non-Newtonian fluids. J. Evolut. Equ. 17, 335–358 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  81. M. Hieber, Y. Shibata, The Fujita-Kato approach to the Navier–Stokes equation in the rotational framework. Math. Z. 265, 481–491 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  82. M. Hieber, W. Stannat, Stochastic stability of the Ekman spiral. Ann. Sc. Norm. Super. Pisa XII, 189–208 (2013)

    Google Scholar 

  83. M. Hieber, A. Hussein, T, Kashiwabara, Global strong L p well-posedness of the 3D primitive equations with heat and salinity diffusion. J. Diff. Equ. 261, 6950–6981 (2016)

    Article  MATH  Google Scholar 

  84. M. Hieber, M. Nesensohn, J. Prüss, K. Schade, Dynamics of nematic lquid crystals: the quasilinear approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 397–408 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  85. M. Hong, J. Li, Z. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in \(\mathbb R^3\). Comm. Partial Differ. Equ. 39, 1284–1328 (2014)

    Google Scholar 

  86. L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. I, II (Springer, Berlin, 1983)

    MATH  Google Scholar 

  87. T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach Spaces, vol. I (Springer, Berlin, 2016)

    Book  MATH  Google Scholar 

  88. T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach Spaces, vol. II (Springer, Berlin, 2017)

    Book  MATH  Google Scholar 

  89. A. Inoue, M. Wakimoto, On existence of solutions of the Navier–Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA 24, 303–319 (1977)

    MathSciNet  MATH  Google Scholar 

  90. N. Kalton, G. Lancien, A solution to the problem of L p-maximal regularity. Math. Z. 235, 559–568 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  91. N. Kalton, L. Weis, The H ∞-calculus and sums of closed operators. Math. Ann. 321, 319–345 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  92. T. Kato, Perturbation Theory of Linear Operators (Springer, Berlin, 1966)

    Book  MATH  Google Scholar 

  93. H. Koba, Nonlinear stability of Ekman boundary layers in rotating stratified fluids. Memoirs Amer. Math. Soc. 228, 1 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  94. M. Köhne, J. Prüss, M. Wilke, On quasilinear parabolic evolution equations in weighted L p-spaces. J. Evol. Equ. 10, 443–463 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  95. P. Kunstmann, L. Weis, Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H ∞-functional calculus, in Functional Analytic Methods for Evolution Equations. ed. by M. Ianelli, R. Nagel, S. Piazzera. Lecture Notes in Mathematics, vol. 1855 (Springer, Berlin, 2004), pp. 65–311

    Google Scholar 

  96. P. Kunstmann, L. Weis, New criteria for the H ∞-calculus and the Stokes operator on bounded Lipschitz domains. J. Evol. Equ. 17, 387–409 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  97. F. M. Leslie, Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  98. J. Li, E. Titi, The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: rigorous justification of the hydrostatic approximation (2017). arXiv:1706.08885

    Google Scholar 

  99. J. Li, E. Titi, Recent advances concerning certain classes of geophysical flows, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. ed. by Y. Giga, A. Novotny, vol. 1 (Springer, Berlin, 2018), pp. 933–972

    Google Scholar 

  100. F. Lin, Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena. Comm. Pure Appl. Math. 42, 789–814 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  101. F. Lin, Ch. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm. Pure Appl. Math. 48, 501–537 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  102. F. Lin, C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. Philos. Trans. R. Soc. Lon. Ser. A, Math. Phys. Eng. Sci. 372, 20130361 (2014)

    Google Scholar 

  103. J.L. Lions, R. Temam, Sh.H. Wang, New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5, 237–288 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  104. J.L. Lions, R. Temam, Sh.H. Wang, On the equations of the large-scale ocean. Nonlinearity 5, 1007–1053 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  105. C. Liu, H. Wu, X. Xu, On the general Ericksen-Leslie system: Parodi’s relation, well-posedness and stability. Arch. Ration. Mech. Anal. 208, 59–107 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  106. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems (Birkhäuser, Basel, 1995)

    Google Scholar 

  107. W. Ma, H. Gong, J. Li, Global strong solutions to incompressible Ericksen-Leslie system in \(\mathbb R^3\). Nonlinear Anal. 109, 230–235 (2014)

    Google Scholar 

  108. J. Málek, J. Necas, M. Ruzicka, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2. Adv. Differ. Equ. 6, 257–302 (2001)

    MathSciNet  MATH  Google Scholar 

  109. J. Malek, V. Prusa, Derivation of equations for incmpressible and compressible fluids, in ed. by Y. Giga, A. Novotny. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, vol. 1 (Springer, Berlin, 2018), pp. 3–72

    Google Scholar 

  110. V. Maslennikova, M. Bogovskii, Elliptic boundary values in unbounded domains with noncompact and nonsmooth boundaries. Rend. Sem. Mat. Fis. Milano 56, 125–138 (1986)

    Article  MathSciNet  Google Scholar 

  111. N. Masmoudi, Ekman layers of rotating fluids: the case of general initial data. Comm. Pure Appl. Math. 53, 432–483 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  112. M. McCracken, The resolvent problem for the Stokes equation on half spaces in L p. SIAM J. Math. Anal. 12, 201–228 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  113. A. McIntosh, Operators which have an H ∞-calculus, in Miniconference on operator theory and partial differential equations. ed. by B. Jefferies, A. McIntosh, W. Ricker. Proceeding Center Mathematica Analysis A.N.U., vol. 14 (1986), pp.210–231

    Google Scholar 

  114. T. Miyakawa, H. Sohr, Weak solutions of Navier–Stokes equations. Math. Z. 199, 455–478 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  115. A. Noll, J. Saal, H ∞-calculus for the Stokes operator on L q-spaces. Math. Z. 244, 651–688 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  116. M. Padula, Asymptotic Stability of Steady Compressible Fluids. Lecture Notes Mathematics, vol. 2024 (Springer, Berlin, 2010)

    Google Scholar 

  117. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)

    Book  MATH  Google Scholar 

  118. J. Pedlovsky, Geophysical Fluid Dynamics (Springer, New York, 1987)

    Book  Google Scholar 

  119. G. Pisier, Some results on Banach spaces without local unconditional structure. Compos. Math. 37, 3–19 (1978)

    MathSciNet  MATH  Google Scholar 

  120. J. Prüss, Maximal regularity for evolution equations in L p-spaces. In: Conf. Semin. Mat. Univ. Bari, (2002)(285), (2003), 1–39.

    Google Scholar 

  121. J. Prüss, G. Simonett, Maximal regularity for evolution equations in weighted L p-spaces. Arch. Math. 82, 415–431 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  122. J. Prüss, G. Simonett, On the two-phase Navier–Stokes equations with surface tension. Inter. Free Bound. 12, 311–345 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  123. J. Prüss, G. Simonett, Analytic solutions for the two-phase Navier–Stokes equations with surface tension and gravity, in Parabolic Problems, Progress. Nonlinear Differential Equations Applications, vol. 80, (Birkhäuser, Basel, 2011), pp. 507–540

    Google Scholar 

  124. J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics, vol. 105 (Birkhäuser, Basel, 2016)

    Google Scholar 

  125. J. Prüss, M. Wilke. Addendum to the paper “On quasilinear parabolic evolution equations in weighted L p-spaces II”. J. Evol. Equ. 17, 1381–1388 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  126. J. Prüss, G. Simonett, R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems. J. Diff. Equ. 246, 3902–3931 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  127. J.-P. Raymond, M. Vanninathan, A fluid-structure model coupling the Navier–Stokes equations and the Lamé system. J. Math. Pures Appl. 102, 546–596 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  128. J. Robinson, J. Rodrigo, W. Sadowski, The Three-Dimensional Navier–Stokes Equations. Cambridge Studies in Advanced Mathematicals, vol. 157 (Cambridge University Press, Cambridge, 2016)

    Google Scholar 

  129. F. Rousset, Stability of large Ekman boundary layers in rotating fluids. Arch. Ration. Mech. Anal. 172, 213–245 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  130. Y. Shibata, S. Shimizu, On a free boundary value problem for the Navier–Stokes equations. Differ. Integral Equ. 20, 241–276 (2007)

    MATH  Google Scholar 

  131. Y. Shibata, S. Shimizu, Report on a local in time solvability of free surface problems for the Navier–Stokes equations with surface tension. Appl. Anal. 90, 201–214 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  132. Y. Shibata, S. Shimizu, Maximal L p-L q regularity for the two-phase Stokes equations; model problems. J. Differ. Equ. 251, 373–419 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  133. C. Simader, H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematicals, vol. 360 (CRC Press, Boca Raton, 1997)

    Google Scholar 

  134. H. Sohr, The Navier–Stokes Equations. An Elementary Functional Analytic Approach (Birkhäuser, Basel, 2001)

    Google Scholar 

  135. V.A. Solonnikov, Estimates for solutions of nonstationary Navier–Stokes equations. J. Soviet Math. 8, 213–317 (1977)

    Google Scholar 

  136. V.A. Solonnikov, Solvability of a problem of evolution of an isolated amount of a viscous incompressible capillary fluid. Zap. Nauchn. Sem. LOMI 140, 179–186 (1984); English transl. in J. Soviet Math. 37 (1987)

    Google Scholar 

  137. V.A. Solonnikov, On the quasistationary approximation in the problem of motion of a capillary drop, in Topics in Nonlinear Analysis. The Herbert Amann Anniversary Volume. ed. by J. Escher, G. Simonett (Birkhäuser, Basel, 1999), pp. 641–671

    Google Scholar 

  138. V.A. Solonnikov, On the stability of nonsymmetric equilibrium figures of a rotating viscous imcompressible liquid. Inter. Free Bound. 6, 461–492 (2004)

    Article  MATH  Google Scholar 

  139. E.M. Stein, Topics in Harmonic Analysis Related to Littlewood-Paley Theory (Princeton University Press, Princeton, 1970)

    Book  MATH  Google Scholar 

  140. E.M. Stein, Harmonic Analysis: Real-Variables Methods, Orthogonality and Oscillatory Integrals (Princeton University Press, Princeton, 1993)

    MATH  Google Scholar 

  141. E.M. Stein, R. Shakarchi, Fourier Analysis: An Introduction (Princeton University Press, Princeton, 2003)

    MATH  Google Scholar 

  142. T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Adv. Differ. Equ. 8, 1499–1532 (2003)

    MathSciNet  MATH  Google Scholar 

  143. T. Takahashi, M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. J. Math. Fluid Mech. 6, 63–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  144. H. Tanabe, Equations of Evolution (Pitman, London, 1979)

    MATH  Google Scholar 

  145. N. Tanaka, Two-phase free boundary problem for viscous incompressible thermocapillary convection. Jpn. J. Math. 21, 1–42 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  146. A. Tani, N. Tanaka, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Ration. Mech. Anal. 130, 303–314 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  147. H. Temam, Navier Stokes Equations. Monographs in Mathematics, vol. 78 (Birkhäuser, Basel, 1992)

    Google Scholar 

  148. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978)

    MATH  Google Scholar 

  149. H. Triebel, Theory of Function Spaces, (Reprint of 1983 edition) (Springer, Berlin, 2010)

    MATH  Google Scholar 

  150. T.-P. Tsai, Lectures on the Navier–Stokes equations, in Graduate Studies in Mathematics (American Mathematical Society,, Providence, 2018)

    Book  Google Scholar 

  151. S. Ukai, A solution formula for the Stokes equation in \(\mathbb R_3^+\), Comm. Pure Appl. Math. 11, 611–621 (1987)

    Google Scholar 

  152. E. G. Virga, Variational Theories for Liquid Crystals (Chapman-Hall, London, 1994)

    Book  MATH  Google Scholar 

  153. W. Wang, P. Zhang, Z. Zhang, Well-posedness of the Ericksen-Leslie system. Arch. Ration. Mech. Anal. 210, 837–855 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  154. L. Weis, Operator valued Fourier multiplier theorems and maximal L p-regularity. Math. Ann. 319, 735–758 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  155. H. Wu, X. Xu, Ch. Liu, On the general Ericksen-Leslie system: Parodi’s relation, well-posedness and stability. Arch. Ration. Mech. Anal. 208, 59–107 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  156. F. Zimmermann, On vector-valued Fourier multiplier theorems. Studia Math. 89, 201–222 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department (FB) of Mathematics, TU Darmstadt, Darmstadt, Germany

    Matthias Hieber

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    Giovanni P. Galdi

  2. Department of Mathematics, Waseda University, Tokyo, Japan

    Yoshihiro Shibata

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Hieber, M. (2020). Analysis of Viscous Fluid Flows: An Approach by Evolution Equations. In: Galdi, G., Shibata, Y. (eds) Mathematical Analysis of the Navier-Stokes Equations. Lecture Notes in Mathematics(), vol 2254. Springer, Cham. https://doi.org/10.1007/978-3-030-36226-3_1

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