Skip to main content Accessibility help
×
Home
The Three-Dimensional Navier–Stokes Equations
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 21
  • Export citation
  • Recommend to librarian
  • Buy the print book

Book description

A rigorous but accessible introduction to the mathematical theory of the three-dimensional Navier–Stokes equations, this book provides self-contained proofs of someof the most significant results in the area, many of which can only be found in researchpapers. Highlights include the existence of global-in-time Leray–Hopf weak solutionsand the local existence of strong solutions; the conditional local regularity results ofSerrin and others; and the partial regularity results of Caffarelli, Kohn, and Nirenberg.Appendices provide background material and proofs of some 'standard results' thatare hard to find in the literature. A substantial number of exercises are included, with fullsolutions given at the end of the book. As the only introductory text on the topic to treatall of the mainstream results in detail, this book is an ideal text for a graduate course ofone or two semesters. It is also a useful resource for anyone working in mathematicalfluid dynamics.

Reviews

'I loved this very well-written book and I highly recommend it.'

Jean C. Cortissoz Source: Mathematical Reviews

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
×

Contents


Page 1 of 2



Page 1 of 2


References
Adams, R.A. & Fournier, J.J.F. (2003) Sobolev spaces. Academic Press, Kidlington, Oxford.
Agmon, S., Douglis, A., & Nirenberg, L. (1964) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17, 35–92.
Aubin, J.P. (1963) Un theoreme de compacite. C. R. Acad. Sci. Pari. 256, 5042–5044.
Bahouri, H., Chemin, J.-Y.,& Danchin, R. (2011) Fourier analysis and nonlinear partial differential equations. Springer, Heidelberg.
Bardos, C., Lopes Filho, M., Niu, D., Nussenzveig Lopes, H., & Titi, E.S. (2013) Stability of viscous, and instability of non-viscous, 2D weak solutions of incompressible fluids under 3D perturbations. SIAM J. Math. Anal. 45, 1871–1885.
Batchelor, G.K. (1999) An introduction to fluid dynamics. Cambridge University Press, Cambridge.
Beale, J.T.,Kato, T.,& Majda, A. (1984) Remarks on the breakdown of smooth solutions for the 3-D Euler equation. Comm. Math. Phys. 94, 61–66.
Beirão da Veiga, H. (1985a) On the suitable weak solutions to the Navier–Stokes equations in the whole space. J. Math. Pures Appl. 64, 77–86.
Beirão da Veiga, H. (1985b) On the construction of suitable weak solutions to the Navier–Stokes equations via a general approximation theorem. J. Math. Pures Appl. 64, 321–334.
Beirão da Veiga, H. (1987) Existence and asymptotic behavior for strong solutions of the Navier–Stokes equations in the whole space. Indiana Univ. Math. J. 36, 149–166.
Beirão da Veiga, H. (1995) A new regularity class for the Navier–Stokes equations in ℝ n . Chinese Ann. Math. Ser. B. 4, 407–412.
Beirão da Veiga, H. (2000) On the smoothness of a class of weak solutions to theNavier– Stokes equations. J. Math. Fluid Mech. 2, 315–323.
Beirão da Veiga, H. & Secchi, P. (1987) L p -stability for the strong solutions of the Navier–Stokes equations in the whole space. Arch. Ration. Mech. Anal. 98, 65–69.
Benedek, A., Calderón, A.P., & Panzone, R. (1962) Convolution operators on Banach space valued functions. Proc. Nat. Acad. Sci. U.S.A. 48, 356–365.
Bergh, J. & Löfström, J. (1976). Interpolation spaces. Springer-Verlag, Berlin.
Berselli, L.C. (2002) On a regularity criterion for the solutions to the 3D Navier–Stokes equations. Differential Integral Equation. 15, 1129–1137.
Berselli, L.C. & Galdi, G.P. (2002) Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations. Proc. Amer.Math. Soc. 130, 3585–3595.
Berselli, L.C. & Spirito, S. (2016a) On the construction of suitable weak solutions to the 3D Navier–Stokes equations in a bounded domain by an artificial compressibility method. Commun. Contemp. Math., to appear.
Berselli, L.C. & Spirito, S. (2016b)Weak solutions to the Navier–Stokes equations constructed by semi-discretization are suitable. Contemp. Math., to appear.
Biot, J.-B. & Savart, F. (1820) Note sure le magnétisme de la pile de Volta. Annales Chim. Phys. 15, 222–223.
Biryuk, A., Craig, W., & Ibrahim, S. (2007) Construction of suitable weak solutions of the Navier–Stokes equations. In “Stochastic analysis and partial differential equations”. Contemp. Math. 49, 1–18.
Bogovskii, M.E. (1986) Decomposition of L p (Ω,ℝ n .) into the direct sum of subspaces of solenoidal and potential vector fields. Soviet Math. Dokl. 33, 161–165.
Bourbaki, N. (2004) Integration. Springer-Verlag, Berlin.
Caffarelli, L., Kohn, R., & Nirenberg, L. (1982) Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure. Appl. Math. 35, 771–931.
Calderón, C.P. (1990) Existence of weak solutions for the Navier–Stokes equations with initial data in L p . Trans. Amer. Math. Soc. 318, 179–200.
Cannone, M. (1995) Ondelettes, paraproduits et Navier–Stokes. Diderot Editeur, Paris.
Cannone, M. (2003) Harmonic analysis tools for solving the incompressible Navier– Stokes equations. In Friedlander, S. & Serre, D. (eds.) Handbook of mathematical fluid dynamics, Vol. 3. Elsevier, Kidlington.
Cao, C. & Titi, E.S. (2008) Regularity criteria for the three-dimensional Navier–Stokes Equations. Indiana Univ. Math. J. 57, 2643–2661.
Castaing, C. (1967) Sur les multi-applications measurables. Rev. France Inform. Rech. Oper. 1, 91–126.
Chemin, J.-Y. (1992) Remarques sur l'existence globale pour le systeme de Navier– Stokes incompressible. SIAM J. Math. Anal. 23, 20–28.
Chemin, J.-Y. & Lerner, N. (1995) Flot de champs de vecteurs non lipschitziens et equations de Navier–Stokes. J. Differential Equation. 121, 314–328.
Chemin, J.-Y., Desjardins, B., Gallagher, I., & Grenier, E. (2006) Mathematical geophysics. Oxford University Press, Oxford.
Chen, C.-C., Strain, R.M., Yau, H.-T., & Tsai, T.-P. (2008) Lower bound on the blow-up rate of the axisymmetric Navier–Stokes equations. Int. Math. Res. Not. article ID rnn016.
Chernyshenko, S.I., Constantin, P., Robinson, J.C., & Titi, E.S. (2007) A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations. J. Math. Phys. 48, 065204.
Chorin, A.J. & Marsden, J.E. (1993) A mathematical introduction to fluid mechanics. Third edition. Springer-Verlag, New York.
Chung, S.-Y. (1999) Uniqueness in the Cauchy problem for the heat equation. Proc. Edinburgh Math. Soc. 42, 455–468.
Constantin, P. (1986) Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations. Comm. Math. Phys. 104, 311–326.
Constantin, P. & Foias, C. (1988) Navier–Stokes equations. University of Chicago Press, Chicago, IL.
Dacorogna, B. (2004) Introduction to the calculus of variations. Imperial College Press, London.
Danchin, R. (2000) Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 41, 579–614.
Dashti, M. & Robinson, J.C. (2008) An a posteriori condition on the numerical approximations of theNavier–Stokes equations for the existence of a strong solution. SIAM J. Numer. Anal. 46, 3136–3150.
Dashti, M. & Robinson, J.C. (2009) A simple proof of uniqueness of the particle trajectories for solutions of the Navier–Stokes equations. Nonlinearit. 22, 735–746.
De Giorgi, E. (1957) Sulla differenziabilita e lanaliticita delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3, 25–43.
De Lellis, C. & Székelyhidi Jr., L. (2010) On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195, 225–260.
DiPerna, R.J. & Lions, P.-L. (1989) Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547.
Doering, C.R. & Gibbon, J.D. (1995) Applied analysis of the Navier–Stokes equations. Cambridge University Press, Cambridge.
Duoandikoetxea, J. (2001) Fourier analysis. Graduate Studies in Mathematics 29. American Mathematical Society, Providence, RI.
Escauriaza, L., Seregin, G., & Šverák, V. (2003) L 3,∞-solutions of Navier–Stokes equations and backward uniqueness. Russian Math. Survey. 58, 211–250.
Evans, L.C. (1998) Partial differential equations. American Mathematical Society, Providence, RI.
Evans, L.C. & Gariepy, R.F. (1992) Measure theory and fine properties of functions. CRC Press, Boca Raton, FL.
Fabes, E.B., Jones, B.F., & Riviere, N.M. (1972) The initial value problem for the Navier–Stokes equations with data in Lp . Arch. Ration. Mech. Anal. 45, 222–240.
Fabes, E.B., Lewis, J.E., & Riviere, N.M. (1977) Singular integrals and hydrodynamic potentials. Amer. J. Math. 99, 601–625.
Falconer, K.J. (1985) The geometry of fractal sets. Cambridge University Press, Cambridge.
Falconer, K.J. (1990) Fractal geometry. Wiley, Chichester.
Farwig, R. & Sohr, H. (1996) Helmholtz decomposition and Stokes resolvent system for aperture domains in Lq -spaces. Analysis. 16, 1–26.
Farwig, R. & Sohr, H. (1996) Helmholtz decomposition and Stokes resolvent system for aperture domains in Lq -spaces. Analysi. 16, 1–26.
Farwig, R., Kozono, H., & Sohr, H. (2005) A Lq . approach to Stokes and Navier–Stokes equations in general domains. Acta Math. 195, 21–53.
Fefferman, C.L. (1971) The multiplier problem for the ball. Ann. of Math. 94, 330–336.
Fefferman, C.L. (2000) Existence and smoothness of the Navier–Stokes equation. In The millennium prize problems, 57–67. Clay Math. Inst., Cambridge, MA.
Feynman, R.P., Leighton, R.B., & Sands, M. (1970) The Feynman lectures on physics, Volume II. Addison Wesley Publishing Co., Reading, MA, London.
Foias, C., Guillopé, C., & Temam, R. (1981) New a priori estimates for Navier– Stokes equations in dimension 3. Comm. Partial Differential Equation. 6, 329–359.
Foias, C., Guillopé, C., & Temam, R. (1985) Lagrangian representation of a flow. J. Differential Equation. 57, 440–449.
Foias, C. & Temam, R. (1989) Gevrey class regularity for the solutions of the Navier– Stokes equations. J. Funct. Anal. 87, 359–369.
Foias, C., Manley, O., Rosa, R., & Temam, R. (2001) Navier–Stokes equations and turbulence. Cambridge University Press, Cambridge.
Folland, G.B. (1999) Real analysis. Modern techniques and their applications. Second edition. John Wiley & Sons, Inc., New York, NY.
Friedlander, F.G. & Joshi, M. (1999) Introduction to the theory of distributions. Cambridge University Press, Cambridge.
Fujita, H. & Kato, T. (1964) On the Navier–Stokes initial value problem. I. Arch. Ration. Mech. Anal. 16, 269–315.
Fujiwara, D. & Morimoto, H. (1977) A Lr -theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24, 685–700.
Galdi, G.P. (2000) An introduction to the Navier–Stokes initial-boundary value problem. In Galdi, G.P., Heywood, J.G., & Rannacher, R. (eds.) Fundamental directions in mathematical fluid dynamics, 1–70. Birkhauser, Basel.
Galdi, G.P. (2011) An introduction to the mathematical theory of Navier–Stokes equations. Steady state problems. Second edition. Springer, New York, NY.
Galdi, G.P. & Maremonti, P. (1986) Monotonic decreasing and asymptotic behaviour of the kinetic energy for weak solutions of the Navier–Stokes equations in exterior domains. Arch. Ration. Mech. Anal. 94, 253–266.
Gallagher, I. (1997) The tridimensional Navier–Stokes equations with almost bidimensional data: stability, uniqueness, and life span. Int. Math. Res. Notice. 18, 919–935.
Giga, Y. (1983) Time and spatial analyticity of solutions of the Navier–Stokes equations. Comm. Partial Differential Equation. 8, 929–948.
Giga, Y. (1986) Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier–Stokes system. J. Differential Equation. 62, 186–212.
Gilbarg, D. & Trudinger, N.S. (1983) Elliptic partial differential equations of second order. Springer, Berlin.
Grafakos, L. (2008) Classical Fourier analysis. Graduate text in mathematics 249. Springer, New York, NY.
Grafakos, L. (2009) Modern Fourier analysis. Graduate text in mathematics 250. Springer, New York, NY.
Guermond, J.-L. (2006) Finite-element-based Faedo–Galerkin weak solutions to the Navier–Stokes equations in the three-dimensional torus are suitable. J.Math. Pures Appl. 85, 451–464.
Guermond, J.-L. (2007) Faedo–Galerkin weak solutions of the Navier–Stokes equations with Dirichlet boundary conditions are suitable. J. Math. Pures Appl. 88, 87–106.
Hale, J.K. (1980) Ordinary differential equations. Kreiger, Malabar, FL.
Hardy, G.H. (1932) A note on two inequalities. J. London Math. Soc. 11, 167–170.
Hartman, P. (1973) Ordinary differential equations. Wiley, Baltimore.
Helmholtz, H. (1858) Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55.
Heywood, J. (1976) On uniqueness questions in the theory of viscous flow. Acta Math. 136, 61–102.
Heywood, J. (1988) Epochs of regularity for weak solutions of the Navier–Stokes equations in unbounded domains. Tohoku Math. J. 40, 293–313.
Hopf, E. (1951) Über die Aufgangswertaufgave fur die hydrodynamischen Grundliechungen. Math. Nachr. 4, 213–231.
Hörmander, L. (1960) Estimates for translation invariant operators in Lp spaces. Acta Math. 104, 93–139.
Iftimie, D., Karch, G., & Lacave, C. (2014) Asymptotics of solutions to the Navier– Stokes system in exterior domains. J. London Math. Soc. 90, 785–806.
James, R.C. (1964) Weakly Compact Sets. Trans. Amer. Math. Soc. 113, 129–140.
Kahane, C. (1969) On the spatial analyticity of solutions of theNavier–Stokes equations. Arch. Ration. Mech. Anal. 33, 386–405.
Kato, T. (1984) Strong Lp -solutions of the Navier–Stokes equations in ℝ m with applications to weak solutions. Math. Zeit. 187, 471–480.
Kiselev, A.A. & Ladyzhenskaya, O.A. (1957) On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid. Izv. Akad. Nauk SSSR. Ser. Mat. 21, 655–680.
Koch, H. & Tataru, D. (2001) Well-posedness for the Navier–Stokes equations. Adv. Math. 157, 22–35.
Kohn, R.V. (1982) Partial regularity and the Navier–Stokes equations. Lecture Notes in Num. Appl. Anal. 5, 101–118. (Nonlinear PDE in Applied Science U.S.–Japan Seminar, Tokyo, 1982.)
Komatsu, G. (1979) Analyticity up to the boundary of solutions of nonlinear parabolic equations. Comm. Pure Appl. Math. 32, 669–720.
Kozono, H. (1998) Uniqueness and regularity of weak solutions to the Navier–Stokes equations. In Recent topics on mathematical theory of viscous incompressible fluid (Tsukuba, 1996). Lecture Notes Numer. Appl. Anal.. 16, 161–208. Kinokuniya, Tokyo.
Kozono, H. & Sohr, H. (1996) Remark on uniqueness of weak solutions to the Navier– Stokes equations. Analysi. 16, 255–271.
Kozono, H. & Taniuchi, T. (2000) Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Comm. Math. Phys. 214, 191–200.
Kozono, H., Ogawa, T., & Taniuchi, T. (2002) The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242, 251–278.
Krylov, N.V. (1996) Lectures on elliptic and parabolic equations in Holder spaces. American Mathematical Society, Providence, RI.
Krylov, N.V. (2001) The heat equation in Lq ((0,T); Lp )-space with weights. SIAM J. Math. Anal. 32, 1117–1141.
Krylov, N.V. (2008) Lectures on elliptic and parabolic equations in Sobolev spaces. American Mathematical Society, Providence, RI.
Kukavica, I. (2008) Regularity for the Navier–Stokes equations with a solution in a Morrey space. Indiana Univ. Math. J. 57, 2843–2860.
Kukavica, I. (2009a) Partial regularity results for solutions of the Navier–Stokes system. In Robinson, J.C. & Rodrigo, J.L. (eds.) Partial differential equations and fluid mechanics, 121–145. Cambridge University Press, Cambridge.
Kukavica, I. (2009b) The fractal dimension of the singular set for solutions of the Navier–Stokes system. Nonlinearit. 22, 2889–2900.
Kukavica, I. & Pei, Y. (2012) An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier–Stokes system. Nonlinearit. 25, 2775–2783.
Ladyzhenskaya, O.A. (1959) Solution “in the large” of the nonstationary boundary value problem for the Navier–Stokes system in two space variables. Comm. Pure Appl. Math. 12, 427–433.
Ladyzhenskaya, O.A. (1967) On uniqueness and smoothness of generalized solutions to the Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5, 169–185; English transl. Sem. Math. V.A. Steklov Math. Inst. Leningra. 5(1969), 60–66.
Ladyzhenskaya, O.A. (1969) The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, NY.
Ladyzhenskaya, O.A. (1970) Unique solvability in the large of three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry. Seminar in Mathematics, V.A. Steklov Mathematical Institute, Leningrad, 7, Boundary value problems of mathematical physics and related aspects of function theory, Part 2, Edited by O.A., Ladyzhenskaya, 70–79.
Ladyzhenskaya, O.A. & Seregin, G.A. (1999) On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations. J. Math. Fluid Mech. 1, 356–387.
Ladyzhenskaya, O.A., Solonnikov, V.A., & Uraltseva, N.N. (1967) Linear and quasilinear equations of parabolic type. American Mathematical Society, Providence, RI.
Lemarié-Rieusset, P.G. (2002) Recent developments in the Navier–Stokes problem. Chapman & Hall/CRC, Boca Raton, FL.
Leray, J. (1934) Essai sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63, 193–248.
Lieberman, G.M. (1996) Second order parabolic differential equations.World Scientific Publishing Co., Inc., River Edge, NJ.
Lin, F. (1998) A new proof of the Caffarelli–Kohn–Nirenberg theorem. Comm. Pure Appl. Math. 51, 241–257.
Lions, J.-L. (1969) Quelques methodes de resolution des problemes aux limites non lineaires. Dunod Gauthier-Villars, Paris.
Lions, J.-L. & Prodi, G. (1959) Un theoreme d'existence et unicite dans les equations de Navier–Stokes en dimension 2. C. R. Acad. Sci. Pari. 248, 3519–3521.
Lions, P.-L. (1994) Mathematical topics in fluid mechanics, Volume 1: Incompressible models. Oxford University Press, Oxford.
Lunardi, A. (2009) Interpolation theory. 2nd edition. Edizioni della Normale, Pisa.
Mahalov, A., Titi, E.S., & Leibovich, S. (1990) Invariant helical subspaces for the Navier–Stokes equations. Arch. Ration. Mech. Anal. 112, 193–222.
Majda, A.J. & Bertozzi, A.L. (2002) Vorticity and incompressible flow. Cambridge University Press, Cambridge.
Marín-Rubio, P., Robinson, J.C., & Sadowski, W. (2013) Solutions of the 3D Navier– Stokes equations for initial data in H ½: robustness of regularity and numerical verification of regularity for bounded sets of initial data in H 1 . J. Math. Anal. Appl. 400, 76–85.
Masuda, K. (1967) On the analyticity and the unique continuation theorem for solutions of the Navier–Stokes equations. Proc. Japan. Acad. 43, 827–832.
Masuda, K. (1984) Weak solutions of Navier–Stokes equations. Tohoku Math. J. (2. 36, 623–646.
McCormick, D.S., Robinson, J.C., & Rodrigo, J.L. (2013) Generalised Gagliardo– Nirenberg inequalities using weak Lebesgue spaces and BMO.Milan J. Math. 81, 265–289.
McCormick, D.S., Olson, E.J., Robinson, J.C., Rodrigo, J.L., Vidal-López, A., & Zhou, Y. (2016a) Lower bounds on blowing-up solutions of the 3D Navier–Stokes equations in Ḣ 3/2, Ḣ 5/2, and Ḃ 5/2 2,1. arXiv:1503.04323. SIAM J. Math. Anal., to appear.
McCormick, D.S., Fefferman, C.L., Robinson, J.C., & Rodrigo, J.L. (2016b) Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. arXiv:1602.02588.
Mihlin, S.G. (1957) Fourier integrals and multiple singular integrals. Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 12, 143–155.(in Russian).
Miyakawa, T. & Sohr, H. (1988) On energy inequality, smoothness and large time behavior in L 2 for weak solutions of the Navier–Stokes equations in exterior domains. Math. Z. 199, 455–478.
Montero, J.A. (2015) Lower bounds for possible blow–up solutions for the Navier– Stokes equations revisited. arXiv:1503.03063.
Mucha, P.B. (2008) Stability of 2D incompressible flows in R3. J. Differential Equations 245, 2355–2367.
Muscalu, C. & Schlag, W. (2013) Classical and multilinear harmonic analysis. Volume I. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge.
Nečas, J., Ružička, M., & Šverák, V. (1996) On Leray's self-similar solutions of the Navier–Stokes equations. Acta Math. 176, 283–294.
Neustupa, J. (1999) Partial regularity of weak solutions to the Navier–Stokes equations in the class L ∞(0,T; L 3(Ω)3). J. Math. Fluid Mech. 1, 309–325.
O'Leary, M. (2003) Conditions for the local boundedness of solutions of the Navier– Stokes system in three dimensions. Comm. Partial Differential Equation. 28, 617–636.
Pettis, B.J. (1938) On integration in vector spaces. Trans. Amer.Math. Soc. 44, 277–304.
Planchon, F. (2003) An extension of the Beale–Kato–Majda criterion for the Euler equations. Comm. Math. Phys. 232, 319–326.
Pooley, B.C. & Robinson, J.C. (2016) Well-posedness for the diffusive 3DBurgers equations with initial data in H 1/2 . In Robinson, J.C., Rodrigo, J.L., Sadowski, W., & Vidal-López, A. (eds.) Recent progress in the theory of the Euler and Navier–Stokes equations, 137–153. Cambridge University Press, Cambridge.
Prodi, G. (1959) Un teorema di unicita per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182.
Raugel, G. & Sell, G.R. (1993) Navier–Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions. J. Amer. Math. Soc. 6, 503–568.
Renardy, M. & Rogers, R.C. (2004) An introduction to partial differential equations. Second Edition. Springer-Verlag, New York, NY.
Robertson, A.P. (1974) On measurable selections. Proc. Roy. Soc. Edinburgh Sect.. 72, 1–7.
Robinson, J.C. (2001) Infinite-dimensional dynamical systems. Cambridge University Press, Cambridge.
Robinson, J.C. (2006) Regularity and singularity in the three-dimensional Navier– Stokes equations. Boletin de SEM. 35, 43–71.
Robinson, J.C. (2011) Dimensions, embeddings, and attractors. Cambridge University Press, Cambridge.
Robinson, J.C. & Sadowski, W. (2007) Decay of weak solutions and the singular set of the three-dimensional Navier–Stokes equations. Nonlinearit. 20, 1185–1191.
Robinson, J.C. & Sadowski, W. (2009) Almost everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier–Stokes equations. Nonlinearit. 22, 2093–2099.
Robinson, J.C. & Sadowski, W. (2014) A local smoothness criterion for solutions of the 3D Navier–Stokes equations. Rend. Semin. Mat. Univ. Padov. 131, 159–178.
Robinson, J.C., Sadowski, W., & Sharples, N. (2013) On the regularity of Lagrangian trajectories corresponding to suitable weak solutions of the Navier–Stokes equations. Procedia IUTA. 7, 161–166.
Robinson, J.C., Sadowski, W., & Silva, R.P. (2012) Lower bounds on blow up solutions of the three-dimensional Navier–Stokes equations in homogeneous Sobolev spaces. J. Math. Phys. 53, 115618.
Roubíček, T. (2013) Nonlinear partial differential equations with applications. Second edition. Birkhauser, Springer, Basel.
Rudin, W. (1991) Functional Analysis. McGraw-Hill, New York, NY.
Scheffer, V. (1976) Turbulence and Hausdorff dimension. In Turbulence and Navier– Stokes equation, Orsay 1975, Springer Lecture Notes in Mathematics 565, 174–183. Springer, Berlin.
Scheffer, V. (1977) Hausdorff measure and the Navier–Stokes equations. Comm. Math. Phys. 55, 97–112.
Scheffer, V. (1980) The Navier–Stokes equations on a bounded domain. Comm. Math. Phys. 73, 1–42.
Scheffer, V. (1985) A solution to the Navier–Stokes inequality with an internal singularity. Comm. Math. Phys. 101, 47–85.
Scheffer, V. (1987) Nearly one-dimensional singularities of solutions to the Navier– Stokes inequality. Comm. Math. Phys. 110, 525–551.
Scheffer, V. (1993) An inviscid flowwith compact support in space–time. J. Geom. Anal. 3, 343–401.
Schonbek, M.E. (1985) L 2 decay for weak solutions to the Navier–Stokes equation. Arch. Ration. Mech. Anal. 88, 209–222.
Seregin, G. (2014) Lecture notes on regularity theory for the Navier–Stokes equations. World Scientific, Hackensack, NJ.
Seregin, G. & Šverák, V. (2002) Navier–Stokes equations with lower bounds on pressure. Arch. Ration. Mech. Anal. 163, 65–86.
Seregin, G. & Zajaczkowski, W. (2007) A sufficient condition of regularity for axially symmetric solutions to the Navier–Stokes equations. SIAM J. Math. Anal. 39, 669–685.
Serrin, J. (1962) On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195.
Serrin, J. (1963) The initial value problem for the Navier–Stokes equations. In Nonlinear Problem. (Proc. Sympos., Madison, Wis.), 69–98. Univ. of Wisconsin Press, Madison, WI.
Shapiro, V.L. (1966) The uniqueness of solutions of the heat equation in an infinite strip. Trans. Amer. Math. Soc. 125, 326–361.
Shatah, J. & Struwe, M. (1994) Well-posedness in the energy space for semilinear wave equations with critical growth. Int. Math. Res. Notice. 1994, 303–309.
Shnirelman, A. (1997) On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math. 50, 1261–1286.
Shnirelman, A. (2000) Weak solutions with decreasing energy of incompressible Euler equations. Comm. Math. Phys. 210, 541–603.
Simader, C.G. & Sohr, H. (1992) A new approach to the Helmholtz decomposition and the Neumann problem in L q-spaces for bounded and exterior domains. In Galdi, G.P. (ed.) Mathematical Problems Relating to the Navier–Stokes Equations. Series on Advances in Mathematics for Applied Sciences, 11, 1–35. World Scientific, Singapore.
Simader, C.G., Sohr, H., & Varnhorn, W. (2014) Necessary and sufficient conditions for the existence of Helmholtz decompositions in general domains. Ann. Univ. Ferrara 60, 245–262.
Simon, J. (1987) Compact sets in the spac. Lp (0,T; B). Ann. Mat. Pura Appl. 146, 65–96.
Sohr, H. (1983) Zur Reguläritatstheorie der instationären Gleichungen von Navier– Stokes. (German) [On the regularity theory of the nonstationary Navier–Stokes equations. Math. Z. 184, 359–375.
Sohr, H. (2001) The Navier–Stokes equations: an elementary functional analytic approach. Modern Birkhauser Classics. Birkhauser/Springer, Basel.
Sohr, H. & vonWahl, W. (1984) On the singular set and the uniqueness of weak solutions of the Navier–Stokes equations. Manuscripta Math. 49, 27–59.
Sohr, H. & von Wahl, W. (1986) On the regularity of the pressure of weak solutions of Navier–Stokes equations. Arch. Math. (Basel. 46, 428–439.
Solonnikov, V.A. (1964) Estimates of the solutions of a nonstationary linearized system of Navier–Stokes equations. Trudy Mat. Inst. Steklov. 70, in: Amer. Math. Soc. Translations, Series 2, Vol. 75, 1–117.
Stein, E.M. (1970) Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, NJ.
Stein, E.M. (1993) Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ.
Struwe, M. (1988) On partial regularity results for the Navier–Stokes equations. Comm. Pure. Appl. Math. 41, 437–458.
Takahashi, S. (1990) On interior regularity criteria for weak solutions of the Navier– Stokes equations. Manuscripta Math. 69, 237–254.
Temam, R. (1977) Navier–Stokes equations. North Holland, Amsterdam. Reprinted by AMS Chelsea, 2001.
Temam, R. (1982) Behaviour at time t = 0 of the solutions of semi-linear evolution equations. J. Differential Equation. 43, 73–92.
Temam, R. (1983) Navier–Stokes equations and nonlinear functional analysis. SIAM, Philadelphia, PA.
Tychonoff, A.N. (1935) Uniqueness theorem for the heat equation. Mat. Sb. 42, 199–216.
Vasseur, A. (2007) A new proof of partial regularity of solutions to Navier–Stokes equations. Nonlinear Diff. Equ. Appl. 14, 753–785.
von Wahl, W. (1980) Regularitatsfragen fur die instationaren Navier–Stokesschen Gleichungen in hoheren Dimensionen. J. Math. Soc. Japa. 32, 263–283.
von Wahl, W. (1982) The equation u' + A(tu) = f in a Hilbert space and Lp -estimates for parabolic equations. J. London Math. Soc. 25, 483–497.
vonWahl, W. (1985) The equations of Navier–Stokes and abstract parabolic equations. Friedr. Vieweg & Sons, Braunschweig.
Weinberger, H.F. (1999) An example of blowup produced by equal diffusions. J. Differential Equation. 154, 225–237.
Yosida, K. (1980) Functional analysis. Springer Classics in Mathematics, Springer, Berlin.
Zhou, Y. & Pokorny, M. (2010) On the regularity of the solutions of the Navier–Stokes equations via one velocity component. Nonlinearit. 23, 1097–1107.
Zuazua, E. (2002) Log-Lipschitz regularity and uniqueness of the flow for a field in ( Wn/p+1,p loc (ℝ n )) n . C. R. Acad. Sci. Pari. 335, 17–22.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.