Acta mathematica scientia,Series B ›› 2018, Vol. 38 ›› Issue (3): 778-790.doi: 10.1016/S0252-9602(18)30783-5
• Articles • Previous Articles Next Articles
Jianlin ZHANG1, Yuming QIN2
Received:2017-01-01
Revised:2017-05-14
Online:2018-06-25
Published:2018-06-25
Supported by:The first author is supported by the Key Research Projects of Henan Higher Education Institutions (18A110036, 18A110038) and the second author is supported by the NNSF of China (11671075).
Jianlin ZHANG, Yuming QIN. EXACT SOLUTIONS FOR THE CAUCHY PROBLEM TO THE 3D SPHERICALLY SYMMETRIC INCOMPRESSIBLE NAVIER-STOKES EQUATIONS[J].Acta mathematica scientia,Series B, 2018, 38(3): 778-790.
| [1] Abidi H. Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes. Bull Sci Math (in French), 2008,132:592-624 [2] Bae H, Jin B. Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations. J Differentail Equations, 2005, 209:365-391 [3] Bourgain J, Pavlović N. Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J Funct Anal, 2008, 255(9):2233-2247 [4] Brandolese L. On the localization of symmetric and asymmetric solutions of the Navier-Stokes equations in Rn. C R Acad Sci Paris, Ser I Math, 2001, 332(2):125-130 [5] Brandolese L. Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations. Rev Mat Iberoamericana, 2004, 20:225-258 [6] Brandolese L. Space-time decay of Navier-Stokes flows invariant under rotations. Math Ann, 2004, 329:685-706 [7] Brandolese L, Vigneron F. New asymptotic profiles of nonstationary solutions of Navier-Stokes system. J Math Pures Appl, 2007, 88:64-86 [8] Cannone M. Ondelettes, paraproduits et Navier-Stokes. Paris:Dierot Editeur, 1995 [9] Cannone M. Harmonic analysis tools for solving the incompressible Navier-Stokes equations//Friedlander S, Serre D. Handbook of Mathematical Fluid Dynamics. North Holland:Elsevier, 2004, 3:161-244 [10] Chemin J Y, Desjardins B, Gallagher I, Grenier E. Fluids with anisotropic viscosity. Modélisation Mathématique et Analyse Numérique, 2000,34:315-335 [11] Chemin J Y, Gallagher I. On the global wellposedness of the 3-D Navier-Stokes equations with large initial data. Annales Scientifiques de l'École Normale Supérieure, 2006, 39(4):679-698 [12] Chemin J Y, Gallagher I. Wellposedness and stability results for the Navier-Stokes equations in R3. Annales De Linstitut Henri Poincaré Non Linear Analysis, 2009, 26(2):599-624 [13] Chemin J Y, Gallagher I. Large, global solutions to the Navier-Stokes equations, slowly varying in one direction. Trans Amer Math Soc, 2010, 362(6):2859-2873 [14] Chemin J Y, Gallagher I, Mullaert C. The role of spectral anisotropy in the resolution of the threedimensional Navier-Stokes equations. Progress in Nonlinear Differential Equations and Their Applications, New York:Springer, 2013, 84:53-79 [15] Chemin J Y, Gallagher I, Paicu M. Global regularity for some classes of large solutions to the Navier-Stokes equations. Ann Math, 2011, 173(2):983-1012 [16] Chemin J Y, Gallagher I, Zhang P. Sums of large global solutions to the incompressible Navier-Stokes equations. Journal für die reine und angewandte Mathematik, 2013, 681:65-82 [17] Chemin J Y, Zhang P. On the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations. Comm Math Phys, 2007, 272:529-566 [18] Chen C C, Strain R M, Yau H T, Tsai T P. Lower bound on the blow-up rate of the axisymmetric NavierStokes equations. Comm in PDE, 2009, 34(3):203-232 [19] Cannone M, Meyer Y, Planchon F. Solutions autosimilaires des équations de Navier-Stokes. Rev roumaine Sci tech sér électrotech énergét, 1994, 2:209-216 [20] Dobrokhotov S Y, Shafarevich A I. Some integral identities and remarks on the decay at infinity of the solutions to the Navier-Stokes equations in the entire space. Russian J Math Phys, 1994, 2(1):133-135 [21] Feng H, Šverák V. On the Cauchy problem for axi-symmetric vortex rings. Arch Ration Mech Anal, 2015, 215:89-123 [22] Fujita H, Kato T. On the Navier-Stokes initial value problem I. Arch Ration Mech Anal, 1964, 16(4):269-315 [23] Gallay T, Šverák V. Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations. Confluentes Math, 2015, 7:67-92 [24] Gallay T, Wayne C E. Long-time asymptotics of the Navier-Stokes and vorticity equations on R3. Phil Trans Roy Soc London, 2002, 360:2155-2188 [25] Giga Y. Solutions of semilinear parabolic equations in Lp and regularity of weak solutions of the NavierStokes system. J Differential Equations, 1986, 62:186-212 [26] Giga Y, Miyakawa T. Navier-Stokes flows in R3 with measures as initial vorticity and the Morrey spaces. Comm in PDE, 1989, 14:577-618 [27] Han P. Long-time behavior for the nonstationary Navier-Stokes flows in L1(Rn+). J Func Anal, 2014, 266:1511-1546 [28] He C, Xin Z. On the decay properties of Solutions to the nonstationary Navier-Stokes Equations in R3. Proc Roy Edinburgh Soc Sect A, 2001, 131:597-619 [29] He C, Miyakawa T. On L1-summability and asymptotic profiles for smooth solutions to Navier-Stokes equations in a 3D exterior domain. Math Z, 2003, 245:387-417 [30] Hopf E. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math Nachr, 1951, 4:213-231 [31] Hou T, Lei Z, Li C. Global regularity of the three-dimensional axisymmetric Navier-Stokes equations with anisotropic data. Comm in PDE, 2008, 33(7-9):1622-1637 [32] Iftimie D. A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity. SIAM J Math Anal, 2002, 33:1483-1493 [33] Kato T. Strong Lp-solutions of the Navier-Stokes equations in Rm, with applications to weak solutions. Math Zeit, 1984, 187:471-480 [34] Kato T. Liapunov functions and monotonicity in the Navier-Stokes equation//Lecture Notes in Mathematics. New York/Berlin:Springer-Verlag, 1990, 1450:53-63 [35] Koch H, Tataru D. Well-posedness for the Navier-Stokes equations. Adv Math, 2001, 157:22-35 [36] Ladyzhenskaya O A. The mathematical theory of viscous incompressible flow. New York:Gordon and Breach Science Publishers, 1969 [37] Lei Z, Lin F H, Zhou Y. Structure of helicity and global solutions of incompressible Navier-Stokes equation. Arch Ration Mech Anal, 2015, 218:1417-1430 [38] Leibovich S, Mahalov A, Titi E. Invariant helical subspaces for the Navier-Stokes equations. Arch Ration Mech Anal, 1990, 112:193-222 [39] Lemarié-Rieusset P G. Recent developments in the Navier-Stokes problems//Research Notes in Mathematics. Chapman & Hall/CRC, 2002 [40] Lemarié-Rieusset P G. The Navier-Stokes equations in the critical Morrey-Campanato space. Rev Mat Iberoam, 2007, 23:1-31 [41] Leonardi S, Málek J, Nečas J, Pokorný M. On axially symmetric flows in R3. Z Anal Anwendungen, 1999, 18:639-649 [42] Leray J. Étude de diverses équations intégrales non linéaires etde quelques problèmes que pose l'hydrodynamique. J Math Pures Appl, 1933, 12:1-82 [43] Leray J. Essai sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math, 1934, 63:193-248 [44] Lions J L. Quelques methodes de résolution des problèmes aux limites non linéaires. Paris:Dunod, 1969 [45] Lions P L. Mathematical topics in fluid dynamics, Vol.1 Incompressible models. Oxford:Oxford Sci Publ, Oxford University Press, 1996 [46] Lions P L. Mathematical topics in fluid dynamics, Vol.2 Compressible models. Oxford:Oxford Sci Publ, Oxford University Press, 1998 [47] Mahalov A, Titi E S, Leibovich S. Invariant helical subspaces for the Navier-Stokes equations. Arch Ration Mech Anal, 1990, 112:193-222 [48] Miyakawa T. On space-time decay properties of nonstationary incompressible Navier-Stokes flows in the whole space. SIAM J Math Anal, 2001, 33:523-544 [49] Miyakawa T. Notes on space-time decay properties of nonstationary incompressible Navier-Stokes flows in Rn. Funkcialaj Ekvacioj, 2002, 45:271-289 [50] Paicu M. Équation anisotrope de Navier-Stokes dans des espaces critiques. Revista Rev Matemtica Iberoamericana, 2005, 21:179-235 [51] Plecháč P, Šverák V. Singular and regular solutions of a non-linear parabolic system. Nonlinearity, 2003, 16:2093-2097 [52] Ponce G, Racke R, Sideris T, Titi E. Global stability of large solutions to the 3D Navier-Stokes equations. Comm Math Phys, 1994, 159:329-341 [53] Qin Y. Nonlinear parabolic-hyperbolic coupled systems and their attractors, Volume 184//Advances in Partial Differential Equations. Basel-Boston-Berlin:Birkhauser Verlag AG, 2008 [54] Qin Y, Huang L. Global well-posedness of nonlinear parabolic-hyperbolic coupled systems//Frontiers in Mathematics. Basel:Springer AG, 2012 [55] Qin Y, Liu X, Wang T. Global existence and uniqueness of nonlinear evolutionary fluid equations//Frontiers in Mathematics. Basel:Springer AG, 2015 [56] Qin Y, Zhang J. Asymptotic behavior of exact solutions for the Cauchy problem to the 3D cylindrically symmetric Navier-Stokes equations. Math Meth Appl Sci, 2017, 40:4538-4547 [57] Qin Y, Zhang J, Su X, Cao J. Global existence and exponential stability of spherically symmetric solutions to the compressible combustion radiative and reactive gas. J Math Fluid Mech, 2016, 18:415-461 [58] Schonbek M E. L2 decay for weak solutions of the Navier-Stokes equations. Arch Ration Mech, 1985, 88(3):200-222 [59] Schonbek M E. Large time behavior of solutions to Navier-Stokes equations. Comm in PDE, 1986, 11(7):733-763 [60] Schonbek M E. Lower bounds of rates of decay for solutions to the Navier-Stokes equations. J Amer Math Soc, 1991, 4:423-449 [61] Schonbek M E. Asymptotic behavior of solutions to the three dimensional Navier-Stokes equations. Indiana Univ Math J, 1992, 41(2):809-823 [62] Skalák Z. On the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component. Nonlinear Analysis, TMA, 2014, 104(12):84-89 [63] Tao T. Finite time blowup for an averaged three-dimensional Navier-Stokes equation. J Amer Math Soc, 2016, 29(3):601-674 [64] Ukhovskii M R, Iudovich V I. Axially symmetric flows of ideal and viscous fluids filling the whole space. J Appl Math Mech, 1968, 32:52-62 [65] Beirão da Veiga H. Existence and asymptotic behavior for strong solution of the Navier-Stokes equations on the whole space. Indiana Univ Math J, 1987, 36:149-166 [66] Weissler F. The Navier-Stokes initial value problem in Lp. Arch Ration Mech Anal, 1980, 74:219-230 [67] Wiegner M. Decay of the L∞-norm of solutions of the Navier-Stokes equations in unbounded domains. Acta Appl Math, 1994, 37:215-219 [68] Wong P. Global wellposedness for a certain class of large initial data for the 3D Navier-Stokes equations. Annales Henri Poincaré, 2014, 15(4):633-643 [69] Zhang J. Remarks on global existence and exponential stability of solutions for the viscous radiative and reactive gas with large initial data. Nonlinearity, 2017, 30:1221-1261 |
|