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Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation

  • * Corresponding author: Jingrui Wang

    * Corresponding author: Jingrui Wang 
The authors were in part supported in part by NSFC (Grant No. 11301338) and Shanghai Scientific Research Innovation Project (15ZZ098)
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  • In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $\dot{H}^{-α}(\mathbb{R}^{3})$ or $\dot{H}^{-α}(\mathbb{T}^{3})$ with $0<α≤ 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $t≥0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlović and Staffilani on (SIMA) in which $α$ is restricted to $0<α<\frac{1}{4}$.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  •   Á. Bényi, T. Oh and O. Pocovinicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, Excursions in Harmonic Analysis, Appl. Numer. Harmon. Anal. , Birkhäuser/Springer, Cham, 4 (2015), 3-25.
      Á. Bényi , T. Oh  and  O. Pocovinicu , On the probabilistic Cauchy theory of the cubic nonlinear Schrodinger equation on $\mathbb{R}^{d}, d≥ 3$, Trans. Amer. Math. Soc. Ser. B, 2 (2015) , 1-50.  doi: 10.1090/btran/6.
      J. Bourgain , Invariant measures for the 2D defocusing nonliear Schrödinger equation, Comm. Math. Phys., 176 (1996) , 421-445.  doi: 10.1007/BF02099556.
      N. Burq  and  N. Tzvetkov , Random Data Cauchy theory for supercritical wave equation Ⅰ:Local theory, Invent. Math., 173 (2008) , 449-475.  doi: 10.1007/s00222-008-0124-z.
      N. Burq  and  N. Tzvetkov , Random Data Cauchy theory for supercritical wave equation Ⅱ:A global existence result, Invent. Math., 173 (2008) , 477-496.  doi: 10.1007/s00222-008-0123-0.
      N. Burq  and  N. Tzvetkov , Probabilistic Well-Posedness for the Cubic Wave Equation, J. Eur. Math. Soc., 16 (2014) , 1-30.  doi: 10.4171/JEMS/426.
      C. Deng  and  S. Cui , Random data Cauchy problem for the Navier-Stokes equation on $\mathbb{T}^{3}$, J.Differential Equation, 251 (2011) , 902-917.  doi: 10.1016/j.jde.2011.05.002.
      E. Hopf , Über die Aufgangswertaufgabe für die hydrodynamischen Grundliechungen, Math. Nachr., 4 (1951) , 213-231.  doi: 10.1002/mana.3210040121.
      P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman Hall/CRC Res. Notes Math. 431, Chapman & Hall/CTC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.
      J. Leray , Essai sur mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934) , 193-248.  doi: 10.1007/BF02547354.
      J. Lührmann  and  D. Mendelson , Random data Cauchy theory for nonlinear wave equations of Power-Type on $\mathbb{R}^{3}$, Comm. Partial Differential Equations, 39 (2014) , 2262-2283.  doi: 10.1080/03605302.2014.933239.
      A. R. Nahmod , N. Pavlović  and  G. Staffilani , Almost sure existence of global weak solutions for supercritical Navier-Stokes equation, SIAM J. Math. Anal., 45 (2013) , 3431-3452.  doi: 10.1137/120882184.
      R. E. A. C. Paley and A. Zygmund, On some series of functions (1)(2)(3), Proc. Camb. Philos. Soc., 26 (1930), 337-357,458-474; 28 (1932), 190-205.
      R. Témam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co. , Amsterdam-New York, 1979.
      T. Zhang  and  D. Fang , Random data Cauchy theory for the generalized incompressible Navier-Stokes equations, J. Math. Fluid Mech., 14 (2012) , 311-324.  doi: 10.1007/s00021-011-0069-7.
      T. Zhang  and  D. Fang , Random data Cauchy theory for the incompressible three dimensional Navier-Stokes Equation, Proc. Amer. Math. Soc., 139 (2011) , 2827-2837.  doi: 10.1090/S0002-9939-2011-10762-7.
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