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September  2017, 37(9): 5003-5019. doi: 10.3934/dcds.2017215

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

1. 

School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China

2. 

School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China

* Corresponding author: Jingrui Wang

Received  June 2016 Revised  April 2017 Published  June 2017

Fund Project: The authors were in part supported in part by NSFC (Grant No. 11301338) and Shanghai Scientific Research Innovation Project (15ZZ098).

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 $0<α<\frac{1}{4}$.

Citation: Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215
References:
[1]

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.  Google Scholar

[2]

Á. BényiT. 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.  Google Scholar

[3]

J. Bourgain, Invariant measures for the 2D defocusing nonliear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.  doi: 10.1007/BF02099556.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

E. Hopf, Über die Aufgangswertaufgabe für die hydrodynamischen Grundliechungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[9]

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.  Google Scholar

[10]

J. Leray, Essai sur mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[11]

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.  Google Scholar

[12]

A. R. NahmodN. 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.  Google Scholar

[13]

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. Google Scholar

[14]

R. Témam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co. , Amsterdam-New York, 1979.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

Á. BényiT. 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.  Google Scholar

[3]

J. Bourgain, Invariant measures for the 2D defocusing nonliear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.  doi: 10.1007/BF02099556.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

E. Hopf, Über die Aufgangswertaufgabe für die hydrodynamischen Grundliechungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[9]

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.  Google Scholar

[10]

J. Leray, Essai sur mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[11]

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.  Google Scholar

[12]

A. R. NahmodN. 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.  Google Scholar

[13]

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. Google Scholar

[14]

R. Témam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co. , Amsterdam-New York, 1979.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

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