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Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials
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 |
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 $0<α<\frac{1}{4}$.
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. |
| [2] |
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. |
| [3] |
J. Bourgain,
Invariant measures for the 2D defocusing nonliear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.
doi: 10.1007/BF02099556. |
| [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. |
| [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. |
| [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. |
| [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. |
| [8] |
E. Hopf,
Über die Aufgangswertaufgabe für die hydrodynamischen Grundliechungen, Math. Nachr., 4 (1951), 213-231.
doi: 10.1002/mana.3210040121. |
| [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. |
| [10] |
J. Leray,
Essai sur mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
| [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. |
| [12] |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [2] |
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. |
| [3] |
J. Bourgain,
Invariant measures for the 2D defocusing nonliear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.
doi: 10.1007/BF02099556. |
| [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. |
| [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. |
| [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. |
| [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. |
| [8] |
E. Hopf,
Über die Aufgangswertaufgabe für die hydrodynamischen Grundliechungen, Math. Nachr., 4 (1951), 213-231.
doi: 10.1002/mana.3210040121. |
| [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. |
| [10] |
J. Leray,
Essai sur mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
| [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. |
| [12] |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
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