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Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation
Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity
1. | Organization for Promotion of Tenure Track, University of Miyazaki, 1-1, Gakuenkibanadai-nishi, Miyazaki, 889-2192, Japan |
2. | Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City, 380-8553, Japan |
References:
[1] |
Á. Bényi, T. Oh and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, Excursion in Harmonic Analysis, Applied and Numerical Harmonic Analysis, 4 (2015), 3-25.
doi: 10.1007/978-3-319-20188-7_1. |
[2] |
Á. Bényi, T. Oh and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbbR^d$, $d \ge 3$, Trans. Amer. Math. Soc. Ser. B, 2 (2015), 1-50.
doi: 10.1090/btran/6. |
[3] |
H. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.
doi: 10.1090/S0002-9947-01-02754-4. |
[4] |
J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.
doi: 10.1007/BF02099299. |
[5] |
J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.
doi: 10.1007/BF02099556. |
[6] |
J. Bourgain, Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices, (1998), 253-283.
doi: 10.1155/S1073792898000191. |
[7] |
J. Bourgain and A. Bulut, Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 1267-1288.
doi: 10.1016/j.anihpc.2013.09.002. |
[8] |
J. Bourgain and A. Bulut, Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: The 3d case, J. Eur. Math. Soc. (JEMS), 16 (2014), 1289-1325.
doi: 10.4171/JEMS/461. |
[9] |
N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations I: local theory, Invent. Math., 173 (2008), 449-475.
doi: 10.1007/s00222-008-0124-z. |
[10] |
N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations II: a global existence result, Invent. Math., 173 (2008), 477-496.
doi: 10.1007/s00222-008-0123-0. |
[11] |
N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS), 16 (2014), 1-30.
doi: 10.4171/JEMS/426. |
[12] |
J. Colliander, J. Delort, C. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325.
doi: 10.1090/S0002-9947-01-02760-X. |
[13] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness result for Schrödigner equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.
doi: 10.1137/S0036141001384387. |
[14] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness result for Schrödigner equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.
doi: 10.1137/S0036141001394541. |
[15] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T.Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbbR^{3}$, Ann. of Math., 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[16] |
J. Colliander and T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\mathbbT)$, Duke Math. J., 161 (2012), 367-414.
doi: 10.1215/00127094-1507400. |
[17] |
C. Deng and S. Cui, Random-data Cauchy problem for the Navier-Stokes equations on $\mathbbT ^3$, J. Differential Equations, 251 (2011), 902-917.
doi: 10.1016/j.jde.2011.05.002. |
[18] |
A. Grünrock, On the Cauchy - and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations,, , ().
|
[19] |
M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré, 26 (2009), 917-941.
doi: 10.1016/j.anihpc.2008.04.002. |
[20] |
N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.
doi: 10.1016/0167-2789(92)90185-P. |
[21] |
N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.
doi: 10.1016/0362-546X(93)90071-Y. |
[22] |
S. Herr, On the Cauchy Problem for the Derivative Nonlinear Schrödinger Equation with Periodic Boundary Condition, Int. Math. Res. Not., 2006.
doi: 10.1155/IMRN/2006/96763. |
[23] |
H. Hirayama, Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data, Comm. Pure Appl. Anal., 13 (2014), 1563-1591.
doi: 10.3934/cpaa.2014.13.1563. |
[24] |
H. Hirayama, Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity, Funkcialaj Ekvacioj, 58 (2015), 431-450.
doi: 10.1619/fesi.58.431. |
[25] |
H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity,, , ().
|
[26] |
S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1 (\mathbbT^3)$, Duke Math. J., 159 (2011), 329-349.
doi: 10.1215/00127094-1415889. |
[27] |
M. Ikeda, N. Kishimoto and M. Okamoto, Well-posedness for a quadratic derivative nonlinear schrödinger system at the critical regularity, Journal of Functional Analysis, 271 (2016), 747-798.
doi: 10.1016/j.jfa.2016.05.009. |
[28] |
J. Lührmann and D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on $\mathbbR^3$, Comm. Partial Differential Equations, 39 (2014), 2262-2283.
doi: 10.1080/03605302.2014.933239. |
[29] |
R. Mosincat and T. Oh, A remark on global well-posedness of the derivative nonlinear Schrödinger equation on the circle, C. R. Math. Acad. Sci. Paris, 353 (2015), 837-841, arXiv:1502.02261v3.
doi: 10.1016/j.crma.2015.06.015. |
[30] |
A. S. Nahmod and G. Staffilani, Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space, J. Eur. Math. Soc. (JEMS), 17 (2015), 1687-1759.
doi: 10.4171/JEMS/543. |
[31] |
H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Eqns., 4 (1999), 561-580. |
[32] |
H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Diff. Eqns., 42 (2001), 1-23. |
[33] |
L. Thomann, Random data Cauchy problem for supercritical Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2385-2402.
doi: 10.1016/j.anihpc.2009.06.001. |
[34] |
Y. Wu, Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space, Anal. PDE, 6 (2013), 1989-2002.
doi: 10.2140/apde.2013.6.1989. |
[35] |
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. |
[36] |
S. Zhong, The Cauchy problem of null form wave equation on $\mathbbT^d$ with random initial data, Funkcial. Ekvac., 55 (2012), 367-403.
doi: 10.1619/fesi.55.367. |
show all references
References:
[1] |
Á. Bényi, T. Oh and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, Excursion in Harmonic Analysis, Applied and Numerical Harmonic Analysis, 4 (2015), 3-25.
doi: 10.1007/978-3-319-20188-7_1. |
[2] |
Á. Bényi, T. Oh and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbbR^d$, $d \ge 3$, Trans. Amer. Math. Soc. Ser. B, 2 (2015), 1-50.
doi: 10.1090/btran/6. |
[3] |
H. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.
doi: 10.1090/S0002-9947-01-02754-4. |
[4] |
J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.
doi: 10.1007/BF02099299. |
[5] |
J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.
doi: 10.1007/BF02099556. |
[6] |
J. Bourgain, Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices, (1998), 253-283.
doi: 10.1155/S1073792898000191. |
[7] |
J. Bourgain and A. Bulut, Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 1267-1288.
doi: 10.1016/j.anihpc.2013.09.002. |
[8] |
J. Bourgain and A. Bulut, Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: The 3d case, J. Eur. Math. Soc. (JEMS), 16 (2014), 1289-1325.
doi: 10.4171/JEMS/461. |
[9] |
N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations I: local theory, Invent. Math., 173 (2008), 449-475.
doi: 10.1007/s00222-008-0124-z. |
[10] |
N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations II: a global existence result, Invent. Math., 173 (2008), 477-496.
doi: 10.1007/s00222-008-0123-0. |
[11] |
N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS), 16 (2014), 1-30.
doi: 10.4171/JEMS/426. |
[12] |
J. Colliander, J. Delort, C. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325.
doi: 10.1090/S0002-9947-01-02760-X. |
[13] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness result for Schrödigner equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.
doi: 10.1137/S0036141001384387. |
[14] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness result for Schrödigner equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.
doi: 10.1137/S0036141001394541. |
[15] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T.Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbbR^{3}$, Ann. of Math., 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[16] |
J. Colliander and T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\mathbbT)$, Duke Math. J., 161 (2012), 367-414.
doi: 10.1215/00127094-1507400. |
[17] |
C. Deng and S. Cui, Random-data Cauchy problem for the Navier-Stokes equations on $\mathbbT ^3$, J. Differential Equations, 251 (2011), 902-917.
doi: 10.1016/j.jde.2011.05.002. |
[18] |
A. Grünrock, On the Cauchy - and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations,, , ().
|
[19] |
M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré, 26 (2009), 917-941.
doi: 10.1016/j.anihpc.2008.04.002. |
[20] |
N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.
doi: 10.1016/0167-2789(92)90185-P. |
[21] |
N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.
doi: 10.1016/0362-546X(93)90071-Y. |
[22] |
S. Herr, On the Cauchy Problem for the Derivative Nonlinear Schrödinger Equation with Periodic Boundary Condition, Int. Math. Res. Not., 2006.
doi: 10.1155/IMRN/2006/96763. |
[23] |
H. Hirayama, Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data, Comm. Pure Appl. Anal., 13 (2014), 1563-1591.
doi: 10.3934/cpaa.2014.13.1563. |
[24] |
H. Hirayama, Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity, Funkcialaj Ekvacioj, 58 (2015), 431-450.
doi: 10.1619/fesi.58.431. |
[25] |
H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity,, , ().
|
[26] |
S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1 (\mathbbT^3)$, Duke Math. J., 159 (2011), 329-349.
doi: 10.1215/00127094-1415889. |
[27] |
M. Ikeda, N. Kishimoto and M. Okamoto, Well-posedness for a quadratic derivative nonlinear schrödinger system at the critical regularity, Journal of Functional Analysis, 271 (2016), 747-798.
doi: 10.1016/j.jfa.2016.05.009. |
[28] |
J. Lührmann and D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on $\mathbbR^3$, Comm. Partial Differential Equations, 39 (2014), 2262-2283.
doi: 10.1080/03605302.2014.933239. |
[29] |
R. Mosincat and T. Oh, A remark on global well-posedness of the derivative nonlinear Schrödinger equation on the circle, C. R. Math. Acad. Sci. Paris, 353 (2015), 837-841, arXiv:1502.02261v3.
doi: 10.1016/j.crma.2015.06.015. |
[30] |
A. S. Nahmod and G. Staffilani, Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space, J. Eur. Math. Soc. (JEMS), 17 (2015), 1687-1759.
doi: 10.4171/JEMS/543. |
[31] |
H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Eqns., 4 (1999), 561-580. |
[32] |
H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Diff. Eqns., 42 (2001), 1-23. |
[33] |
L. Thomann, Random data Cauchy problem for supercritical Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2385-2402.
doi: 10.1016/j.anihpc.2009.06.001. |
[34] |
Y. Wu, Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space, Anal. PDE, 6 (2013), 1989-2002.
doi: 10.2140/apde.2013.6.1989. |
[35] |
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. |
[36] |
S. Zhong, The Cauchy problem of null form wave equation on $\mathbbT^d$ with random initial data, Funkcial. Ekvac., 55 (2012), 367-403.
doi: 10.1619/fesi.55.367. |
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