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December  2016, 36(12): 6943-6974. doi: 10.3934/dcds.2016102

## 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

Received  February 2016 Revised  March 2016 Published  October 2016

We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\mathbb{R} ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|\xi | \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R} ^d)$ with $s> \max \left( \frac{d-1}{d} s_c , \frac{s_c}{2}, s_c - \frac{d}{2(d+1)} \right)$ for $d+m \ge 5$, where $s$ is below the scaling critical regularity $s_c := \frac{d}{2}-\frac{1}{m-1}$.
Citation: Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102
##### References:
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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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