Almost conservation laws for stochastic nonlinear Schrödinger equations. (English) Zbl 1476.35230
Summary: In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the \(I\)-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on \(\mathbb{R}^3\) with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the \(I\)-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
Keywords:
stochastic nonlinear Schrödinger equation; global well-posedness; \(I\)-method; almost conservation lawReferences:
| [1] | Bényi, Á.; Oh, T.; Pocovnicu, O., On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on \({\mathbb{R}}^d, d \ge 3\), Trans. Amer. Math. Soc. Ser. B, 2, 1-50 (2015) · Zbl 1339.35281 · doi:10.1090/btran/6 |
| [2] | Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3, 2, 107-156 (1993) · Zbl 0787.35097 · doi:10.1007/BF01896020 |
| [3] | Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal, 3, 3, 209-262 (1993) · Zbl 0787.35098 · doi:10.1007/BF01895688 |
| [4] | J. Bourgain, Refinements of Strichartz’s inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not., 1998, 253-283. · Zbl 0917.35126 |
| [5] | T. Cazenave, F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), 18-29, Lecture Notes in Math., 1394. · Zbl 0694.35170 |
| [6] | M. Christ, J. Colliander, T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, arXiv:math/0311048 [math.AP]. · Zbl 1048.35101 |
| [7] | Cheung, K.; Mosincat, R., Stochastic nonlinear Schrödinger equations on tori, Stoch. PDE: Anal. Comp., 7, 2, 169-208 (2019) · Zbl 1447.60094 · doi:10.1007/s40072-018-0125-x |
| [8] | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett., 9, 5-6, 659-682 (2002) · Zbl 1152.35491 · doi:10.4310/MRL.2002.v9.n5.a9 |
| [9] | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211, 1, 173-218 (2004) · Zbl 1062.35109 · doi:10.1016/S0022-1236(03)00218-0 |
| [10] | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Sharp global well-posedness for KdV and modified KdV on \({\mathbb{R}}\) and \({\mathbb{T}} \), J. Amer. Math. Soc., 16, 3, 705-749 (2003) · Zbl 1025.35025 · doi:10.1090/S0894-0347-03-00421-1 |
| [11] | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Resonant decompositions and the \(I\)-method for the cubic nonlinear Schrödinger equation on \({\mathbb{R}}^2\), Discrete Contin. Dyn. Syst., 21, 3, 665-686 (2008) · Zbl 1147.35095 · doi:10.3934/dcds.2008.21.665 |
| [12] | Colliander, J.; Oh, T., Almost sure well-posedness of the cubic nonlinear Schrödinger equation below \(L^2({\mathbb{T}}) \), Duke Math. J., 161, 3, 367-414 (2012) · Zbl 1260.35199 · doi:10.1215/00127094-1507400 |
| [13] | Da Prato, G.; Debussche, A., Strong solutions to the stochastic quantization equations, Ann. Probab., 31, 4, 1900-1916 (2003) · Zbl 1071.81070 · doi:10.1214/aop/1068646370 |
| [14] | G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014. xviii+493 pp. · Zbl 1317.60077 |
| [15] | de Bouard, A.; Debussche, A., The stochastic nonlinear Schrödinger equation in \(H^1\), Stochastic Anal. Appl., 21, 1, 97-126 (2003) · Zbl 1027.60065 · doi:10.1081/SAP-120017534 |
| [16] | de Bouard, A.; Debussche, A.; Tsutsumi, Y., White noise driven Korteweg-de Vries equation, J. Funct. Anal., 169, 2, 532-558 (1999) · Zbl 0952.60062 · doi:10.1006/jfan.1999.3484 |
| [17] | Ginibre, J.; Tsutsumi, Y.; Velo, G., On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151, 2, 384-436 (1997) · Zbl 0894.35108 · doi:10.1006/jfan.1997.3148 |
| [18] | J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), no. 1, 1-32. · Zbl 0396.35028 |
| [19] | Ginibre, J.; Velo, G., Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144, 1, 163-188 (1992) · Zbl 0762.35008 · doi:10.1007/BF02099195 |
| [20] | L. Grafakos, Classical Fourier analysis, Third edition. Graduate Texts in Mathematics, 249. Springer, New York, 2014. xviii+638 pp. · Zbl 1304.42001 |
| [21] | Gubinelli, M.; Koch, H.; Oh, T., Renormalization of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc., 370, 10, 7335-7359 (2018) · Zbl 1400.35240 · doi:10.1090/tran/7452 |
| [22] | M. Gubinelli, H. Koch, T. Oh, Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, arXiv:1811.07808 [math.AP]. |
| [23] | M. Gubinelli, H. Koch, T. Oh, L. Tolomeo, Global dynamics for the two-dimensional stochastic nonlinear wave equations,arXiv:2005.10570 [math.AP]. |
| [24] | I. Karatzas, S. Shreve, Brownian motion and stochastic calculus, Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. · Zbl 0734.60060 |
| [25] | Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46, 1, 113-129 (1987) · Zbl 0632.35038 |
| [26] | Kenig, C.; Ponce, G.; Vega, L., A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9, 2, 573-603 (1996) · Zbl 0848.35114 · doi:10.1090/S0894-0347-96-00200-7 |
| [27] | Keel, M.; Tao, T., Endpoint Strichartz estimates, Amer. J. Math., 120, 5, 955-980 (1998) · Zbl 0922.35028 · doi:10.1353/ajm.1998.0039 |
| [28] | Kishimoto, N., A remark on norm inflation for nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 18, 3, 1375-1402 (2019) · Zbl 1409.35191 · doi:10.3934/cpaa.2019067 |
| [29] | Oh, T., Periodic stochastic Korteweg-de Vries equation with additive space-time white noise, Anal. PDE, 2, 3, 281-304 (2009) · Zbl 1190.35202 · doi:10.2140/apde.2009.2.281 |
| [30] | Oh, T., A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60, 259-277 (2017) · Zbl 1382.35273 · doi:10.1619/fesi.60.259 |
| [31] | Oh, T.; Okamoto, M., On the stochastic nonlinear Schrödinger equations at critical regularities, Stoch. Partial Differ. Equ. Anal. Comput, 8, 44, 869-894 (2020) · Zbl 1452.35192 |
| [32] | Oh, T.; Pocovnicu, O.; Wang, Y., On the stochastic nonlinear Schrödinger equations with non-smooth additive noise, Kyoto J. Math., 60, 4, 1227-1243 (2020) · Zbl 1454.35349 · doi:10.1215/21562261-2019-0060 |
| [33] | M. Reed, B. Simon, Methods of modern mathematical physics. I. Functional analysis, Second edition. Academic Press, Inc., New York, 1980. xv+400 pp. · Zbl 0459.46001 |
| [34] | B. Simon, The \(P(\varphi )_2\) Euclidean (quantum) field theory, Princeton Series in Physics. Princeton University Press, Princeton, N.J., 1974. xx+392 pp. · Zbl 1175.81146 |
| [35] | Strichartz, RS, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44, 3, 705-714 (1977) · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1 |
| [36] | T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp. · Zbl 1106.35001 |
| [37] | Thomann, L.; Tzvetkov, N., Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, 23, 11, 2771-2791 (2010) · Zbl 1204.35154 · doi:10.1088/0951-7715/23/11/003 |
| [38] | Tsutsumi, Y., \(L^2\)- solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30, 1, 115-125 (1987) · Zbl 0638.35021 |
| [39] | Yajima, K., Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110, 3, 415-426 (1987) · Zbl 0638.35036 · doi:10.1007/BF01212420 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
