-
Previous Article
A characterization of Sierpiński carpet rational maps
- DCDS Home
- This Issue
-
Next Article
Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions
Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type
Mathematics Department, Princeton University, Princeton, New Jersey, 08544, USA |
We study a class of $3D$ quadratic Schrödinger equations as follows, $(\partial_t -i Δ) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by Germain-Masmoudi-Shatah in [2], the interaction of $u$ and $\bar{u}$ is very strong at the low frequency part, e.g., $1× 1 \to 0$ type interaction (the size of input frequency is "1" and the size of output frequency is "0"). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again cause the growth of profile in the medium frequency part due to the $1× 0\to 1$ type interaction. The issue of strong $1× 1\to 0$ type interaction makes the global existence problem very delicate.
In this paper, we show that, as long as there are "$ε$" derivatives inside the quadratic term $Q (u, \bar{u})$, there exists a global solution for small initial data. As a byproduct, we also give a simple proof for the almost global existence of the small data solution of $(\partial_t -i Δ)u = |u|^2 = u\bar{u}$, which was first proved by Ginibre-Hayashi [3]. Instead of using vector fields, we consider this problem purely in Fourier space.
References:
| [1] |
T. Cazenave and F. Weissler,
Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529. |
| [2] |
P. Germain, N. Masmoudi and J. Shatah,
Global Solutions for $3D$ Quadratic Schrödinger Equations, Int. Math. Res. Notice, 2009 (2009), 414-432.
doi: 10.1093/imrn/rnn135. |
| [3] |
J. Ginibre and N. Hayashi,
Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions, Math. Z., 219 (1995), 119-140.
doi: 10.1007/BF02572354. |
| [4] |
Z. Guo, L. Peng and B. Wang,
Decay estimates for a class of wave equations, J. Func, Anal., 254 (2008), 1642-1660.
doi: 10.1016/j.jfa.2007.12.010. |
| [5] |
N. Hayashi and P. Naumkin,
On the quadratic nonlinear Schrödinger equation in three space dimensions, Int. Math. Res. Notice, 2000 (2000), 115-132.
doi: 10.1155/S1073792800000088. |
| [6] |
M. Ikeda and T. Inui,
Small data blow-up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015), 571-581.
doi: 10.1007/s00028-015-0273-7. |
| [7] |
Y. Kawahara,
Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in $3D$, Differential Integral Equations, 18 (2005), 169-194.
|
| [8] |
S. Klainerman and G. Ponce,
Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure. Appl. Math., 36 (1983), 133-141.
doi: 10.1002/cpa.3160360106. |
| [9] |
W. Strauss,
Nonlinear scattering theory at low energy, J. Fun. Anal., 41 (1981), 110-133.
doi: 10.1016/0022-1236(81)90063-X. |
show all references
References:
| [1] |
T. Cazenave and F. Weissler,
Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529. |
| [2] |
P. Germain, N. Masmoudi and J. Shatah,
Global Solutions for $3D$ Quadratic Schrödinger Equations, Int. Math. Res. Notice, 2009 (2009), 414-432.
doi: 10.1093/imrn/rnn135. |
| [3] |
J. Ginibre and N. Hayashi,
Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions, Math. Z., 219 (1995), 119-140.
doi: 10.1007/BF02572354. |
| [4] |
Z. Guo, L. Peng and B. Wang,
Decay estimates for a class of wave equations, J. Func, Anal., 254 (2008), 1642-1660.
doi: 10.1016/j.jfa.2007.12.010. |
| [5] |
N. Hayashi and P. Naumkin,
On the quadratic nonlinear Schrödinger equation in three space dimensions, Int. Math. Res. Notice, 2000 (2000), 115-132.
doi: 10.1155/S1073792800000088. |
| [6] |
M. Ikeda and T. Inui,
Small data blow-up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015), 571-581.
doi: 10.1007/s00028-015-0273-7. |
| [7] |
Y. Kawahara,
Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in $3D$, Differential Integral Equations, 18 (2005), 169-194.
|
| [8] |
S. Klainerman and G. Ponce,
Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure. Appl. Math., 36 (1983), 133-141.
doi: 10.1002/cpa.3160360106. |
| [9] |
W. Strauss,
Nonlinear scattering theory at low energy, J. Fun. Anal., 41 (1981), 110-133.
doi: 10.1016/0022-1236(81)90063-X. |
| [1] |
Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021026 |
| [2] |
Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 |
| [3] |
Norisuke Ioku. Some space-time integrability estimates of the solution for heat equations in two dimensions. Conference Publications, 2011, 2011 (Special) : 707-716. doi: 10.3934/proc.2011.2011.707 |
| [4] |
Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323 |
| [5] |
Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009 |
| [6] |
Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385 |
| [7] |
Yuming Zhang. On continuity equations in space-time domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212 |
| [8] |
Marjan Uddin, Hazrat Ali. Space-time kernel based numerical method for generalized Black-Scholes equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2905-2915. doi: 10.3934/dcdss.2020221 |
| [9] |
Xenia Kerkhoff, Sandra May. Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021054 |
| [10] |
Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1 |
| [11] |
Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845 |
| [12] |
Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033 |
| [13] |
Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613 |
| [14] |
Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499 |
| [15] |
Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004 |
| [16] |
Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108 |
| [17] |
Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100 |
| [18] |
Vincent Astier, Thomas Unger. Galois extensions, positive involutions and an application to unitary space-time coding. Advances in Mathematics of Communications, 2019, 13 (3) : 513-516. doi: 10.3934/amc.2019032 |
| [19] |
Susanne Pumplün, Thomas Unger. Space-time block codes from nonassociative division algebras. Advances in Mathematics of Communications, 2011, 5 (3) : 449-471. doi: 10.3934/amc.2011.5.449 |
| [20] |
Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]