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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. |
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