We consider the mass-critical nonlinear fourth-order Schrödinger equations with random initial data. We prove almost sure local well-posedness and probabilistic small data global well-posedness below L2-space. We also prove probabilistic blow-up for the equation with non-gauge invariance and rough random initial data.
REFERENCES
1.
M.
Ben-Artzi
, H.
Koch
, and J. C.
Saut
, “Dispersion estimates for fourth-order Schrödinger equations
,” C. R. Acad. Sci.
330
(1
), 87
–92
(2000
).2.
A.
Bényi
, T.
Oh
, and O.
Pocovnicu
, “Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS
,” in Excursions in Harmonic Analysis, Volume 4, Applied and Numerical Harmonic Analysis
(Birkhäuser, Cham
, 2015
), pp. 3
–25
.3.
A.
Bényi
, T.
Oh
, and O.
Pocovnicu
, “On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on
,” Trans. Am. Math. Soc., Ser. B
2
, 1
–50
(2015
).4.
Á.
Bényi
, T.
Oh
, and O.
Pocovnicu
, “Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on
,” Trans. Am. Math. Soc., Ser. B
6
(4
), 114
–160
(2019
).5.
T.
Boulenger
and E.
Lenzmann
, “Blowup for biharmonic NL4S
,” Ann. Sci. Éc. Norm. Supér.
50
(3
), 503
–544
(2017
).6.
J.
Bourgain
, “Invariant measures for the 2D-defocusing nonlinear Schrödinger equation
,” Commun. Math. Phys.
176
(2
), 421
–445
(1996
).7.
N.
Burq
and N.
Tzvetkov
, “Random data Cauchy theory for supercritical wave equations. I. Local theory
,” Invent. Math.
173
(3
), 449
–475
(2008
).8.
N.
Burq
, L.
Thomann
, and N.
Tzvetkov
, “Long time dynamics for the one dimensional nonlinear Schrödinger equation
,” Ann. Inst. Fourier
63
(6
), 2137
–2198
(2013
).9.
T.
Cazenave
, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics Vol. 10
(American Mathematical Society; Courant Institute of Mathematical Sciences
, 2003
).10.
M.
Chen
and S.
Zhang
, “Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities
,” Nonlinear Anal.
190
, 111608
(2020
).11.
Y.
Cho
and S.
Lee
, “Strichartz estimates in spherical coordinates
,” Indiana Univ. Math. J.
62
(3
), 991
–1020
(2013
).12.
Y.
Deng
, “Two-dimensional nonlinear Schrödinger equation with random radial data
,” Anal. PDE
5
(5
), 913
–960
(2012
).13.
V. D.
Dinh
, “Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces
,” Int. J. Appl. Math.
31
(4
), 483
–535
(2018
).14.
V. D.
Dinh
, “On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation
,” Bull. Belg. Math. Soc. Simon Stevin
25
(3
), 415
–437
(2018
).15.
V. D.
Dinh
, “Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space
,” Nonlinear Anal.
172
, 115
–140
(2018
).16.
V. D.
Dinh
, “On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation
,” J. Dyn. Differ. Equations
31
(4
), 1793
–1823
(2019
).17.
V. D.
Dinh
, “Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations
,” Nonlinearity
34
, 776
(2021
).18.
V. D.
Dinh
, “Random data theory for the cubic fourth-order nonlinear Schrödinger equation
,” Commun. Pure Appl. Anal.
20
(2
), 651
–680
(2021
).19.
B.
Dodson
, J.
Lührmann
, and D.
Mendelson
, “Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation
,” Adv. Math.
347
, 619
–676
(2019
).20.
Q.
Guo
, “Scattering for the focusing L2-supercritical and -subcritical biharmonic NLS equations
,” Commun. Partial Differ. Equations
41
(2
), 185
–207
(2016
).21.
Z.
Guo
and Y.
Wang
, “Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations
,” J. Anal. Math.
124
(1
), 1
–38
(2014
).22.
H.
Hirayama
and M.
Okamoto
, “Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity
,” Discrete Contin. Dyn. Syst.
36
(12
), 6943
–6974
(2016
).23.
H.
Hirayama
and M.
Okamoto
, “Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity
,” arXiv:1505.06497 (2015
).24.
Z.
Huo
and Y.
Jia
, “The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament
,” J. Differ. Equations
214
, 1
–35
(2005
).25.
G.
Hwang
, “Probabilistic well-posedness of the mass-critical NLS with radial data below
,” J. Math. Anal. Appl.
475
, 1842
–1854
(2019
).26.
M.
Ikeda
and T.
Inui
, “Some non-existence results for semilinear Schrödinger equation without gauge invariance
,” J. Math. Anal. Appl.
425
, 758
–773
(2015
).27.
M.
Ikeda
and Y.
Wakasugi
, “Small data blow-up of L2-solution for the nonlinear Schrödinger equation without gauge invariance
,” Differ. Int. Equations
26
(11/12
), 1265
–1285
(2013
), available at http://projecteuclid.org/euclid.die/1378327426.28.
V. I.
Karpman
, “Stabiliztion of solition instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations
,” Phys. Rev. E
53
(2
), 1336
–1339
(1996
).29.
V. I.
Karpman
and A. G.
Shagalov
, “Stability of solition described by nonlinear Schrödinger-type equations with higher-order dispersion
,” Physica D
144
, 194
–210
(2000
).30.
Y.
Ke
, “Remark on the Strichartz estimates in the radial case
,” J. Math. Anal. Appl.
387
, 857
–861
(2012
).31.
R.
Killip
, J.
Murphy
, and M.
Visan
, “Almost sure scattering for the energy-critical NLS with radial data below
,” Commun. Partial Differ. Equations
44
(1
), 51
–71
(2019
).32.
J.
Lührmann
and D.
Mendelson
, “Random data Cauchy theory for the nonlinear wave equations of power-type on
,” Commun. Partial Differ. Equations
39
(12
), 2262
–2283
(2014
).33.
C.
Miao
, G.
Xu
, and L.
Zhao
, “Global well-posedness and scattering for the defocusing energy critical nonlinear Schrödinger equation of fourth order in the radial case
,” J. Differ. Equations
246
, 3715
–3749
(2009
).34.
C.
Miao
, G.
Xu
, and L.
Zhao
, “Global well-posedness and scattering for the focusing energy critical nonlinear Schrödinger equations of fourth order in dimensions d ≥ 9
,” J. Differ. Equations
251
, 3381
–3402
(2011
).35.
C.
Miao
, H.
Wu
, and J.
Zhang
, “Scattering theory below energy for the cubic fourth-order Schrödinger equation
,” Math. Narchr.
288
(7
), 798
–823
(2015
).36.
T.
Oh
, M.
Okamoto
, and O.
Pocovnicu
, “On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
,” Discrete Contin. Dyn. Syst.
39
(6
), 3479
–3520
(2019
).37.
B.
Pausader
, “Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case
,” Dyn. Partial Differ. Equations
4
(3
), 197
–225
(2007
).38.
B.
Pausader
, “The focusing energy-critical fourth-order Schrödinger equation with radial data
,” Discrete Contin. Dyn. Syst.
24
, 1275
–1292
(2009
).39.
B.
Pausader
, “The cubic fourth-order Schrödinger equation
,” J. Funct. Anal.
256
, 2473
–2517
(2009
).40.
B.
Pausader
and S.
Shao
, “The mass-critical fourth-order Schrödinger equation in high dimensions
,” J. Hyper. Differ. Equations
07
, 651
–705
(2010
).41.
B.
Pausader
and S.
Xia
, “Scattering theory for the fourth-order Schrödinger equation in low dimensions
,” Nonlinearity
26
, 2175
–2191
(2013
).42.
S.
Zhang
and S.
Xu
, “The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities
,” Commun. Pure Appl. Anal.
19
(6
), 3367
–3385
(2020
).43.
S.
Zhu
, H.
Yang
, and J.
Zhang
, “Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation
,” Dyn. Partial Differ. Equations
7
, 187
–205
(2010
).© 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.