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Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere
Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$
1. | Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China |
2. | Institute of Applied Physics and Computational Mathematics, Beijing, 100094 |
3. | China Academy of Aerospace Aerodynamics, Beijing 100074, China |
Moreover, in 4-D and more higher dimension, it is shown that it is locally well-posed in $H^1(\mathbb{R}^n)$ with $n\geq 4$.
The method in this paper combine the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequality). We mainly extend the spaces $\mathbf{F}^s$ and $\mathbf{N}^s$ in one dimension [4] to higher dimension.
References:
[1] |
J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations, part II: the KdV equation, Geom. Funct. Anal., 3 (1993), 107-156.
doi: 10.1007/BF01896020. |
[2] |
A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012. |
[3] |
A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798.
doi: 10.1090/S0894-0347-06-00551-0. |
[4] |
A. D. Ionescu, C. E. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304.
doi: 10.1007/s00222-008-0115-0. |
[5] |
C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[6] |
C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.
doi: 10.1215/S0012-7094-93-07101-3. |
[7] |
C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.
doi: 10.1090/S0894-0347-96-00200-7. |
[8] |
E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286. |
[9] |
D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in phase space analysis with applications to PDEs, 181-213, Progr. Nonlinear Differential Equations Appl., 84, Birkhäuser/Springer, New York, 2013.
doi: 10.1007/978-1-4614-6348-1_10. |
[10] |
F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565.
doi: 10.3934/dcds.2009.24.547. |
[11] |
F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., 260 (2011), 1060-1085.
doi: 10.1016/j.jfa.2010.11.005. |
[12] |
L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.
doi: 10.1016/j.anihpc.2013.12.003. |
[13] |
F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.
doi: 10.1137/110850566. |
[14] |
T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908.
doi: 10.1353/ajm.2001.0035. |
show all references
References:
[1] |
J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations, part II: the KdV equation, Geom. Funct. Anal., 3 (1993), 107-156.
doi: 10.1007/BF01896020. |
[2] |
A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012. |
[3] |
A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798.
doi: 10.1090/S0894-0347-06-00551-0. |
[4] |
A. D. Ionescu, C. E. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304.
doi: 10.1007/s00222-008-0115-0. |
[5] |
C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[6] |
C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.
doi: 10.1215/S0012-7094-93-07101-3. |
[7] |
C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.
doi: 10.1090/S0894-0347-96-00200-7. |
[8] |
E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286. |
[9] |
D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in phase space analysis with applications to PDEs, 181-213, Progr. Nonlinear Differential Equations Appl., 84, Birkhäuser/Springer, New York, 2013.
doi: 10.1007/978-1-4614-6348-1_10. |
[10] |
F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565.
doi: 10.3934/dcds.2009.24.547. |
[11] |
F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., 260 (2011), 1060-1085.
doi: 10.1016/j.jfa.2010.11.005. |
[12] |
L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.
doi: 10.1016/j.anihpc.2013.12.003. |
[13] |
F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.
doi: 10.1137/110850566. |
[14] |
T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908.
doi: 10.1353/ajm.2001.0035. |
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