doi: 10.3934/dcds.2021147
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Local well-posedness for the Zakharov system in dimension d ≤ 3

Universität Bielefeld, Fakultät für Mathematik, Postfach 10 01 31, 33501 Bielefeld, Germany

Received  April 2021 Revised  August 2021 Early access October 2021

The Zakharov system in dimension $ d\leqslant 3 $ is shown to be locally well-posed in Sobolev spaces $ H^s \times H^l $, extending the previously known result. We construct new solution spaces by modifying the $ X^{s,b} $ spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.

Citation: Akansha Sanwal. Local well-posedness for the Zakharov system in dimension d ≤ 3. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021147
References:
[1]

I. BejenaruZ. GuoS. Herr and K. Nakanishi, Well-posedness and scattering for the Zakharov system in four dimensions, Anal. PDE, 8 (2015), 2029-2055.  doi: 10.2140/apde.2015.8.2029.  Google Scholar

[2]

I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system, J. Func. Anal., 261 (2011), 478-506.  doi: 10.1016/j.jfa.2011.03.015.  Google Scholar

[3]

I. BejenaruS. HerrJ. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$ initial data, Nonlinearity, 22 (2009), 1063-1089.  doi: 10.1088/0951-7715/22/5/007.  Google Scholar

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I. BejenaruS. Herr and D. Tataru, A convolution estimate for two dimensional hypersurfaces, Rev. Mat. Iberoam., 26 (2010), 707-728.  doi: 10.4171/RMI/615.  Google Scholar

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J. BennettA. Carbery and J. Wright, A non-linear generalisation of the Loomis-Whitney inequality and applications, Math. Res. Lett., 12 (2005), 443-457.  doi: 10.4310/MRL.2005.v12.n4.a1.  Google Scholar

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H. A. Biagioni and F. Linares, Ill-posedness for the Zakharov system with generalised nonlinearity, Proc. Am. Math. Soc., 131 (2003), 3113-3121.  doi: 10.1090/S0002-9939-03-06898-9.  Google Scholar

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T. Candy, S. Herr and K. Nakanishi, The Zakharov system in dimension $d\geqslant 4$, preprint, arXiv: 1912.05820, (to appear in Journal of The European Mathematical Society). Google Scholar

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Z. Chen and S. Wu, Local well-posedness for the Zakharov system in dimension $d = 2, 3$, Commun. Pure Appl. Anal., doi: 10.3934/cpaa.2021161. Google Scholar

[10]

J. CollianderJ. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Am. Math. Soc., 360 (2008), 4619-4638.  doi: 10.1090/S0002-9947-08-04295-5.  Google Scholar

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L. Domingues and R. Santos, A note on $C^2$ ill-posedness results for the Zakharov system in arbitrary dimension, preprint, arXiv: 1910.06486. Google Scholar

[12]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[13]

F. Grube, Zur Regularität der Flussabbildung des Zakharov-Systems, Master's thesis, Bielefeld University, 2020. Google Scholar

[14]

Z. Guo and K. Nakanishi, Small energy scattering for the Zakharov system with radial symmetry, Int. Math. Res. Not. IMRN, 2014 (2014), 2327-2342.  doi: 10.1093/imrn/rns296.  Google Scholar

[15]

J. Holmer, Local ill-posedness for the 1D Zakharov system, Electron. J. Differ. Equ., (2007), 1-22.  Google Scholar

[16]

N. Kishimoto, A remark on norm inflation for nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 18 (2019), 1375-1402.  doi: 10.3934/cpaa.2019067.  Google Scholar

[17]

L. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Am. Math. Soc., 55 (2005), 961-962.  doi: 10.1090/S0002-9904-1949-09320-5.  Google Scholar

[18]

L. Molinet and S. Vento, Improvement of the energy method for strongly nonresonant dispersive equations and applications, Anal. PDE, 8 (2015), 1455-1495.  doi: 10.2140/apde.2015.8.1455.  Google Scholar

[19]

T. Ozawa and Y. Tsutsumi, Existence and smoothing effects of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., 28 (1992), 329-361.  doi: 10.2977/prims/1195168430.  Google Scholar

[20]

H. Pecher, Global well-posedness below energy space for the 1-dimensional Zakharov system, Int. Math. Res. Not., 2001 (2001), 1027-1056.  doi: 10.1155/S1073792801000496.  Google Scholar

[21]

H. Pecher, An improved local well-posedness result for the one-dimensional Zakharov system, J. Math. Anal. Appl., 342 (2008), 1440-1454.  doi: 10.1016/j.jmaa.2008.01.035.  Google Scholar

[22]

A. Rubel, Eine Normalformreduktion für das Zakharov System, Master's thesis, Bielefeld University, 2016. Google Scholar

[23]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, Volume 106, American Mathematical Society, 2006. doi: 10.1090/cbms/106.  Google Scholar

[24]

V. E. Zakharov et al., Collapse of Lamgmuir waves, Sov. Phys. JETP, 35) (1972), 908-914.   Google Scholar

show all references

References:
[1]

I. BejenaruZ. GuoS. Herr and K. Nakanishi, Well-posedness and scattering for the Zakharov system in four dimensions, Anal. PDE, 8 (2015), 2029-2055.  doi: 10.2140/apde.2015.8.2029.  Google Scholar

[2]

I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system, J. Func. Anal., 261 (2011), 478-506.  doi: 10.1016/j.jfa.2011.03.015.  Google Scholar

[3]

I. BejenaruS. HerrJ. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$ initial data, Nonlinearity, 22 (2009), 1063-1089.  doi: 10.1088/0951-7715/22/5/007.  Google Scholar

[4]

I. BejenaruS. Herr and D. Tataru, A convolution estimate for two dimensional hypersurfaces, Rev. Mat. Iberoam., 26 (2010), 707-728.  doi: 10.4171/RMI/615.  Google Scholar

[5]

J. BennettA. Carbery and J. Wright, A non-linear generalisation of the Loomis-Whitney inequality and applications, Math. Res. Lett., 12 (2005), 443-457.  doi: 10.4310/MRL.2005.v12.n4.a1.  Google Scholar

[6]

H. A. Biagioni and F. Linares, Ill-posedness for the Zakharov system with generalised nonlinearity, Proc. Am. Math. Soc., 131 (2003), 3113-3121.  doi: 10.1090/S0002-9939-03-06898-9.  Google Scholar

[7]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Int. Math. Res. Not., 1996 (1996), 515-546.  doi: 10.1155/S1073792896000359.  Google Scholar

[8]

T. Candy, S. Herr and K. Nakanishi, The Zakharov system in dimension $d\geqslant 4$, preprint, arXiv: 1912.05820, (to appear in Journal of The European Mathematical Society). Google Scholar

[9]

Z. Chen and S. Wu, Local well-posedness for the Zakharov system in dimension $d = 2, 3$, Commun. Pure Appl. Anal., doi: 10.3934/cpaa.2021161. Google Scholar

[10]

J. CollianderJ. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Am. Math. Soc., 360 (2008), 4619-4638.  doi: 10.1090/S0002-9947-08-04295-5.  Google Scholar

[11]

L. Domingues and R. Santos, A note on $C^2$ ill-posedness results for the Zakharov system in arbitrary dimension, preprint, arXiv: 1910.06486. Google Scholar

[12]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[13]

F. Grube, Zur Regularität der Flussabbildung des Zakharov-Systems, Master's thesis, Bielefeld University, 2020. Google Scholar

[14]

Z. Guo and K. Nakanishi, Small energy scattering for the Zakharov system with radial symmetry, Int. Math. Res. Not. IMRN, 2014 (2014), 2327-2342.  doi: 10.1093/imrn/rns296.  Google Scholar

[15]

J. Holmer, Local ill-posedness for the 1D Zakharov system, Electron. J. Differ. Equ., (2007), 1-22.  Google Scholar

[16]

N. Kishimoto, A remark on norm inflation for nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 18 (2019), 1375-1402.  doi: 10.3934/cpaa.2019067.  Google Scholar

[17]

L. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Am. Math. Soc., 55 (2005), 961-962.  doi: 10.1090/S0002-9904-1949-09320-5.  Google Scholar

[18]

L. Molinet and S. Vento, Improvement of the energy method for strongly nonresonant dispersive equations and applications, Anal. PDE, 8 (2015), 1455-1495.  doi: 10.2140/apde.2015.8.1455.  Google Scholar

[19]

T. Ozawa and Y. Tsutsumi, Existence and smoothing effects of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., 28 (1992), 329-361.  doi: 10.2977/prims/1195168430.  Google Scholar

[20]

H. Pecher, Global well-posedness below energy space for the 1-dimensional Zakharov system, Int. Math. Res. Not., 2001 (2001), 1027-1056.  doi: 10.1155/S1073792801000496.  Google Scholar

[21]

H. Pecher, An improved local well-posedness result for the one-dimensional Zakharov system, J. Math. Anal. Appl., 342 (2008), 1440-1454.  doi: 10.1016/j.jmaa.2008.01.035.  Google Scholar

[22]

A. Rubel, Eine Normalformreduktion für das Zakharov System, Master's thesis, Bielefeld University, 2016. Google Scholar

[23]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, Volume 106, American Mathematical Society, 2006. doi: 10.1090/cbms/106.  Google Scholar

[24]

V. E. Zakharov et al., Collapse of Lamgmuir waves, Sov. Phys. JETP, 35) (1972), 908-914.   Google Scholar

2]">Figure 1.  Region of well-posedness for $ d = 3 $ in [2]
Figure 2.  New region of well-posedness for $ d\leq 3 $
Table 1.  $ H \times L\rightarrow L $
$ |\tau_1| $ $ |\tau_2| $ $ |\tau_0| $ Conclusion
$ \lesssim \lambda_1 $ $ \lesssim \lambda_1 $ $ \lesssim \lambda_1 $ $ \lambda_0^2\lesssim \lambda_1 $
$ \lesssim \lambda_1 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ |{\tau _W}|\sim|{\tau _0}|,\;\;{\lambda _1} \ll \lambda _0^2 $
$ \lambda_1\ll\cdot \ll \lambda_1^2 $ $ \lesssim \lambda_1 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ |\tau_S|\sim |\tau_0|, \lambda_1 \ll \lambda_0^2 $
$ \lambda_1 \ll\cdot \ll \lambda_1^2 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ |\tau_S|,|\tau_W|\sim \lambda_0^2, \lambda_1 \ll \lambda_0^2 $
$ \lambda_1 \ll\cdot \ll \lambda_1^2 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ \ll \lambda_1 $ $ \lambda_0^2 \ll \lambda_1 $
$ |\tau_1| $ $ |\tau_2| $ $ |\tau_0| $ Conclusion
$ \lesssim \lambda_1 $ $ \lesssim \lambda_1 $ $ \lesssim \lambda_1 $ $ \lambda_0^2\lesssim \lambda_1 $
$ \lesssim \lambda_1 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ |{\tau _W}|\sim|{\tau _0}|,\;\;{\lambda _1} \ll \lambda _0^2 $
$ \lambda_1\ll\cdot \ll \lambda_1^2 $ $ \lesssim \lambda_1 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ |\tau_S|\sim |\tau_0|, \lambda_1 \ll \lambda_0^2 $
$ \lambda_1 \ll\cdot \ll \lambda_1^2 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ |\tau_S|,|\tau_W|\sim \lambda_0^2, \lambda_1 \ll \lambda_0^2 $
$ \lambda_1 \ll\cdot \ll \lambda_1^2 $ $ \lambda_1 \ll \cdot \ll \lambda_1^2 $ $ \ll \lambda_1 $ $ \lambda_0^2 \ll \lambda_1 $
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