## Analysis & PDE

• Anal. PDE
• Volume 8, Number 8 (2015), 2029-2055.

### Well-posedness and scattering for the Zakharov system in four dimensions

#### Abstract

The Cauchy problem for the Zakharov system in four dimensions is considered. Some new well-posedness results are obtained. For small initial data, global well-posedness and scattering results are proved, including the case of initial data in the energy space. None of these results are restricted to radially symmetric data.

#### Article information

Source
Anal. PDE, Volume 8, Number 8 (2015), 2029-2055.

Dates
Accepted: 3 September 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843187

Digital Object Identifier
doi:10.2140/apde.2015.8.2029

Mathematical Reviews number (MathSciNet)
MR3441212

Zentralblatt MATH identifier
1331.35093

#### Citation

Bejenaru, Ioan; Guo, Zihua; Herr, Sebastian; Nakanishi, Kenji. Well-posedness and scattering for the Zakharov system in four dimensions. Anal. PDE 8 (2015), no. 8, 2029--2055. doi:10.2140/apde.2015.8.2029. https://projecteuclid.org/euclid.apde/1510843187

#### References

• I. Bejenaru and S. Herr, “Convolutions of singular measures and applications to the Zakharov system”, J. Funct. Anal. 261:2 (2011), 478–506.
• I. Bejenaru, S. Herr, J. Holmer, and D. Tataru, “On the 2D Zakharov system with $L\sp 2$-Schrödinger data”, Nonlinearity 22:5 (2009), 1063–1089.
• J. Bourgain and J. Colliander, “On wellposedness of the Zakharov system”, Int. Math. Res. Not. 1996:11 (1996), 515–546.
• B. Dodson, “Global well-posedness and scattering for the focusing, energy-critical nonlinear Schr ödinger problem in dimension $d=4$ for initial data below a ground state threshold”, preprint, 2014.
• J. Ginibre and G. Velo, “Scattering theory for the Zakharov system”, Hokkaido Math. J. 35:4 (2006), 865–892.
• J. Ginibre, Y. Tsutsumi, and G. Velo, “On the Cauchy problem for the Zakharov system”, J. Funct. Anal. 151:2 (1997), 384–436.
• L. Glangetas and F. Merle, “Existence of self-similar blow-up solutions for Zakharov equation in dimension two, I”, Comm. Math. Phys. 160:1 (1994), 173–215.
• L. Glangetas and F. Merle, “Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two, II”, Comm. Math. Phys. 160:2 (1994), 349–389.
• Z. Guo, “Sharp spherically averaged Strichartz estimates for the Schrödinger equation”, preprint, 2014. http://msp.org/idx/arx/1406.2525arXiv 1406.2525
• Z. Guo and K. Nakanishi, “Small energy scattering for the Zakharov system with radial symmetry”, Int. Math. Res. Not. 2014:9 (2014), 2327–2342.
• 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 (2014), 1–38.
• Z. Guo, K. Nakanishi, and S. Wang, “Global dynamics below the ground state energy for the Zakharov system in the 3D radial case”, Adv. Math. 238 (2013), 412–441.
• Z. Guo, S. Lee, K. Nakanishi, and C. Wang, “Generalized Strichartz estimates and scattering for 3D Zakharov system”, Comm. Math. Phys. 331:1 (2014), 239–259.
• Z. Guo, K. Nakanishi, and S. Wang, “Global dynamics below the ground state energy for the Klein–Gordon–Zakharov system in the 3D radial case”, Comm. Partial Differential Equations 39:6 (2014), 1158–1184.
• Z. Guo, K. Nakanishi, and S. Wang, “Small energy scattering for the Klein–Gordon–Zakharov system with radial symmetry”, Math. Res. Lett. 21:4 (2014), 733–755.
• Z. Hani, F. Pusateri, and J. Shatah, “Scattering for the Zakharov system in 3 dimensions”, Comm. Math. Phys. 322:3 (2013), 731–753.
• I. Kato and K. Tsugawa, “Scattering and well-posedness for the zakharov system at a critical space in four and more spatial dimensions”, in preparation.
• M. Keel and T. Tao, “Endpoint Strichartz estimates”, Amer. J. Math. 120 (1998), 955–980.
• C. E. Kenig and F. Merle, “Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case”, Invent. Math. 166:3 (2006), 645–675.
• R. Killip and M. Visan, “The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher”, Amer. J. Math. 132 (2010), 361–424.
• N. Kishimoto, “Local well-posedness for the Zakharov system on the multidimensional torus”, J. Anal. Math. 119 (2013), 213–253.
• N. Masmoudi and K. Nakanishi, “Energy convergence for singular limits of Zakharov type systems”, Invent. Math. 172:3 (2008), 535–583.
• F. Merle, “Blow-up results of virial type for Zakharov equations”, Comm. Math. Phys. 175:2 (1996), 433–455.
• T. Ozawa and Y. Tsutsumi, “The nonlinear Schrödinger limit and the initial layer of the Zakharov equations”, Differential Integral Equations 5:4 (1992), 721–745.
• T. Ozawa and Y. Tsutsumi, “Global existence and asymptotic behavior of solutions for the Zakharov equations in three space dimensions”, Adv. Math. Sci. Appl. 3 (1993/94), 301–334.
• S. H. Schochet and M. I. Weinstein, “The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence”, Comm. Math. Phys. 106:4 (1986), 569–580.
• A. Shimomura, “Scattering theory for Zakharov equations in three-dimensional space with large data”, Commun. Contemp. Math. 6:6 (2004), 881–899.
• C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation: self-focusing and wave collapse, Applied Mathematical Sciences 139, Springer, New York, 1999.
• H. Takaoka, “Well-posedness for the Zakharov system with the periodic boundary condition”, Differential Integral Equations 12:6 (1999), 789–810.
• V. E. Zakharov, “Collapse of Langmuir waves”, Zh. Eksp. Teor. Fiz. 62 (1972), 1745–1759. In Russian; translated in Sov. Physics JETP 35:5 (1972), 908–914.