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

@article{Bejenaru2015WellposednessAS,
title={Well-posedness and scattering for the Zakharov system in four dimensions},
author={Ioan Bejenaru and Zihua Guo and Sebastian Herr and Kenji Nakanishi},
journal={arXiv: Analysis of PDEs},
year={2015}
}
• Published 5 April 2015
• Mathematics
• arXiv: Analysis of PDEs
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 is restricted to radially symmetric data.
Minimal non-scattering solutions for the Zakharov system
. We consider the Zakharov system in the energy critical dimension d = 4 with energy below the ground state. It is known that below the ground state solutions exist globally in time, and scatter in
Norm inflation for the Zakharov system
. We prove norm inﬂation in new regions of Sobolev regularities for the scalar Zakharov system in the spatial domain R d for arbitrary d ∈ N . To this end, we apply abstract considerations of
Local well-posedness of the Cauchy problem for the degenerate Zakharov system
X iv :2 10 3. 05 34 0v 1 [ m at h. A P] 9 M ar 2 02 1 LOCAL WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE DEGENERATE ZAKHAROV SYSTEM ISAO KATO Abstract. The aim of this paper is to investigate
The Zakharov system in 4D radial energy space below the ground state
• Mathematics
American Journal of Mathematics
• 2021
We prove dynamical dichotomy into scattering and blow-up (in a weak sense) for all radial solutions of the Zakharov system in the energy space of four spatial dimensions that have less energy than
Randomized final-state problem for the Zakharov system in dimension three
• Martin Spitz
• Mathematics
Communications in Partial Differential Equations
• 2021
Abstract We consider the final-state problem for the Zakharov system in the energy space in three space dimensions. For without any size restriction, symmetry assumption or additional angular
Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions
• Mathematics
• 2015
We study the Cauchy problem for the Zakharov system in spatial dimension $d\ge 4$ with initial datum $(u(0), n(0), \partial_t n(0)) \in H^k(\mathbb{R}^d) \times \dot{H}^l(\mathbb{R}^d)\times A P ] 1 3 O ct 2 01 8 THE ZAKHAROV SYSTEM IN 4 D RADIAL ENERGY SPACE BELOW THE GROUND STATE • Mathematics • 2018 We prove dynamical dichotomy into scattering and blow-up (in a weak sense) for all radial solutions of the Zakharov system in the energy space of four spatial dimensions that have less energy than ## References SHOWING 1-10 OF 40 REFERENCES Scattering for the Zakharov System in 3 Dimensions • Mathematics • 2012 We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the Generalized Strichartz Estimates and Scattering for 3D Zakharov System • Mathematics • 2013 We obtain scattering for the 3D Zakharov system with non-radial small data in the energy space with angular regularity of degree one. The main ingredient is a generalized Strichartz estimate for the Local well-posedness for the Zakharov system on the multidimensional torus The initial value problem of the Zakharov system on the two dimensional torus with general period is shown to be locally well posed in the Sobolev spaces of optimal regularity, including the energy Scattering theory for the Zakharov system • Mathematics • 2005 We study the theory of scattering for the Zakharov system in space dimension 3. We prove in particular the existence of wave operators for that system with no size restriction on the data in larger Global well - posedness and scattering for the focusing, energy - critical nonlinear Schr\"odinger problem in dimension$d = 4$for initial data below a ground state threshold In this paper we prove global well - posedness and scattering for the focusing, energy - critical nonlinear Schr\"odinger initial value problem in four dimensions. Previous work proved this in five On the 2 d Zakharov system with L 2 Schrödinger data • Mathematics • 2009 We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L ×H−1/2 ×H−3/2. This is the space of optimal regularity in the sense that the Small energy scattering for the Klein-Gordon-Zakharov system with radial symmetry • Mathematics • 2012 We prove small energy scattering for the 3D Klein-Gordon-Zakharov system with radial symmetry. The idea of proof is the same as the Zakharov system studied in \cite{GN}, namely to combine the normal Sharp spherically averaged Strichartz estimates for the Schrödinger equation We prove generalized Strichartz estimates with weaker angular integrability for the Schrödinger equation. Our estimates are sharp except some endpoints. Then we apply these new estimates to prove Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions • Mathematics • 2015 We study the Cauchy problem for the Zakharov system in spatial dimension$d\ge 4$with initial datum$(u(0), n(0), \partial_t n(0)) \in H^k(\mathbb{R}^d) \times \dot{H}^l(\mathbb{R}^d)\times
Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations
• Mathematics
• 2010
We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain the full radial Strichartz estimates up to some endpoints for the