## Differential and Integral Equations

### Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions

#### Abstract

We study the Cauchy problem for the Zakharov system in spatial dimension $d\ge 4$ with initial datum $(u(0), n(0), \partial_t n(0) )\in H^k(\mathbb R^d)\times \dot{H}^l(\mathbb R^d)\times \dot{H}^{l-1}(\mathbb R^d)$. According to Ginibre, Tsutsumi and Velo ([9]), the critical exponent of $(k,l)$ is $((d-3)/2,(d-4)/2 ).$ We prove the small data global well-posedness and the scattering at the critical space. It seems difficult to get the crucial bilinear estimate only by applying the $U^2,\ V^2$ type spaces introduced by Koch and Tataru ([23], [24]). To avoid the difficulty, we use an intersection space of $V^2$ type space and the space-time Lebesgue space $E:=L^2_tL_x^{2d/(d-2)}$, which is related to the endpoint Strichartz estimate.

#### Article information

Source
Differential Integral Equations, Volume 30, Number 9/10 (2017), 763-794.

Dates
First available in Project Euclid: 27 May 2017

https://projecteuclid.org/euclid.die/1495850426

Mathematical Reviews number (MathSciNet)
MR3656486

Zentralblatt MATH identifier
06770141

#### Citation

Kato, Isao; Tsugawa, Kotaro. Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions. Differential Integral Equations 30 (2017), no. 9/10, 763--794. https://projecteuclid.org/euclid.die/1495850426