Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

Abstract

Abstract. We prove low regularity global well-posedness for the 1d Zakharov system and the 3d Klein-Gordon-Schrödinger system, which are systems in two variables u :The Zakharov system is known to be locally well-posed in (u, n) ∈ L 2 × H −1/2 and the Klein-GordonSchrödinger system is known to be locally well-posed in (u, n) ∈ L 2 ×L 2 . Here, we show that the Zakharov and Klein-Gordon-Schrödinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the L 2 norm of u and controlling the growth of n via the estimates in the local theory.

On the Cauchy problem for the magnetic Zakharov system

“…Furthermore, J. Ginibre, Y. Tsutsumi and G. Velo [9] established local well-posedness theory in lower regularity Sobolev spaces. For more well-posedness results for the Zakharov system (1.3), we refer to [4,5,11,16] and the references therein. However, the system (1.3) ignores the effect of the magnetic filed which is generated in the laser plasma.…”

Well-posedness for the Schroedinger-Improved Boussinesq System and Related Bilinear Estimates

“…In particular, the existence time of the solution is estimated from below by the L 2 -norms of the initial data. We employ the detailed continuation argument given in [8]. The key observation is that independently of the initial data, we can take time length for which the L 2 -norm of the solution for (1.10) becomes double in size.…”

“…Thus, to prove Theorem 1.2, we only use the wave energy, which enables us to prove the case m ¼ À1. More precisely, we apply the idea of [8]. Since the S-imBq system is regularized version of the Zakharov system, the proof of Theorem 1.2 is simpler than that of the Zakharov system.…”

“…In [8], Colliander, Holmer and Tzirakis have proved that the Cauchy problem for the Zakharov system with data (1.7) is globally well-posed when n ¼ 1, s 1 ¼ 0, s 2 ¼ À1=2, while for n ¼ 2, there exists a blow-up solution (see [11]). …”

Untitled

“…In this case it is somehow easier to establish the well-posedness of the system due to the dispersive effects of the solution waves. We cite the following papers [Added and Added 1984;1988, Bejenaru andHerr 2011;Bejenaru et al 2009;Bourgain and Colliander 1996;Colliander et al 2008;Ginibre et al 1997;Kenig et al 1995;Sulem and Sulem 1979] as a historical summary of the results. It is expected that (see, e.g., [Kishimoto 2011]) the optimal regularity range for local well-posedness is on the line s 1 D s 0 1 2 because the two equations in the Zakharov system equally share the loss of derivative.…”