Construction of blow-up solutions for Zakharov system on \( T^{2} \)

Abstract

We consider the Zakharov system in two space dimension with periodic boundary condition:We prove the existence of finite time blow-up solutions of (Z). Further, we show there exists no minimal mass blow-up solution. RésuméNous considérons le système de Zakharov dans l'espace à deux dimensions avec la condition périodique au bord :Nous prouvons l'existence de solutions de (Z) explosant au temps fini. En outre, nous prouvons qu'il n'y a aucune solution explosive de masse minimale.

Local well-posedness for the Zakharov system on the multidimensional torus

“…The global existence of solution and the blow-up problem for the Zakharov system on T 2 will be discussed further in our forthcoming paper [16]. In particular, it will turn out that we do not have to assume n 1 (0) = 0, and that the assumption u 0 L 2 ≪ 1 can be replaced with u 0 L 2 (T 2 ) ≤ Q L 2 (R 2 ) , which is the optimal threshold in the sense that there exists a finitetime blow-up solution starting from an initial datum with u 0 L 2 greater than but arbitrarily close to Q L 2 (R 2 ) .…”

Sharp constants in RiemannianLp-Gagliardo–Nirenberg inequalities

“…Recently, the Riemannian Gagliardo-Nirenberg optimal constants studied in [8] where applied by [22] to obtain global existence theorems for Zakharov system in T 2 . A particularly important family of applications of optimal Gagliardo-Nirenberg inequalities is the transition to optimal Entropy inequalities, in the spirit of [10,15].…”

The Zakharov system in dimension $d \geqslant 4$

“…) is a global (non-dispersive) solution of (1.1). This connection has been used to analyze the blow-up behaviour [14,15,29] in dimension d = 2, and also in the periodic case [27]. Furthermore, we can write the Zakharov energy as…”