Uniqueness results for Zakharov-Kuznetsov equation

Abstract

In this paper we study uniqueness properties of solutions to the Zakharov-Kuznetsov equation of plasma physic.Given two sufficiently regular solutions u 1 , u 2 , we prove that, if u 1´u2 decays fast enough at two distinct times, then u 1 " u 2 .The equation was introduced in the context of plasma physic by Zakharov and Kuznetsov in [38], where they formally deduced that the propagation of nonlinear ion-acoustic waves in magnetized plasma is governed by this mathematical model. A rigorous derivation of equation (1) was given by Lannes, Linares and Saut in [29].The problem of local and global well-posedness for the Cauchy problem associated to (1) has extensively been studied. Up to date the best local well-posedness result available in the literature was obtained independently by Molinet and Pilod [33] and Grünrock and Herr [16] for initial data in H s pR 2 q, s ą 1 2 . Then the global theory follows by standard arguments based on L 2 and H 1 conservation laws. We refer to [14,[30][31][32] and references therein for other results of this type and several additional remarks concerning with properties of this equation.Our main goal is to prove uniqueness properties from two distinct times for equation (1). More precisely we want to deduce sufficient conditions on the behavior of the difference u 1´u2 of two solutions u 1 , u 2 of (1) at two different times, t 0 " 0 and t 1 " 1, which guarantee that u 1 " u 2 . This kind of results is inspired to the program performed in [7][8][9][10][11][12] for Schrödinger and KdV (see also [37] and Remark 1.1 below for further details).The main motivation for our study is a recent work by Bustamante, Isaza and Mejía [4] where an upper bound for the possible decay at two different times of a non-trivial difference of two solutions of (1) was given. More precisely they prove the following: 1 ( [4] ). Suppose that for some small ε ą 0 u 1 , u 2 P C`r0, 1s; H 4 pR 2 q X L 2 pp1`x 2`y2 q 4 3`ε dxdyq˘X C 1`r 0, 1s; L 2 pR 2 q˘, are solutions of (1). Then there exists a universal constant a 0 ą 0, such that if for some a ą a 0 u 1 p0q´u 2 p0q, u 1 p1q´u 2 p1q P L 2 pe apx 2`y2 q 3{4 dxdyq, then u 1 " u 2 .

On the unique continuation property of solutions to the two-dimensional Zakharov-Kuznetsov equation

“…3/2. This claim found support and confirmation in [4] where the following unique continuation principle was obtained for the symmetric ZK equation (4).…”

“…In this work we are mainly interested in the unique continuation setting of problems linked to the equation. At this regard we should mention the recent work [4]. The main achievement in [4] has been to obtain sharp sufficient conditions on the (spatial) decay of the difference of two solutions of (1) at two different times which guarantee that both solutions coincide.…”

“…At this regard we should mention the recent work [4]. The main achievement in [4] has been to obtain sharp sufficient conditions on the (spatial) decay of the difference of two solutions of (1) at two different times which guarantee that both solutions coincide. More precisely in [4] the following theorem is proved.…”

“…Nevertheless one realizes that this choice does not work as well as (5). In order to see that, first we stress that, even if Theorem 2 is of its independent interest, the original purpose in [4] was to pass through the proof of Theorem 2 only as an intermediate step to get Theorem 1. In other words, the final goal in [4] was to deduce Theorem 1 from Theorem 2 once one gets back to the original variables.…”

On uniqueness results for the Benjamin equation

“…Also in this direction, in [2], for the Ostrovsky equation and in [3], for the Zakharov-Kuznetsov (ZK) equation, Bustamante et al, showed that if the difference of two sufficiently regular solutions u 1 and u 2 have a suitable decay in two different times, then they are equal. Recently, the results in [3] was improved in [6].…”