Title: On the near soliton dynamics for the 2D cubic Zakharov-Kuznetsov equations Show Chinese title

Title: 关于二维三次Zakharov-Kuznetsov方程的孤子动力学近似

Abstract: In this article, we consider the Cauchy problem for the cubic (mass-critical) Zakharov-Kuznetsov equations in dimension two: $$\partial_t u+\partial_{x_1}(\Delta u+u^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^{2}.$$ For initial data in $H^1$ close to the soliton with a suitable space-decay property, we fully describe the asymptotic behavior of the corresponding solution. More precisely, for such initial data, we show that only three possible behaviors can occur: 1) The solution leaves a tube near soliton in finite time; 2) the solution blows up in finite time; 3) the solution is global and locally converges to a soliton. In addition, we show that for initial data near a soliton with non-positive energy and above the threshold mass, the corresponding solution will blow up as described in Case 2. Our proof is inspired by the techniques developed for mass-critical generalized Korteweg-de Vries equation (gKdV) equation in a similar context by Martel-Merle-Rapha\"el. More precisely, our proof relies on refined modulation estimates and a modified energy-virial Lyapunov functional. The primary challenge in our problem is the lack of coercivity of the Schr\"odinger operator which appears in the virial-type estimate. To overcome the difficulty, we apply a transform, which was first introduced in Kenig-Martel [13], to perform the virial computations after converting the original problem to the adjoint one. Th coercivity of the Schr\"odinger operator in the adjoint problem has been numerically verified by Farah-Holmer-Roudenko-Yang [9]. Show Chinese abstract

Abstract: 本文中，我们研究二维空间中质量临界三次Zakharov-Kuznetsov方程的Cauchy问题：$$\partial_t u+\partial_{x_1}(\Delta u+u^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^{2}.$$对于初值数据在$H^1$附近且具有适当的衰减性质的孤立子，我们完全描述了相应解的渐近行为。 更具体地说，对于这样的初值数据，我们证明只有三种可能的行为可以发生：1) 解在有限时间内离开孤立子附近的管状区域；2) 解在有限时间内爆破；3) 解是整体存在的，并且局部收敛到一个孤立子。 此外，我们证明了对于初值数据接近一个孤立子且能量非正且超过阈值质量的情况，相应的解将按照情形2中描述的方式爆破。 我们的证明受到Martel-Merle-Raphaël在类似背景下为质量临界广义Korteweg-de Vries（gKdV）方程所开发技术的启发。 更确切地说，我们的证明依赖于精细的调制估计和修正的能量-动量李雅普诺夫泛函。 我们问题的主要挑战在于动量型估计中出现的薛定谔算子缺乏强制性。 为了克服这个困难，我们应用了一种变换，该变换最初由Kenig-Martel [13]引入，在将原始问题转换为伴随问题后执行动量计算。 伴随问题中薛定谔算子的强制性已经被Farah-Holmer-Roudenko-Yang [9]通过数值验证。