论文标题
$ l^2- $批判性广义Zakharov-kuznetsov方程的Cauchy问题3
The Cauchy problem for the $L^2-$critical generalized Zakharov-Kuznetsov equation in dimension 3
论文作者
论文摘要
我们证明了$ l^2 $批判性广义Zakharov-kuznetsov方程的本地良好性，（3/4,1）。$我们还证明了该方程的初始数据$ _0 $ u_0 \ in H^s，\ s \ s \ s \ in [1,2），$ u_0 $ u_0 $ u_0 $ u_0几乎是“拟定的”。 $ c（[0，t]：h^s（\ mathbb {r}^3））\ cap x^s_t $，并且在该类中是唯一的，我们可以在唯一更大的空间中确保数据对解决方案的连续性。 We also prove that solutions satisfy the expected conservation of $L^2-$mass for the whole $s \in (3/4,2)$ range, and energy for $s \in (1,2).$ By a limiting argument, this implies, in particular, global existence for small initial data in $H^1.$ Finally, we study the question of almost everywhere (a.e.) convergence of solutions of the initial value problem to initial data.
We prove local well-posedness for the $L^2$ critical generalized Zakharov-Kuznetsov equation in $H^s, \, s \in (3/4,1).$ We also prove that the equation is "almost well-posedness" for initial data $u_0 \in H^s, \, s \in [1,2),$ in the sense that the solution belongs to a certain intersection $C([0,T] : H^s(\mathbb{R}^3)) \cap X^s_T$ and is unique within that class, where we can ensure continuity of the data-to-solution map in an only slightly larger space. We also prove that solutions satisfy the expected conservation of $L^2-$mass for the whole $s \in (3/4,2)$ range, and energy for $s \in (1,2).$ By a limiting argument, this implies, in particular, global existence for small initial data in $H^1.$ Finally, we study the question of almost everywhere (a.e.) convergence of solutions of the initial value problem to initial data.
