Local Existence of Strong Solutions to the 3D Zakharov-Kuznetsov Equation in a Bounded Domain

Abstract

We consider here the local existence of strong solutions for the Zakharov-Kuznetsov (ZK) equation posed in a limited domain \(\mathcal{M}=(0,1)_{x}\times(-\pi/2, \pi/2)^{d}\), d=1,2. We prove that in space dimensions 2 and 3, there exists a strong solution on a short time interval, whose length only depends on the given data. We use the parabolic regularization of the ZK equation as in Saut et al. (J. Math. Phys. 53(11):115612, 2012) to derive the global and local bounds independent of ϵ for various norms of the solution. In particular, we derive the local bound of the nonlinear term by a singular perturbation argument. Then we can pass to the limit and hence deduce the local existence of strong solutions.