Initial-boundary value problems for the three-dimensional Zakharov-Kuznetsov equation

A. V. Faminskii

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 51, Tome 536 (2024), pp. 336-378

Résumé

Faminskii A. V. Initial-boundary value problems for the three-dimensional Zakharov–Kuznetsov equation. Initial-boundary value problems are considered for the Zakharov–Kuznetsov equation $u_t + b u_x + \Delta u_x + uu_x = f$ in the case of three spatial variables $(x,y,z)$ posed on a domain $\mathbb R_+ \times\Omega$, where $\Omega$ — is a bounded domain with respect to $(y,z)$ with sufficiently smooth boundary. For $t>0$ on the left boundary $x=0$ the non-homogeneous Dirichlet boundary condition is set, while on $\partial\Omega$ — homogeneous either Dirichlet or Neumann condition. Results on existence of global in time weak and strong solutions, as well as on uniqueness of strong solutions are established. An initial function is assumed to belong to weighted (at $+\infty$) spaces $L_2$ in the case of weak solutions and $H^1$ in the case of strong solutions. Both power and exponential weights are allowed. In the case of Dirichlet boundary condition large-time decay of small solutions is also obtained.