# American Institute of Mathematical Sciences

July  2018, 17(4): 1561-1572. doi: 10.3934/cpaa.2018074

## On special regularity properties of solutions of the Zakharov-Kuznetsov equation

 1 IMPA, Estrada Dona Castorina 110, Rio de Janeiro 22460-320, Brazil 2 Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

Received  December 2016 Revised  April 2017 Published  April 2018

We study special regularity properties of solutions to the initial value problem associated to the Zakharov-Kuznetsov equation in three dimensions. We show that the initial regularity of the data in a family of half-spaces propagates with infinite speed. By dealing with the finite envelope of a class of these half-spaces we extend the result to the complement of a family of cones in $\mathbb{R}^3$.

Citation: Felipe Linares, Gustavo Ponce. On special regularity properties of solutions of the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1561-1572. doi: 10.3934/cpaa.2018074
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##### References:
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