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

Local Existence of Strong Solutions to the 3 D Zakharov-Kuznetsov Equation in a Bounded Domain 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

Wang, Chuntian
Applied Mathematics and Optimization , Volume 69 (1) – Feb 1, 2014
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References (21)

Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer Science+Business Media New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-013-9212-6
Publisher site
See Article on Publisher Site

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.

Journal

Applied Mathematics and Optimization – Springer Journals

Published: Feb 1, 2014

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