November  2014, 34(11): 4897-4910. doi: 10.3934/dcds.2014.34.4897

The existence of strong solutions to the $3D$ Zakharov-Kuznestov equation in a bounded domain

1. 

Department of Mathematics and The Institute, for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405-5701, United States

Received  October 2013 Revised  February 2014 Published  May 2014

We consider the Zakharov-Kuznestov (ZK) equation posed in a limited domain $\mathcal{M}=(0,1)_{x}\times(-\pi /2, \pi /2)^d,$ $ d=1,2$ supplemented with suitable boundary conditions. We prove that there exists a solution $u \in \mathcal C ([0, T]; H^1(\mathcal{M})) $ to the initial and boundary value problem for the ZK equation in both dimensions $2$ and $3$ for every $T>0$. To the best of our knowledge, this is the first result of the global existence of strong solutions for the ZK equation in $3D$.
    More importantly, the idea behind the application of anisotropic estimation to cancel the nonlinear term, we believe, is not only suited for this model but can also be applied to other nonlinear equations with similar structures.
    At the same time, the uniqueness of solutions is still open in $2D$ and $3D$ due to the partially hyperbolic feature of the model.
Citation: Chuntian Wang. The existence of strong solutions to the $3D$ Zakharov-Kuznestov equation in a bounded domain. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4897-4910. doi: 10.3934/dcds.2014.34.4897
References:
[1]

E. S. Baykova and A. Faminskii, On initial-boundary-value problems in a strip for the generalized two-dimensional Zakharov-Kuznetsov equation, Adv. Differential Equations, 18 (2013), 663-686.  Google Scholar

[2]

J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A, 302 (1981), 457-510. doi: 10.1098/rsta.1981.0178.  Google Scholar

[3]

J. L. Bona, W. G. Pritchard and L. R. Scott, A comparison of solutions of two model equations for long waves, Fluid Dynamics in Astrophysics and Geophysics (Chicago, Ill., 1981), Amer. Math. Soc., (1983), 235-267.  Google Scholar

[4]

J. L. Bona, S. M. Sun and B. Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436. doi: 10.1081/PDE-120024373.  Google Scholar

[5]

T. Colin and J. M. Ghidaglia, An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval, Adv. Differential Equations, 6 (2001), 1463-1492.  Google Scholar

[6]

T. Colin and M. Gisclon, An initial-boundary-value problem that approximate the quarter-plane problem for the Korteweg-de Vries equation, Nonlinear Anal., 46 (2001), 869-892. doi: 10.1016/S0362-546X(00)00155-3.  Google Scholar

[7]

C. Cao and E. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.  Google Scholar

[8]

G. G. Doronin and N. A. Larkin, Exponential decay for the linear Zakharov-Kuznetsov equation without critical domain restrictions, Appl. Math. Lett., 27 (2014), 6-10. doi: 10.1016/j.aml.2013.08.010.  Google Scholar

[9]

A. V. Faminskii, On the nonlocal well-posedness of a mixed problem for the Zakharov-Kuznetsov equation, Sovrem. Mat. Prilozh., 38 (2006), 135-148. doi: 10.1007/s10958-007-0491-9.  Google Scholar

[10]

A. V. Faminskii, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Differential Equations, (2008), 23pp.  Google Scholar

[11]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.  Google Scholar

[12]

D. Lannes, F. Linares and J. C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, in Progress in Nonlinear Differential Equations and their Applications (eds. M. Cicognani, F. Colombini and D. Del Santo), Birkaüser, 2013, 181-213. doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[13]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[14]

E. W. Laedke and K. H. Spatschek, Growth rates of bending solitons, J. Plasma Phys., 28 (1982), 469-484. doi: 10.1017/S0022377800000428.  Google Scholar

[15]

O. A. Ladyenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.  Google Scholar

[16]

N. A. Larkin and E. Tronco, Regular solutions of the 2D Zakharov-Kuznetsov equation on a half-strip, J. Differential Equations, 254 (2013), 81-101. doi: 10.1016/j.jde.2012.08.023.  Google Scholar

[17]

Z. Qin and R. Temam, Penalty method for the KdV equation, Appl. Anal., 91 (2012), 193-211. doi: 10.1080/00036811.2011.579564.  Google Scholar

[18]

J. C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Differential Equations, 15 (2010), 1001-1031.  Google Scholar

[19]

J. C. Saut, R. Temam and C. Wang, An initial and boundary-value problem for the Zakharov-Kuznestov equation in a bounded domain, J. Math. Phys., 53 (2012), 115612, 29pp. doi: 10.1063/1.4752102.  Google Scholar

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[21]

C. Wang, Local existence of strong solutions to the 3D Zakharov-Kuznetsov equation in a bounded domain, Appl. Math. Optim., 69 (2014), 1-19, arXiv:1307.6827. doi: 10.1007/s00245-013-9212-6.  Google Scholar

[22]

V. E. Zakharov and E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP, 30 (1974), 285-286. Google Scholar

show all references

References:
[1]

E. S. Baykova and A. Faminskii, On initial-boundary-value problems in a strip for the generalized two-dimensional Zakharov-Kuznetsov equation, Adv. Differential Equations, 18 (2013), 663-686.  Google Scholar

[2]

J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A, 302 (1981), 457-510. doi: 10.1098/rsta.1981.0178.  Google Scholar

[3]

J. L. Bona, W. G. Pritchard and L. R. Scott, A comparison of solutions of two model equations for long waves, Fluid Dynamics in Astrophysics and Geophysics (Chicago, Ill., 1981), Amer. Math. Soc., (1983), 235-267.  Google Scholar

[4]

J. L. Bona, S. M. Sun and B. Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436. doi: 10.1081/PDE-120024373.  Google Scholar

[5]

T. Colin and J. M. Ghidaglia, An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval, Adv. Differential Equations, 6 (2001), 1463-1492.  Google Scholar

[6]

T. Colin and M. Gisclon, An initial-boundary-value problem that approximate the quarter-plane problem for the Korteweg-de Vries equation, Nonlinear Anal., 46 (2001), 869-892. doi: 10.1016/S0362-546X(00)00155-3.  Google Scholar

[7]

C. Cao and E. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.  Google Scholar

[8]

G. G. Doronin and N. A. Larkin, Exponential decay for the linear Zakharov-Kuznetsov equation without critical domain restrictions, Appl. Math. Lett., 27 (2014), 6-10. doi: 10.1016/j.aml.2013.08.010.  Google Scholar

[9]

A. V. Faminskii, On the nonlocal well-posedness of a mixed problem for the Zakharov-Kuznetsov equation, Sovrem. Mat. Prilozh., 38 (2006), 135-148. doi: 10.1007/s10958-007-0491-9.  Google Scholar

[10]

A. V. Faminskii, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Differential Equations, (2008), 23pp.  Google Scholar

[11]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.  Google Scholar

[12]

D. Lannes, F. Linares and J. C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, in Progress in Nonlinear Differential Equations and their Applications (eds. M. Cicognani, F. Colombini and D. Del Santo), Birkaüser, 2013, 181-213. doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[13]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[14]

E. W. Laedke and K. H. Spatschek, Growth rates of bending solitons, J. Plasma Phys., 28 (1982), 469-484. doi: 10.1017/S0022377800000428.  Google Scholar

[15]

O. A. Ladyenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.  Google Scholar

[16]

N. A. Larkin and E. Tronco, Regular solutions of the 2D Zakharov-Kuznetsov equation on a half-strip, J. Differential Equations, 254 (2013), 81-101. doi: 10.1016/j.jde.2012.08.023.  Google Scholar

[17]

Z. Qin and R. Temam, Penalty method for the KdV equation, Appl. Anal., 91 (2012), 193-211. doi: 10.1080/00036811.2011.579564.  Google Scholar

[18]

J. C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Differential Equations, 15 (2010), 1001-1031.  Google Scholar

[19]

J. C. Saut, R. Temam and C. Wang, An initial and boundary-value problem for the Zakharov-Kuznestov equation in a bounded domain, J. Math. Phys., 53 (2012), 115612, 29pp. doi: 10.1063/1.4752102.  Google Scholar

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[21]

C. Wang, Local existence of strong solutions to the 3D Zakharov-Kuznetsov equation in a bounded domain, Appl. Math. Optim., 69 (2014), 1-19, arXiv:1307.6827. doi: 10.1007/s00245-013-9212-6.  Google Scholar

[22]

V. E. Zakharov and E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP, 30 (1974), 285-286. Google Scholar

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