# American Institute of Mathematical Sciences

June  2014, 19(4): 1047-1085. doi: 10.3934/dcdsb.2014.19.1047

## Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise

 1 Department of Mathematics, Virginia Polytechnic and State University, Blacksburg, VA 24061, United States 2 Department of Mathematics and The Institute, for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, United States, United States

Received  July 2013 Revised  January 2014 Published  April 2014

We study in this article the stochastic Zakharov-Kuznetsov equation driven by a multiplicative noise. We establish, in space dimensions two and three the global existence of martingale solutions, and in space dimension two the global pathwise uniqueness and the existence of pathwise solutions. New methods are employed to deal with a special type of boundary conditions and to verify the pathwise uniqueness of martingale solutions with a lack of regularity, where both difficulties arise due to the partly hyperbolic feature of the model.
Citation: Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047
##### 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, 8 (2013), 663-686. [2] A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304. doi: 10.1007/BF00996149. [3] P. Billingsley, Probability and Measure, $2^{nd}$ edition, John Wiley & Sons Inc., New York, 1986. [4] 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. [5] J. L. Bona, W. G. Pritchard and L. R. Scott, A comparison of solutions of two model equations for long waves, In Fluid Dynamics in Astrophysics and Geophysics (Chicago, Ill., 1981), Amer. Math. Soc., 1983, 235-267. [6] A. de Bouard and A. Debussche, On a stochastic Korteweg-de Vries equation with homogeneous noise, in Séminaire: Équations aux Dérivées Partielles. 2007-2008, École Polytech., Exp. No. V, 2009, 15pp. [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. [8] A. Debussche, N. Glatt-Holtz and and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011),1123-1144. doi: 10.1016/j.physd.2011.03.009. [9] A. Debussche and J. Printems, Effect of a localized random forcing term on the Korteweg-de Vries equation, J. Comput. Anal. Appl., 3 (2001), 183-206. doi: 10.1023/A:1011596026830. [10] 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. [11] A. V. Faminskii, On the nonlocal well-posedness of a mixed problem for the Zakharov-Kuznetsov equation, Sovrem. Mat. Prilozh., (2006), 135-148; Translation in J. Math. Sci. (N. Y.), 147 (2007), 6524-6537. doi: 10.1007/s10958-007-0491-9. [12] A. V. Faminskii, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Differential Equations, (2008), 23pp. [13] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467. [14] F. Flandoli, An introduction to 3D stochastic fluid dynamics, in SPDE in Hydrodynamic: Recent Progress and Prospects, Springer Berlin/Heidelberg, (2008), 51-150. doi: 10.1007/978-3-540-78493-7_2. [15] W. Gao and J. Bao, Exact solutions for a $(2+1)$-dimensional stochastic KdV equation, J. Jilin Univ. Sci., 44 (2006), 46-49. [16] N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600. [17] I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833. [18] R. Herman and A. Rose, Numerical realizations of solutions of the stochastic KdV equation, Math. Comput. Simulation, 80 (2009), 164-172. doi: 10.1016/j.matcom.2009.06.008. [19] A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Teor. Veroyatnost. i Primenen., 42 (1997), 209-216. doi: 10.4213/tvp1769. [20] 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, FL Colombini and D. Del Santo), Birkaüser, (2013), 183-215. arXiv:1205.5080. doi: 10.1007/978-1-4614-6348-1_10. [21] E. W. Laedke and K. H. Spatschek, Growth rates of bending solitons, J. Plasma Phys., 26 (1982), 469-484. doi: 10.1017/S0022377800000428. [22] 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. [23] Q. Liu, A modified Jacobi elliptic function expansion method and its application to Wick-type stochastic KdV equation, Chaos Solitons Fractals, 32 (2007), 1215-1223. doi: 10.1016/j.chaos.2005.11.043. [24] R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310. doi: 10.1137/S0036141002409167. [25] M. Ondreját, Stochastic nonlinear wave equations in local Sobolev spaces, Electron. J. Probab., 15 (2010), 1041-1091. doi: 10.1214/EJP.v15-789. [26] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, Berlin, 2007. [27] J. C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Differential Equations, 15 (2010), 1001-1031. [28] 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, 29. doi: 10.1063/1.4752102. [29] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, $2^{nd}$ edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050. [30] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167. [31] V. E. Zakharov and E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP, 30 (1974), 285-286. [32] S. Zhang and H. Q. Zhang, Fan sub-equation method for Wick-type stochastic partial differential equations, Phys. Lett. A, 374 (2010), 4180-4187. doi: 10.1016/j.physleta.2010.08.023.

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##### 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, 8 (2013), 663-686. [2] A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304. doi: 10.1007/BF00996149. [3] P. Billingsley, Probability and Measure, $2^{nd}$ edition, John Wiley & Sons Inc., New York, 1986. [4] 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. [5] J. L. Bona, W. G. Pritchard and L. R. Scott, A comparison of solutions of two model equations for long waves, In Fluid Dynamics in Astrophysics and Geophysics (Chicago, Ill., 1981), Amer. Math. Soc., 1983, 235-267. [6] A. de Bouard and A. Debussche, On a stochastic Korteweg-de Vries equation with homogeneous noise, in Séminaire: Équations aux Dérivées Partielles. 2007-2008, École Polytech., Exp. No. V, 2009, 15pp. [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. [8] A. Debussche, N. Glatt-Holtz and and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011),1123-1144. doi: 10.1016/j.physd.2011.03.009. [9] A. Debussche and J. Printems, Effect of a localized random forcing term on the Korteweg-de Vries equation, J. Comput. Anal. Appl., 3 (2001), 183-206. doi: 10.1023/A:1011596026830. [10] 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. [11] A. V. Faminskii, On the nonlocal well-posedness of a mixed problem for the Zakharov-Kuznetsov equation, Sovrem. Mat. Prilozh., (2006), 135-148; Translation in J. Math. Sci. (N. Y.), 147 (2007), 6524-6537. doi: 10.1007/s10958-007-0491-9. [12] A. V. Faminskii, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Differential Equations, (2008), 23pp. [13] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467. [14] F. Flandoli, An introduction to 3D stochastic fluid dynamics, in SPDE in Hydrodynamic: Recent Progress and Prospects, Springer Berlin/Heidelberg, (2008), 51-150. doi: 10.1007/978-3-540-78493-7_2. [15] W. Gao and J. Bao, Exact solutions for a $(2+1)$-dimensional stochastic KdV equation, J. Jilin Univ. Sci., 44 (2006), 46-49. [16] N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600. [17] I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833. [18] R. Herman and A. Rose, Numerical realizations of solutions of the stochastic KdV equation, Math. Comput. Simulation, 80 (2009), 164-172. doi: 10.1016/j.matcom.2009.06.008. [19] A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Teor. Veroyatnost. i Primenen., 42 (1997), 209-216. doi: 10.4213/tvp1769. [20] 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, FL Colombini and D. Del Santo), Birkaüser, (2013), 183-215. arXiv:1205.5080. doi: 10.1007/978-1-4614-6348-1_10. [21] E. W. Laedke and K. H. Spatschek, Growth rates of bending solitons, J. Plasma Phys., 26 (1982), 469-484. doi: 10.1017/S0022377800000428. [22] 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. [23] Q. Liu, A modified Jacobi elliptic function expansion method and its application to Wick-type stochastic KdV equation, Chaos Solitons Fractals, 32 (2007), 1215-1223. doi: 10.1016/j.chaos.2005.11.043. [24] R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310. doi: 10.1137/S0036141002409167. [25] M. Ondreját, Stochastic nonlinear wave equations in local Sobolev spaces, Electron. J. Probab., 15 (2010), 1041-1091. doi: 10.1214/EJP.v15-789. [26] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, Berlin, 2007. [27] J. C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Differential Equations, 15 (2010), 1001-1031. [28] 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, 29. doi: 10.1063/1.4752102. [29] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, $2^{nd}$ edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050. [30] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167. [31] V. E. Zakharov and E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP, 30 (1974), 285-286. [32] S. Zhang and H. Q. Zhang, Fan sub-equation method for Wick-type stochastic partial differential equations, Phys. Lett. A, 374 (2010), 4180-4187. doi: 10.1016/j.physleta.2010.08.023.
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