Skip to main content
SpringerLink
Log in

Well-posedness of Hall-magnetohydrodynamics system forced by L\(\acute{\mathrm{e}}\)vy noise

  • Published:
Stochastics and Partial Differential Equations: Analysis and Computations Aims and scope Submit manuscript

Abstract

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for the Hall-magnetohydrodynamics system that is inviscid, resistive, and forced by multiplicative L\(\acute{\mathrm {e}}\)vy noise in the three dimensional space. Moreover, when the initial data is sufficiently small, we prove that the solution exists globally in time in probability; that is, the probability of the global existence of a unique smooth solution may be arbitrarily close to one given the initial data of which its expectation in a certain Sobolev norm is sufficiently small. The proofs go through for the two and a half dimensional case as well. To the best of the authors’ knowledge, an analogous result is absent in the deterministic case due to the lack of viscous diffusion, exhibiting the regularizing property of the noise. Our result may also be considered as a physically meaningful special case of the extension of work of Kim (J Differ Equ 250:1650–1684, 2011) and Mohan and Sritharan (Pure Appl Funct Anal 3:137–178, 2018) from the first-order quasilinear to the second-order quasilinear system because the Hall term elevates the Hall-magnetohydrodynamics system to the quasilinear class, in contrast to the Naiver–Stokes equations that has most often been studied and is semilinear.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Applebaum, D.: L\(\acute{\rm e}\)vy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, New York (2009)

    Book  Google Scholar 

  2. Acheritogaray, M., Degond, P., Frouvelle, A., Liu, J.-G.: Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinet. Relat. Models 4, 901–918 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barbu, V., Da Prato, G.: Existence and ergodicity for the two-dimensional stochastic magneto-hydrodynamics equations. Appl. Math. Optim. 56, 145–168 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Batchelor, G.K.: On the spontaneous magnetic field in a conducting liquid in turbulent motion. Proc. R. Soc. Lond. Ser. A 201, 405–416 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bensoussan, A., Temam, R.: Equations stochastiques du type Navier–Stokes. J. Funct. Anal. 13, 195–222 (1973)

    Article  MATH  Google Scholar 

  6. Campos, L.M.B.C.: On hydromagnetic waves in atmospheres with application to the sun. Theor. Comput. Fluid Dyn. 10, 37–70 (1998)

    Article  MATH  Google Scholar 

  7. Cao, C., Wu, J., Yuan, B.: The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion. SIAM J. Math. Anal. 46, 588–602 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Capiński, M., Peszat, S.: Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations. Nonlinear Differ. Equ. Appl. 4, 185–200 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chae, D., Degond, P., Liu, J.-G.: Wel-posedness for Hall-magneto hydrodynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 555–565 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chae, D., Lee, J.: On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics. J. Differ. Equ. 256, 3835–3858 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chae, D., Schonbek, M.: On the temporal decay for the Hall-magnetohydrodynamic equations. J. Differ. Equ. 255, 3971–3982 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chae, D., Wan, R., Wu, J.: Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion. J. Math. Fluid Mech. 17, 627–638 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chae, D., Weng, S.: Singularity formation for the in compressible Hall-MHD equations without resistivity. Ann. Inst. Henri Poincaré Anal. Non Linéaire 33, 1009–1022 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chae, D., Wolf, J.: On partial regularity for the 3D nonstationary Hall magnetohydrodynamics equations on the plane. SIAM J. Math. Anal. 48, 443–469 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chandrasekhar, S.: The invariant theory of isotropic turbulence in magneto-hydrodynamics. Proc. R. Soc. Lond. Ser. A 204, 435–449 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chueshov, I., Millet, A.: Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl. Math. Optim. 61, 379–420 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Donato, S., Servidio, S., Dmitruk, P., Carbone, V., Shay, M.A., Cassak, P.A., Matthaeus, W.H.: Reconnection events in two-dimensional Hall magnetohydrodynamic turbulence. Phys. Plasmas 19, 092307 (2012)

    Article  Google Scholar 

  18. Eyink, G.L.: Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models. J. Math. Phys. 50, 083102 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fan, J., Malaikah, H., Monaquel, S., Nakamura, G., Zhou, Y.: Global Cauchy problem of 2D generalized MHD equations. Monatsch. Math. 175, 127–131 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fefferman, C.L., McCormick, D.S., Robinson, J.C., Rodrigo, J.L.: Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. J. Funct. Anal. 267, 1035–1056 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hausenblas, E.: Burkholder–Davis–Gundy typeinequalities of the Itô stochastic integral withrespect to Lévy noise on Banach spaces. arXiv:0902.2114v3 [math.PR]

  22. Homann, H., Grauer, R.: Bifurcation analysis of magnetic reconnection in Hall-MHD-systems. Phys. D 208, 59–72 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jiu, Q., Zhao, J.: Global regularity of 2D generalized MHD equations with magnetic diffusion. Z. Angew. Math. Phys. 66, 677–687 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jiu, Q., Zhao, J.: A remark on global regularity of 2D generalized magnetohydrodynamic equations. J. Math. Anal. Appl. 412, 478–484 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)

    MATH  Google Scholar 

  26. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kim, J.U.: On the stochastic quasi-linear symmetric hyperbolic system. J. Differ. Equ. 250, 1650–1684 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Leray, J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lighthill, M.J.: Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Philos. Trans. R. Soc. Lond. Ser. A 252, 397–430 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mahajan, S.M., Krishan, V.: Exact solution of the incompressible Hall magnetohydrodynamics. Mon. Not. R. Astron. Soc. 359, L27–L29 (2005)

    Article  Google Scholar 

  31. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  32. Mandrekar, V., Rüdiger, B.: Existence and uniqueness of path wise solutions for stochasticintegral equations driven by Lévy noise on separableBanach spaces. Stochastics 78, 189–212 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Manna, U., Mohan, M.T.: Two-dimensional magneto-hydrodynamic system with jump processes: well posedness and invariant measures. Commun. Stoch. Anal. 7, 153–178 (2013)

    MathSciNet  Google Scholar 

  34. Manna, U., Mohan, M.T., Sritharan, S.S.: Stochastic non-resistive magnetohydrodynamic system with L\(\acute{\rm e}\)vy noise. Random Oper. Stoch. Equ. 25, 155–194 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Miura, H., Hori, D.: Hall effects on local structures in decaying MHD turbulence. J. Plasma Fusion Res. Ser. 8, 73–76 (2009)

    Google Scholar 

  36. Mohan, M.T., Sritharan, S.S.: Stochastic Euler equations of fluid dynamics with L\(\acute{\rm e}\)vy noise. Asymptot. Anal. 99, 67–103 (2016)

    Article  MathSciNet  Google Scholar 

  37. Mohan, M.T., Sritharan, S.S.: Stochastic quasilinear symmetric hyperbolic system perturbed by L\(\acute{\rm e}\)vy noise. Pure Appl. Funct. Anal. 3, 137–178 (2018)

    MathSciNet  Google Scholar 

  38. Mohan, M.T., Sritharan, S.S.: Stochastic Navier–Stokes equation perturbed by L\(\acute{\rm e}\)vy noise with hereditary viscosity (submitted)

  39. Rüdiger, B., Ziglio, G.: Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces. Stochastics 78, 377–410 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sango, M.: Magnetohydrodynamic turbulent flows: existence results. Phys. D. 239, 912–923 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  42. Skorokhod, A.V.: Studies in the Theory of Random Processes. Dover Publications Inc, New York (1965)

    MATH  Google Scholar 

  43. Sritharan, S.S., Sundar, P.: The stochastic magneto-hydrodynamic system. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2, 241–265 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sundar, P.: Stochastic magneto-hydrodynamic system perturbed by general noise. Commun. Stoch. Anal. 8, 413–437 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Tran, C.V., Yu, X., Zhai, Z.: On global regularity of 2D generalized magnetohydrodynamics equations. J. Differ. Equ. 254(10), 4194–4216 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wardle, M.: Star formation and the Hall effect. Astrophys. Space Sci. 292, 317–323 (2004)

    Article  Google Scholar 

  47. Yamazaki, K.: Remarks on the global regularity of two-dimensional magnetohydrodynamics system with zero dissipation. Nonlinear Anal. 94, 194–205 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Yamazaki, K.: On the global regularity of two-dimensional generalized magnetohydrodynamics system. J. Math. Anal. Appl. 416, 99–111 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. Yamazaki, K.: Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system. Adv. Differ. Equ. 21, 1085–1116 (2016)

    MathSciNet  MATH  Google Scholar 

  50. Yamazaki, K.: Global martingale solution for the stochastic Boussinesq system with zero dissipation. Stoch. Anal. Appl. 34, 404–426 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. Yamazaki, K.: Stochastic Hall-magneto-hydrodynamics system in three and two and a half dimensions. J. Stat. Phys. 166, 368–397 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  52. Yamazaki, K.: Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete Contin. Dyn. Syst. Ser. B 23, 913–938 (2018)

    MathSciNet  MATH  Google Scholar 

  53. Ye, Z., Xu, X.: Global regularity of two-dimensional incompressible generalized magnetohydrodynamics system. Nonlinear Anal. 100, 86–96 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yuan, B., Bai, L.: Remarks on global regularity of 2D generalized MHD equations. J. Math. Anal. Appl. 413, 633–640 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We express our deep gratitude to the anonymous referees for their comments which improved this manuscript significantly, in particular for pointing out an issue in our initial proof of Proposition 4.4. The second author also expresses deep gratitude to Prof. Jiahong Wu for valuable discussions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

    Kazuo Yamazaki

  2. Department of Mathematics, Indian Institute of Technology Roorkee, Haridwar Highway, Roorkee, Uttarakhand, 247667, India

    Manil T. Mohan

Authors
  1. Kazuo Yamazaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manil T. Mohan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuo Yamazaki.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yamazaki, K., Mohan, M.T. Well-posedness of Hall-magnetohydrodynamics system forced by L\(\acute{\mathrm{e}}\)vy noise. Stoch PDE: Anal Comp 7, 331–378 (2019). https://doi.org/10.1007/s40072-018-0129-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40072-018-0129-6

Keywords

Mathematics Subject Classification

Search

Navigation