Abstract
We introduce the stochastic Hall-magneto-hydrodynamics (Hall-MHD) system in three and two and a half dimensions with infinite-dimensional multiplicative noise, white in time, and prove the global existence of a martingale solution via a stochastic Galerkin approximation and applications of Prokhorov’s, Skorokhod’s and martingale representation theorems, as well as the pressure term through de Rham’s theorem adapted to processes. The Hall term represents mathematically a very singular nonlinear term, unprecedented in the previous work. The results extend many others on the deterministic Hall-MHD and stochastic MHD systems and Navier–Stokes equations. In contrast to the stochastic MHD system, the path-wise uniqueness in the two and a half dimensional case is an open problem.
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Yamazaki, K. Stochastic Hall-Magneto-hydrodynamics System in Three and Two and a Half Dimensions. J Stat Phys 166, 368–397 (2017). https://doi.org/10.1007/s10955-016-1683-9
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DOI: https://doi.org/10.1007/s10955-016-1683-9
Keywords
- Hall-magneto-hydrodynamics system
- Martingale representation theorem
- Prokhorov’s theorem
- Skorokhod’s theorem