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PreprintCC BY 4.0arXiv

On the existence of a weak martingale solution for a stochastic magnetohydrodynamics system with noise acting in the magnetic field

  • Jiří Púček
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Abstract

We prove the existence of solutions to the stochastic magnetohydrodynamics (MHD) system, where randomness is introduced through random initial data and a stochastic integral appearing solely in the induction equation, while the fluid equations remain deterministic. We focus on the case where the adiabatic exponent γ satisfies γ> 3/2. The existence proof is carried out using the penalization method. We define the notion of a martingale solution and establish sufficient conditions for its existence. The proof then proceeds by means of the stochastic compactness method. Using an energy inequality, we derive a priori estimates in terms of expectation. Due to the stochastic nature of the problem, we demonstrate convergence in law of the penalized solutions and verify that the limiting object is a martingale solution.

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Cite

BibTeXRIS
BibTeX
@misc{pucek2026existence,
  title = {{On the existence of a weak martingale solution for a stochastic magnetohydrodynamics system with noise acting in the magnetic field}},
  author = {Jiří Púček},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29747},
  eprint = {2609.29747},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29747},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the existence of a weak martingale solution for a stochastic magnetohydrodynamics system with noise acting in the magnetic field
AU  - Jiří Púček
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29747
UR  - https://arxiv.org/abs/2609.29747
AB  - We prove the existence of solutions to the stochastic magnetohydrodynamics (MHD) system, where randomness is introduced through random initial data and a stochastic integral appearing solely in the induction equation, while the fluid equations remain deterministic. We focus on the case where the adiabatic exponent γ satisfies γ> 3/2. The existence proof is carried out using the penalization method. We define the notion of a martingale solution and establish sufficient conditions for its existence. The proof then proceeds by means of the stochastic compactness method. Using an energy inequality, we derive a priori estimates in terms of expectation. Due to the stochastic nature of the problem, we demonstrate convergence in law of the penalized solutions and verify that the limiting object is a martingale solution.
ER  -