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The non-relativistic limit for the e-MHD equations.

Hongli Wang1, Jie Zhao2

  • 1College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China.

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This study proves unique solutions for electron Magnetohydrodynamics (e-MHD) equations in a 3D torus. The findings rigorously justify the non-relativistic limit to incompressible Euler equations under specific conditions.

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Area of Science:

  • Plasma Physics
  • Fluid Dynamics
  • Mathematical Analysis

Background:

  • Electron Magnetohydrodynamics (e-MHD) describes plasma behavior.
  • Understanding the non-relativistic limit is crucial for simplifying complex plasma models.
  • The incompressible Euler equations are a fundamental model in fluid dynamics.

Purpose of the Study:

  • To investigate the non-relativistic limit of 3D e-MHD equations.
  • To establish the existence and uniqueness of solutions for small parameter problems.
  • To rigorously justify the convergence to the incompressible Euler equations.

Main Methods:

  • Analysis of e-MHD equations in a three-dimension unit periodic torus.
  • Utilizing prepared initial data for the investigation.
  • Employing mathematical techniques to analyze the limit behavior.

Main Results:

  • Demonstrated unique solutions for the small parameter e-MHD problems within finite time intervals.
  • Showed that these solutions correspond to smooth solutions of the incompressible Euler equations.
  • Provided rigorous justification for the formal limit.

Conclusions:

  • The non-relativistic limit of 3D e-MHD equations is rigorously established.
  • The study confirms the validity of approximating e-MHD with incompressible Euler equations under specific conditions.
  • Unique solutions are guaranteed for the e-MHD system when the limit system exhibits smooth solutions.