Abstract
We consider the compressible Navier-Stokes system coupled with the Maxwell equations governing the time evolution of the magnetic field. We introduce a relative entropy functional along with the related concept of dissipative solution. As an application of the theory, we show that for small values of the Mach number and large Reynolds number, the global in time weak (dissipative) solutions converge to the ideal MHD system describing the motion of an incompressible, inviscid, and electrically conducting fluid. The proof is based on frequency localized Strichartz estimates for the Neumann Laplacean on unbounded domains.
Highlights
Weak and dissipative solutionsWe start with the standard definition of weak solutions to the compressible MHD system (1.1 - 1.7), supplemented with the initial conditions: cf. [8]
We consider the compressible Navier-Stokes system coupled with the Maxwell equations governing the time evolution of the magnetic field
The proof is based on frequency localized Strichartz estimates for the Neumann Laplacean on unbounded domains
Summary
We start with the standard definition of weak solutions to the compressible MHD system (1.1 - 1.7), supplemented with the initial conditions: cf. [8]. For any τ ∈ [0, T ] and any test function φ ∈ Cc∞([0, T ] × Ω). |B|2divxφ dx dt for any τ ∈ [0, T ] and any test function φ ∈ Cc∞([0, T ] × Ω, R3), φ · n|∂Ω = 0. For the weak solutions to exist globally in time, the initial data must satisfy certain compatibility conditions:. Under the hypotheses (2.7), (2.8), the existence of global-in-time weak solutions to the compressible MHD system (1.1 - 1.7), (2.1) can be shown by the methods developed in Lions [30] and [15], cf [8]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
