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Magnetohydrodynamic Equations Around Couette Flow

Knobel, Niklas 1
1 Institut für Analysis (IANA), Karlsruher Institut für Technologie (KIT)

Abstract:

This thesis studies the stability of the magnetohydrodynamic equations around an affine shear flow and constant magnetic field. The dynamics of the equations change drastically depending on the fluid viscosity ν ≥ 0 and magnetic resistivity κ ≥ 0. The main goal of this thesis is to establish results on (nonlinear) stability and instability for selected dissipation regimes. These stability and instability results are established in Chapters 3-5.
In the first part, we consider the inviscid ν = 0 and resistive κ > 0 case. We linearize around explicit low-frequency solutions of traveling waves to infer the main growth model. Small data in Gevrey 2 spaces are necessary and sufficient for this main growth model to ensure stability.
In the second part, we consider the MHD equations with viscosity and horizontal resistivity ν = κ_x > 0 and κ_y = 0. We show that small initial data in Sobolev spaces ensure stability. Furthermore, we show that for the viscous ν > 0 and non-resistive κ = 0 case, for the linearized MHD equations, there exists initial data such that the magnetic field grows unbounded as t → ∞. Thus,
global in time stability of the magnetic field in Sobolev spaces cannot hold for the nonlinear system.
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Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Publikationstyp Hochschulschrift
Publikationsdatum 29.07.2024
Sprache Englisch
Identifikator KITopen-ID: 1000172809
Verlag Karlsruher Institut für Technologie (KIT)
Umfang vii, 157 S.
Art der Arbeit Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Analysis (IANA)
Prüfungsdatum 24.07.2024
Referent/Betreuer Zillinger, Christian
Liao, Xian

Volltext §
DOI: 10.5445/IR/1000172809
Veröffentlicht am 29.07.2024
Seitenaufrufe: 102
seit 29.07.2024
Downloads: 66
seit 29.07.2024
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