Hu, Xianpeng
(2010)
WEAK SOLUTIONS AND INCOMPRESSIBLE LIMITS OFMULTI-DIMENSIONAL MAGNETOHYDRODYNAMIC FLOWS.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
This dissertation addresses mathematical issues regarding the existence of global weak so-lutions of isentropic compressible magnetohydrodynamic flows (MHD), the limit behaviorof isentropic compressible MHD as Mach number vanishes, and the hydrodynamic limit ofVlasov-Maxwell-Boltzemann equations. More precisely, in the first part, global existenceof weak solutions with large initial data to the Cauchy problem of the three-dimensionalcompressible MHD is established through an invading method for the adiabatic exponent&Gamma > 3/2 and constant viscosity coefficients. In the second part, we focus on the connectionbetween the incompressible MHD and the compressible isentropic MHD; it is showed that asMach number vanishes, the compressible isentropic MHD will converge to the incompress-ible MHD. In the third part, using relative entropy estimate about an absolute Maxwellian,we establish an incompressible Electron-Magnetohydrodynamics-Fourier limit for solutionsof the Vlasov-Maxwell-Blotzmann equation considered over any periodic spatial domain inR^3. It is shown that any properly scaled sequence of renormalized solutions of Vlasov-Maxwell-Boltzmann equations has fluctuations that (in the weak L2 topology) converge toan in¯nitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussi-nesq relations. It is shown that every limit point and the magnetic field are governed by aweak solution of an incompressible electron-magnetohydrodynamics system for all time.
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| Item Type: |
University of Pittsburgh ETD
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| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
30 September 2010 |
| Date Type: |
Completion |
| Defense Date: |
6 April 2010 |
| Approval Date: |
30 September 2010 |
| Submission Date: |
14 July 2010 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
Cauchy problem; Global weak solutions; Incompress-; Renormalized solutions |
| Other ID: |
http://etd.library.pitt.edu/ETD/available/etd-07142010-090003/, etd-07142010-090003 |
| Date Deposited: |
10 Nov 2011 19:51 |
| Last Modified: |
15 Nov 2016 13:45 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/8373 |
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