Jing, Tian
(2023)
Interface Problems in Two-Phase Magnetohydrodynamic Flows.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
We study the motion of two incompressible, conductive fluids in a magnetic field. The viscosity and surface tension are considered. The study includes the existence of varifold solutions, strong solutions, and their weak-strong uniqueness. To obtain varifold solutions, we approximate the equations using the Galerkin method. Using solution operators and the Schauder fixed-point theorem, we can obtain the approximate solutions. The weak convergence method is then used for studying the limit of approximate solutions. Varifolds are used for describing the interface. To find a strong solution, we apply the Hanzawa transformation to the equations, which are transformed into a fixed-interface problem for a short time. The new equations are divided into principal parts and nonlinear parts, which are studied separately. The solution is obtained using the fixed point theory of contraction mappings. When the strong solution exists, all varifold solutions coincide with it. This is proved by estimating the error between strong and varifold solutions using the relative entropy. An inequality of the relative entropy is derived and controlled by utilizing the Gronwall’s inequality.
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Details
| Item Type: |
University of Pittsburgh ETD
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| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
11 May 2023 |
| Date Type: |
Publication |
| Defense Date: |
16 March 2023 |
| Approval Date: |
11 May 2023 |
| Submission Date: |
3 April 2023 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
145 |
| 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: |
3-D MHD, two-phase, varifold solutions, strong solutions, weak-strong uniqueness. |
| Date Deposited: |
11 May 2023 13:02 |
| Last Modified: |
11 May 2023 13:02 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/44411 |
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