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Rigorous Results in Fluid and Kinetic Models / Neel Patel.
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- Book
- Thesis/Dissertation
- Author/Creator:
- Patel, Neel, author.
- Language:
- English
- Subjects (All):
- Mathematics--Penn dissertations.
- Penn dissertations--Mathematics.
- Local Subjects:
- Mathematics--Penn dissertations.
- Penn dissertations--Mathematics.
- Genre:
- Academic theses.
- Physical Description:
- 1 online resource (139 pages)
- Contained In:
- Dissertation Abstracts International 78-12B(E).
- Place of Publication:
- [Philadelphia, Pennsylvania]: University of Pennsylvania ; Ann Arbor : ProQuest Dissertations & Theses, 2017.
- Language Note:
- English
- System Details:
- Mode of access: World Wide Web.
- text file
- Summary:
- In the following, we will consider two different physical systems and their respective PDE models. In the first chapter, we prove time decay of solutions to the Muskat equation, which describes a fluid interface between two incompressible, immiscible fluids with different densities. The authors introduce the norms ∥f∥s= def∫ R2 |xi|s| fˆ(xi)| dxi in order to prove global existence of solutions to the Muskat problem. In this paper, for the 3D Muskat problem, given initial data f0 ∈ H l( R2) for some l ≥ 3 such that ∥f 0∥1 < k 0 for a constant k0 ≈ 1/5, we prove uniform in time bounds of ∥ f∥ s(t) for --2 < s < l -- 1 and assuming ∥f 0∥nu < infinity we prove time decay estimates of the form ∥f∥ s( t) ≤ (1+t)--s+nu for 0 ≤ s ≤ l -- 1 and --2 ≤ nu < s. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We prove analogous results in 2D.
- In the remaining chapters, we consider sufficient conditions, called continuation criteria, for global existence and uniqueness of classical solutions to the three-dimensional relativistic Vlasov-Maxwell system. In the compact momentum support setting, we prove that ∥p(18/5r)--1+beta 0f∥Linfinity tLr xL1p≤ 1 where 1 ≤ r ≤ 2 and beta > 0 is arbitrarily small, is a continuation criteria. The previously best known continuation criteria in the compact setting is ∥p(4/r )--1+beta 0f∥Linfinity tLr xL1p≤ 1, where 1 ≤ r 0 is arbitrarily small, due to Kunze. Our continuation criteria is an improvement in the 1 ≤ r ≤ 2 range. We also consider sufficient conditions for a global existence result to the three-dimensional relativistic Vlasov-Maxwell system without compact support in momentum space. In Luk-Strain, it was shown that ∥ ptheta0f∥L 1xL 1p≤ 1 is a continuation criteria for the relativistic Vlasov-Maxwell system without compact support in momentum space for theta > 5. We improve this result to theta > 3. We also build on another result by Luk-Strain in, in which the authors proved the existence of a global classical solution in the compact regime if there exists a fixed two-dimensional plane on which the momentum support of the particle density remains bounded. We prove well-posedness even if the plane varies continuously in time.
- Notes:
- Source: Dissertation Abstracts International, Volume: 78-12(E), Section: B.
- Advisors: Robert M. Strain; Committee members: Charles Epstein; Philip T. Gressman.
- Department: Mathematics.
- Ph.D. University of Pennsylvania 2017.
- Local Notes:
- School code: 0175
- ISBN:
- 9780355110265
- Access Restriction:
- Restricted for use by site license.
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