2021-10-15T23:08:10Zhttps://keep.lib.asu.edu/oai/requestoai:keep.lib.asu.edu:node-1346322021-08-11T21:09:57Zoai_pmh:alloai_pmh:repo_items134632
https://hdl.handle.net/2286/R.I.42898
http://rightsstatements.org/vocab/InC/1.0/
2017-05
50 pages
eng
Kofroth, Collin Michael
Jones, Don
Smith, Hal
School of Mathematical and Statistical Sciences
Barrett, The Honors College
Text
The dissipative shallow-water equations (SWE) possess both real-world application and extensive analysis in theoretical partial differential equations. This analysis is dominated by modeling the dissipation as diffusion, with its mathematical representation being the Laplacian. However, the usage of the biharmonic as a dissipative operator by oceanographers and atmospheric scientists and its underwhelming amount of analysis indicates a gap in SWE theory. In order to provide rigorous mathematical justification for the utilization of these equations in simulations with real-world implications, we extend an energy method utilized by Matsumura and Nishida for initial value problems relating to the equations of motion for compressible, vsicous, heat-conductive fluids ([6], [7]) and applied by Kloeden to the diffusive SWE ([4]) to prove global time existence of classical solutions to the biharmonic SWE. In particular, we develop appropriate a priori growth estimates that allow one to extend the solution's temporal existence infinitely under sufficient constraints on initial data and external forcing, resulting in convergence to steady-state.
Mathematics
Partial Differential Equation Analysis
Fluid Dynamics
Asymptotic Stability of Biharmonic Shallow Water Equations