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Three-dimensional incompressible convective Brinkman-Forchheimer equations
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Hajduk, Karol (2019) Three-dimensional incompressible convective Brinkman-Forchheimer equations. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3449805~S1
Abstract
This thesis presents a mathematical analysis of the incompressible convective Brinkman-Forchheimer equations in three-dimensional space,
∂tu - μ∆u + (u ▽)u + αu + β|u|r-1 u + ▽p = f; div u = 0;
where α; β ≥ 0, and r ≥ 1. These equations describe the motion of a fluid in a saturated porous medium. They can be seen also as the incompressible Navier-Stokes equations with the additional linear and nonlinear terms αu and β|u|r-1u. For simplicity, we neglect the linear term throughout the thesis, but all the results presented in this thesis hold also for general α > 0. In the thesis we study the influence of the nonlinear term on the existence of weak and strong solutions of the CBF equations and some of their properties.
In particular, we establish that all weak solutions of the `critical' problem (r = 3) verify the Energy Equality
½ || u(T)||2 + μ ⌠T0||▽u(t)||2 dt + β⌠ T0||u(t)||r+1Lr+1 ds = ½||u(0)||2
both on the torus T3 and on bounded domains ΩCR3 with smooth boundary. From this fact, we infer the existence of a strong global attractor in the phase space H → L2 using theory of evolutionary systems developed by Cheskidov [2009].
Moreover, we prove the existence of global-in-time strong solutions on the torus T3, for two cases: r > 3, and r = 3 provided that the product of viscosity (μ) and porosity (β) coefficients is not too small, 4μβ≥ 1. We also establish that strong solutions are unique in the larger class of weak solutions (`weak-strong uniqueness'). Additionally, we provide a `robustness of regularity' condition for strong solutionsof the convective Brinkman {Forchheimer equations when r ϵ [1; 3].
We also give two general methods of simultaneous approximation in Lebesgue and Sobolev spaces using semigroup theory and finite-dimensional eigenspaces of operators. Furthermore, we provide a simple proof of known characterisation of the domains of the fractional powers of the Laplace and Stokes operators, using the theory of real interpolation spaces. This characterisation is needed to apply our approximation method in the proof of the energy equality on bounded domains.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Navier-Stokes equations, Differential equations, Partial, Fluid dynamics | ||||
Official Date: | July 2019 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Robinson, James C. (James Cooper), 1969- | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | v, 126 leaves | ||||
Language: | eng |
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