UBC Theses and Dissertations
Inviscid damping phenomena in some fluid models Jo, Min Jun
Inviscid damping phenomena in mathematical fluid dynamics have been intensively studied for the last decade, as the hydrodynamic analogue of Landau damping for the Vlasov equations. In its full generality, inviscid damping accounts for the extra stabilization mechanism that emerges near target steady states, within the fluid systems that are not inherently energy dissipative. In Chapter 2, we introduce the 2D inviscid IPM (incompressible porous medium) equation and then prove the quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation on the 2D whole space. The quantification is performed with respect to the intensity of stratification. Our proof robustly applies to other fundamental domains in the case of the purely linear density stratification; we prove the analogous results on the 2D torus and the horizontally periodic strip. The obtained temporal decay rates are all sharp, reaching the level of the linearized equations. In Chapter 3 and Chapter 4, we generalize the concept of inviscid damping as the extra stabilizing mechanism that appears in the vicinity of certain stationary solutions to the fluid equations that are fully or at least partially non-dissipative. Such an encompassing notion allows us to view various phenomena through the window of inviscid damping. More precisely, we prove the Lyapunov stability of the nonzero constant background magnetic field for the non-resistive 2D MHD (magnetohydrodynamics) equations in Chapter 3. Then we investigate the validity of the QG (quasi-geostrophic) dynamics as the legit approximation for the inviscid rotating stratified Boussinesq flows in Chapter 4; we determine between convergence and non-convergence with respect to the rotation-stratification ratio.
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