Tai-Peng Tsai

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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The present dissertation is split in three parts.The first considers the (unrestricted) Green tensor of Stokes system in the half-space. We derive the first ever pointwise estimates of such tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, and explore some applications of the pointwise estimates.The second part of this dissertation considers the magnetohydrodynamics equations (MHD equations) and the viscoelastic Navier–Stokes equations with damping. We construct self-similar and discretely self-similar solutions of both the MHD equations and the viscoelastic Navier–Stokes equations with damping with large initial data in the critical weak Lebesgue space.The third part of this dissertation deals with the Patlak–Keller–Segel–Navier–Stokes system. We prove the global existence of free-energy solutions with critical and subcritical mass.

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It has been of great interest in recent decades to know whether the incompressible Euler equations are well-posed in the borderline spaces. In order to understand the behavior of solutions in these spaces, the logarithmically regularized 2D Euler equations were introduced. In the borderline Sobolev space, the local wellposedness was proved by Chae-Wu when the regularization is sufficiently strong, while strong ill-posedness of the unregularized case was established by Bourgain-Li. The first part of the dissertation closes the gap between the two results, by establishing the strong ill-posedness in the remaining intermediate regime of the regularization.The second part of the thesis considers the Cauchy problem of incompressible3D Navier-Stokes equations with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the existence of a time-global weak solution has been known. However, such data do not include constants, and the only known global solutions for non-decaying data are either for perturbations of constants or when the velocity gradients are in L^p with finite p. This work presents how to construct global weak solutions for non-decaying initial data whose local oscillations decay, no matter how slowly.

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The present thesis is split in two parts. The first deals with the focusing Nonlinear Schrödinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish thedegenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.The second considers the perturbed energy critical focusing NonlinearSchrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. The construction relies on the resolvent expansion, which is singular due to the presence of a resonance. Specializing to pure power focusing subcritical perturbations we demonstrate, via variational arguments, and for a certain range of powers, the existence of a ground state soliton, which is then shown to be the previously constructed solution. Finally, we present a dynamical theorem which characterizes the fate of radially-symmetric solutions whose initial data are below the action of the ground state. Such solutions will either scatter or blow-up in finite time depending on the sign of a certain function of their initial data.

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The Schrödinger equation, an equation central to quantum mechanics, is a dispersive equation which means, very roughly speaking, that its solutions have a wave-like nature, and spread out over time. We will consider global behaviour of solutions of two nonlinear variations of the Schrödinger equation. In particular, we consider the nonlinear magnetic Schrödinger equation. [Formulas omitted] We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H¹, then the solution of the above equation decomposes uniquely into a standing wave part, which converges as t → ∞, and a dispersive part, which scatters. We also consider the Schrödinger map equation into the 2-sphere. We obtain a global well-posedness result for this equation with radially symmetric initial data without any size restriction on the initial data. Our technique involves translating the Schrödinger map equation into a cubic, non-local Schrödinger equation via the generalized Hasimoto transform. There, we also show global well-posedness for the non-local Schrödinger equation with radially-symmetric initial data in the critical space L²(ℝ²), using the framework of Kenig-Merle and Killip-Tao-Visan.

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No abstract available.