# Tai-Peng Tsai

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## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - May 2021)

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.

## Publications

- Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities (2021)

Nonlinear Analysis, 211, 112409 - Existence, Uniqueness, and Regularity Results for Elliptic Equations with Drift Terms in Critical Weak Spaces (2020)

SIAM Journal on Mathematical Analysis, 52 (2), 1146--1191 - Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations (2020)

Communications in Partial Differential Equations, 45 (9), 1168--1201 - Global Navier–Stokes Flows for Non-decaying Initial Data with Slowly Decaying Oscillation (2020)

Communications in Mathematical Physics, 375 (3), 1665--1715 - Short Time Regularity of Navier–Stokes Flows with Locally L3 Initial Data and Applications (2020)

International Mathematics Research Notices, - Discretely self-similar solutions to theNavier–Stokes equations with data in Lloc2 satisfying the local energyinequality (2019)

Analysis & PDE, 12 (8), 1943--1962 - Discretely Self-Similar Solutions to the Navier–Stokes Equations with Besov Space Data (2018)

Archive for Rational Mechanics and Analysis, 229 (1), 53--77 - Green tensor of the Stokes system and asymptotics of stationary Navier–Stokes flows in the half space (2018)

Advances in Mathematics, 323, 326--366 - Lectures on Navier-Stokes Equations (2018)

Graduate Studies in Mathematics, - Self-Similar Solutions to the Nonstationary Navier-Stokes Equations (2018)

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, , 461--507 - Forward discretely self-similar solutions of the Navier-Stokes equations II (2017)

Ann. Henri Poincaré, - Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations (2017)

Discrete Contin. Dyn. Syst., - Rotationally corrected scaling invariant solutions to the Navier--Stokes equations (2017)

Comm. Partial Differential Equations, - Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations (2017)

Applied Mathematics Research eXpress, 2017 (2), 431--487 - Forward self-similar solutions of the Navier-Stokes equations in the half space (2016)

Anal. PDE, - Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations (2015)

Proc. Roy. Soc. Edinburgh Sect. A, - Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations (2015)

Funkcial. Ekvac., - Remark on Luo-Hou's ansatz for a self-similar solution to the 3D Euler equations (2015)

J. Nonlinear Sci., - Forward discretely self-similar solutions of the Navier-Stokes equations (2014)

Comm. Math. Phys., - Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations (2014)

Nonlinearity, - On discretely self-similar solutions of the Euler equations (2014)

Math. Res. Lett., - Asymptotics of small exterior Navier-Stokes flows with non-decaying boundary data (2012)

Comm. Partial Differential Equations, - Local Dynamics Near Unstable Branches of NLS Solitons (2012)
- Point singularities of 3D stationary Navier-Stokes flows (2012)

J. Math. Fluid Mech., - Small solutions of nonlinear Schrödinger equations near first excited states (2012)

J. Funct. Anal., - Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on $\Bbb R^2$ (2010)

Comm. Math. Phys., - Global existence and blow-up for harmonic map heat flow (2009)

J. Differential Equations, - Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II (2009)

Comm. Partial Differential Equations, - Scattering theory for the Gross-Pitaevskii equation in three dimensions (2009)

Commun. Contemp. Math., - Stability in $H^1/2$ of the sum of $K$ solitons for the Benjamin-Ono equation (2009)

J. Math. Phys., - Asymptotic stability of harmonic maps under the Schrödinger flow (2008)

Duke Math. J., - Global questions for map evolution equations (2008)

Singularities in PDE and the calculus of variations, - Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations (2008)

Int. Math. Res. Not. IMRN, - Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions (2007)

Ann. Henri Poincaré, - Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations (2007)

Comm. Math. Phys., - Schrödinger flow near harmonic maps (2007)

Comm. Pure Appl. Math., - Spectra of linearized operators for NLS solitary waves (2007)

SIAM J. Math. Anal., - Transient effects in oilfield cementing flows: qualitative behaviour (2007)

European J. Appl. Math., - Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary (2006)

J. Differential Equations, - Scattering for the Gross-Pitaevskii equation (2006)

Math. Res. Lett., - Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations (2006)

Duke Math. J., - Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves (2004)

Int. Math. Res. Not., - Soliton dynamics of nonlinear Schrödinger equations (2004)

Second International Congress of Chinese Mathematicians, - Asymptotic dynamics of nonlinear Schrödinger equations with many bound states (2003)

J. Differential Equations, - Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions (2002)

Comm. Pure Appl. Math., - Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data (2002)

Adv. Theor. Math. Phys., - On the point-particle (Newtonian) limit of the non-linear Hartree equation (2002)

Comm. Math. Phys., - Relaxation of excited states in nonlinear Schrödinger equations (2002)

Int. Math. Res. Not., - Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations (2002)

Comm. Math. Phys., - Stable directions for excited states of nonlinear Schrödinger equations (2002)

Comm. Partial Differential Equations, - On a classical limit of quantum theory and the non-linear Hartree equation (2000)

Conférence Moshé Flato 1999, Vol. I (Dijon), - On the spatial decay of 3-D steady-state Navier-Stokes flows (2000)

Comm. Partial Differential Equations, - Erratum: ``On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates'' [Arch.\ Rational Mech.\ Anal.\ \bf 143 (1988), no.\ 1, 29--51; MR1643650 (99j:35171)] (1999)

Arch. Ration. Mech. Anal., - On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates (1998)

Arch. Rational Mech. Anal., - On problems arising in the regularity theory for the Navier-Stokes equations (1998)

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