Year: 2022

The inviscid limit problem for Navier-Stokes equations

ttn12 boundary layers, fluid dynamics , , ,

A longstanding open problem is to establish the inviscid limit of classical solutions to the incompressible Navier-Stokes equations for smooth initial data on a domain with boundaries. The question is of great physical and mathematical interest, and it deeply links to the transition to turbulence in fluids that may possibly take place faster than expected due to the presence of a boundary. In this article, I shall give a quick overview of this subject, and then highlight some recent works with my former student, Trinh T. Nguyen, (currently a Van Vleck Assistant Professor at University of Wisconsin, Madison), whose main results establish the inviscid limit for smooth data that are only required to be analytic locally near the boundary. This may be the best possible type of positive results that one can hope for, given the known violent instabilities at the boundary, which I shall discuss below. Before getting on, this picture should already hint at the great delicacy in studying boundary layers (source internet):

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Dispersion for Schrödinger equations

ttn12 Lecture notes, Schrodinger ,

The Schrödinger equation forms the basic principles of quantum mechanics (like that of Newton’s second law in classical mechanics). It also plays an important role in describing waves at an appropriate regime in classical fluid dynamics (e.g., water waves) and plasma physics (e.g., Langmuir’s waves or oscillations in a plasma!). In this quick note, I shall present a few basic properties and classical results for the Schrödinger equations, focusing mainly on the defocusing cubic nonlinear equations

\displaystyle i\partial_t u + \Delta u = |u|^2 u \ \ \ \ \ (1)

on {\mathbb{R}_+ \times \mathbb{R}^d}, {d\ge 1} (also known as the Gross-Pitaevskii equation). These notes are rather introductory and classical (e.g., Tao’s lecture notes), which I’m using as part of my lectures at the summer school that P. T. Nam and I are running this week on “the Mathematics of interacting Bose gases” at VIASM, Hanoi, Vietnam (August 1-5, 2022)!

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A roadmap to nonuniqueness of L^p weak solutions to Euler

ttn12 fluid dynamics, Instabilities

In this post, I discuss a rather classical roadmap to obtain the non-uniqueness of {L^p} weak solutions to the classical incompressible Euler equations; namely, focusing on the two-dimensional case, which reads in the vorticity formulation for vorticity function {\omega}:

\displaystyle \partial_t \omega + v \cdot \nabla \omega = 0, \qquad v = \nabla^\perp \Delta^{-1}\omega \ \ \ \ \ (1)

on {\mathbb{R}^2}, with initial vorticity in {L^p} (hence, vorticity remains in {L^p} for all times). It’s known, going far back to Yudovich ’63, that weak solutions with bounded vorticity are unique, leaving open the question of uniqueness of solutions whose vorticity is only in {L^p} for {p\in [1,\infty)}. This blog post is to discuss the possible quick roadmap to proving nonuniqueness arising from the instability nature of fluid models, focusing on the Euler equations (1).

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