Georgia Institute of Technology College of Sciences

 In the realm of mathematical fluid dynamics, a formidable challenge lies in establishing inviscid limits from the Navier-Stokes equations to the Euler equations. The pursuit of solving this intricate problem, particularly concerning singular solutions, persists in both compressible and incompressible scenarios. In particular, compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general.

In this talk, we will discuss the recent proof on the unique vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations, for the general Cauchy Problem. Moreover, we extend our findings by establishing the well-posedness of such solutions within the broader class of inviscid limits of Navier-Stokes equations with locally bounded energy initial values.  This is a joint work with Kang and Vasseur, which can be found on arXiv:2401.09305.

The uniqueness and L2 stability of Euler equations, done by Chen-Krupa-Vasseur, will also be discussed in this talk.

The existence of global solutions for the Schrödinger equation
 i\partial_t u + \Delta u = P_d(u),
with nonlinearity $P_d$ homogeneous of degree $d$, has been extensively studied. Most results focus on the case with gauge invariant nonlinearity, where the solution satisfies several conservation laws. However, the problem becomes more complicated as we consider a general nonlinearity $P_d$. So far, global well-posedness for small data is known for $d$ strictly greater than the Strauss exponent. In dimension $3$, this Strauss exponent is $2$, making NLS with quadratic nonlinearity an interesting topic.

In this talk, I will present a result that shows the global existence and scattering for systems of quadratic NLS for small, localized data. To tackle the challenge presented by the $u\Bar{u}$-type nonlinearity, we require an $\epsilon$ regularization for the terms of this type in the system.
 

Stochastic flows of an advective-diffusive nature are ubiquitous in biology and the physical sciences. Of particular interest is the problem to reconcile observed marginal distributions with a given prior posed by E. Schroedinger in 1932/32 and known as the Schroedinger Bridge Problem (SBP). It turns out that Schroedinger’s problem can be viewed as a problem in large deviations, a modeling problem, as well as a control problem. Due to the fundamental significance of this problem, interest in SBP and in its deterministic (zero-noise limit) counterpart of Optimal Transport (OT) has in recent years enticed a broad spectrum of disciplines, including physics, stochastic control, computer science, and geometry. Yet, while the mathematics and applications of SBP/OT have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention; the problem to interpolate between “unbalanced” marginals has been approached by introducing source/sink terms into the transport equations, in an adhoc manner, chiefly driven by applications in image registration. Nevertheless, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schroedinger’s dictum; that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated law represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this talk is to present recent results on stochastic evolutions with losses, whereupon particles are “killed” (jump into a coffin/extinction state) according to a probabilistic law, and thereby mass is gradually lost along their stochastically driven flow. Through a suitable embedding we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism in the style of Schroedinger, given a prior law that allows for losses, we explore the most probable evolution of particles along with the most likely killing rate as the particles transition between the specified marginals. Our approach differs sharply from previous work involving a Feynman-Kac multiplicative reweighing of the reference measure which, as we argue, is far from Schroedinger’s quest. We develop a suitable Schroedinger system of coupled PDEs' for this problem, an iterative Fortet-IPF-Sinkhorn algorithm for computations, and finally formulate and solve a related fluid-dynamic control problem for the flow of one-time marginals where both the drift and the new killing rate play the role of control variables. Joint work with Tryphon Georgiou and Michele Pavon.

We consider classical scalar fields in dimension 1+1 with a
self-interaction potential being a symmetric double-well. Such a model
admits non-trivial static solutions called kinks and antikinks. A kink
cluster is a solution approaching, for large positive times, a
superposition of alternating kinks and antikinks whose velocities
converge to 0 and mutual distances grow to infinity. Our main result is
a determination of the asymptotic behaviour of any kink cluster at the
leading order.
Our results are partially inspired by the notion of "parabolic motions"
in the Newtonian n-body problem. I will present this analogy and mention
its limitations. I will also explain the role of kink clusters as
universal profiles for formation of multi-kink configurations.
This is a joint work with Andrew Lawrie.