Potomac Region
PDE Seminar

Avoiding vacuum in superfluidity

At low pressures and very low temperatures, Helium-4 is composed of two interacting phases: the superfluid and the normal fluid. We discuss some recent mathematical results in the analysis of this system.

At the micro-scale, the nonlinear Schrödinger equation is coupled with the incompressible inhomogeneous Navier-Stokes equations through a bidirectional nonlinear relaxation mechanism that facilitates mass and momentum exchange between phases. For small initial data, we construct solutions that are either global or almost-global in time, depending on the strength of the superfluid's self-interactions. The primary challenge lies in controlling inter-phase mass transfer to prevent vacuum formation within the normal fluid. Two approaches are employed: one based on energy estimates alone, and another combining energy estimates with maximal regularity. These results are part of joint work with Juhi Jang and Igor Kukavica.

If time permits, we will also discuss a macro-scale model, where both phases are governed by the incompressible Euler equations, coupled through a nonlinear and singular interaction term. We construct unique local-in-time analytic solutions. To address the singularity in the coupling, we ensure the absence of vorticity vacuum, while the derivative loss due to the nonlinearity is offset by trading the analytic radius for dissipation.

Zero viscosity limit of 1D viscous conservation laws at the point of first shock formation

Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small-scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. We provide an asymptotic expansion in viscosity of the viscous solution via the help of matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. Furthermore, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. This is joint work with John Anderson and Cole Graham.

Unique ergodicity for the damped-driven stochastic KdV equation

We discuss the existence, uniqueness, and regularity of invariant measures for the damped-driven stochastic Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems such as the 2D Navier-Stokes equations, can nevertheless be applied in this weakly dissipative setting to establish elementary proofs of both unique ergodicity, albeit without mixing rates, as well as regularity of the support of the invariant measure. Under the assumption of large damping, however, we are able to deduce the existence of a spectral gap with respect to a Wasserstein distance-like function. This is joint work with Nathan Glatt-Holtz (Indiana University) and Geordie Richards (Guelph University).

Spatial decay for coherent states of the Benjamin-Ono equation

We consider solutions to the Benjamin-Ono equation that are localized in a reference frame moving to the right with constant speed. We show that any such solution that decays at least like $\langle x \rangle^{-1-\epsilon}$ for some $\epsilon > 0$ in a comoving coordinate frame must in fact decay like $\langle x \rangle^{-2}$. In view of the explicit soliton solutions, this decay rate is sharp.

Our proof has two main ingredients. The first is microlocal dispersive estimates for the Benjamin-Ono equation in a moving frame, which allow us to prove spatial decay of the solution provided the nonlinearity has sufficient decay. The second is a careful normal form analysis, which allows us to obtain rapid decay of the nonlinearity for a transformed equation while assuming only modest decay of the solution. Our arguments are entirely time-dependent, and do not require the solution to be an exact traveling wave.

Almost sure local well-posedness of Nonlinear Schrödinger equation

During the talk, we will discuss the local solutions of the super-critical cubic Schrödinger equation (NLS) on the whole space with general differential operator. Although such a problem is known to be ill-posed, we show that the random initial data yield almost sure local well-posedness. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation in various directions: higher dimensions, more general operators, weaker regularity assumptions on the initial conditions. In particular, we show that in 3D, the classical cubic NLS is stochastically, locally well-posed for any initial data with regularity in $H^\varepsilon$ for any $\varepsilon > 0$, compared to the known results $\varepsilon > 1/6$ . The proofs are based on precise estimates in frequency space using various tools from Harmonic analysis. This is a joint project with Jean-Baptise Casteras (Lisbon University), Itamar Oliviera (University of Birmingham), and Gennady Uraltsev (University of Virginia, University of Arkansas).

Spectral asymptotics for the Schrödinger equation with bounded, unstructured potentials

High energy spectral asymptotics for Schrödinger operators on compact manifolds have been well studied since the early 1900s and it is now well known that they are intimately related to the structure of periodic geodesics. In this talk, we discuss analogous questions for Schrödinger operators, -\Delta +V on R^d, where V is bounded together with all of its derivatives. Since the geodesic flow on \R^d has no periodic trajectories (or indeed looping trajectories) one might guess that the spectral projector has a full asymptotic expansion. Indeed, for (quasi) periodic V this has been known since the work of Parnovski–Shterenberg in 2016. We show that when d=1, full asymptotic expansions continue to hold for any such V. When d=2, we give a large class of potentials whose spectral projectors have full asymptotics. Nevertheless, in d\geq 2, we construct examples where full asymptotics fail. Based on joint work with L. Parnovski and R. Shterenberg.

Modified gamma calculus for degenerate diffusions

Markov diffusions, and the associated gamma operators are related to myriad properties of a diffusion and its generator, including hypercontractivity, regularity, and long-time behavior. Many of these relationships are communicated via inequalities involving a distance which may be viewed as intrinsic to the gamma operators. Originally, these tools were only available for diffusions with elliptic generators. But, in the past twenty years, growing interest in degenerate diffusions has given rise to adaptations that work in a variety of degenerate models. Still, some basic degenerate models of interest have remained inaccessible to these tools until more recently. We’ll discuss these developments, including a proposed intrinsic distance for some of these more recent models.

TBA

TBA

Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton. 

This is joint work with S. Cao.