Potomac Region
PDE Seminar

Stochastic extinction and persistence in differential equations

We will discuss the stability of $\mathcal{M}_0$, an invariant subset of a Markov process $(X_t)_{t\geq 0}$ on a metric space $\mathcal{M}$. By building the theory of average Lyapunov functions, we formulate general criteria based on the signs of Lyapunov exponents that guarantee extinction ($X_t \to \mathcal{M}_0$ as $t \to \infty$). Additionally, we provide applications to a stochastic SIS epidemic model on a network with regime-switching, a stochastic differential equation version of the Lorenz system, a general class of discrete-time ecological models, and stochastic Kolmogorov systems. This is a joint project with Declan Stacy.

Well-posedness for Dispersive PDE with Combined Nonlinearities

We discuss local and global well-posedness for the 1D NLS with combined nonlinearities in a weighted Sobolev space, scattering for initial data with a quadratic phase, and discuss a possible threshold for the global vs finite time existence. We then comment on how the same quadratic phase can lead to blow up solutions by showing numerical examples for exponential type nonlinearities which motivate analytical results. Finally, we emphasize the flexibility of the weighted space technique by presenting higher dimensional studies and extensions to other dispersive PDE such as the nonlinear Klein-Gordon equation.

Global solution to the one-phase Muskat problem with surface tension

The one-phase Muskat problem describes the motion of the interface separating a wet region from a dry region within a porous medium, a process governed by Darcy’s law. Although physically essential, the inclusion of surface tension introduces an additional challenge. We prove small-data global well-posedness of the one-phase Muskat problem with subcritical initial data $H^s$, $s>d/2+1$. Moreover, the solution converges to zero in Lipschitz norm as $t\rightarrow\infty$. To the best of our knowledge, this work constitutes the first global well-posedness result for the one-phase Muskat problem with surface tension. This is based on joint work with Hongjie Dong.

Sticky particle solutions as energy minimizers for pressureless Euler system

Sticky Particles (SP) solutions to the Pressureless Euler system were introduced by Zeldovich in 1970 to describe large scale formations in the universe by accretion of matter. Since then, a significant amount of work was dedicated to studying various quantitative and qualitative properties of such solutions. In this talk we will discuss a new criterion for singling out SP solutions as the unique kinetic energy minimizers among particle solutions to the system.

Dispersive decay for the Modified Benjamin-Ono equation with small data 

We study the modified Benjamin-Ono equation for small, localized initial data where no solitons arise.  Unlike the standard Benjamin-Ono equation, the modified version is not known to be completely integrable, so we cannot rely on integrable systems tools. First, to manage the nonlocal Hilbert transform in the linear part of the equation, we split it using positive and negative frequency projections. We then combine wave packet testing of Ifrim–Tataru, the Klainerman–Sobolev vector field method, a normal form transformation, and a bootstrap argument, to establish a global-in-time dispersive decay estimate.

Analysis of nonhomogeneous initial-boundary value problems for nonlinear dispersive equations

A variety of physical phenomena are modeled by dispersive partial differential equations, whose study has been at the center of interest within the broader PDE/harmonic analysis/nonlinear waves communities during the past fifty years or so. In this talk, we will focus on the analysis of dispersive equations in the presence of nonzero boundary conditions. Such conditions are directly motivated by applications that arise in domains with a boundary, either in nature or in the laboratory. The relevant problems are known as initial-boundary value problems and can be significantly more involved than the (more standard) initial value problems, which take place on fully unbounded domains. The nonlinear Schrödinger (NLS) equation, which is a central dispersive PDE, will help guide us through the general methodology of our analysis.

Regularity of optimal transport with Coulomb cost

Optimal transport with Coulomb cost arises naturally in the strong-interaction limit of density functional theory. Its repulsive nature allows optimal maps to develop jump discontinuities even for smooth densities. In this talk, I will present a partial regularity result showing that, for bounded, strictly positive, Hölder continuous densities on $\mathbb{R}^d$, the optimal map is a $C^{1,\alpha}$ diffeomorphism outside a closed set of measure zero. I will explain the main ideas of the proof, which adapts a variational approach to the regularity of optimal transport based on almost-minimality and $\varepsilon$-regularity.

The talk is based on joint work with Gero Friesecke, building on earlier work with Felix Otto and Maxime Prod’homme.

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