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.

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