Georgia Institute of Technology College of Sciences

In this talk, we consider the Cauchy problem of a modified two-component Camassa-Holm shallow water system. We first establish local well-possedness of the Cauchy problem of the system. Then we present several blow-up results of strong solutions to the system. Moreover, we show the existence of global weak solutions to the system. Finally, we address global conservative solutions to the system. This talk is based on several joint works with C. Guan, K. H. Karlsen, K. Yan and W. Tan.

We study a class of linear delay-differential equations, with a singledelay, of the form$$\dot x(t) = -a(t) x(t-1).\eqno(*)$$Such equations occur as linearizations of the nonlinear delay equation$\dot x(t) = -f(x(t-1))$ around certain solutions (often around periodicsolutions), and are key for understanding the stability of such solutions.Such nonlinear equations occur in a variety of scientific models, anddespite their simple appearance, can lead to a rather difficultmathematical analysis.We develop an associated linear theory to equation (*) by taking the$m$-fold we

An interesting problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. A particular interest is so called physical vacuum which naturally arises in physical problems. The main difficulty lies in the fact that the physical systems become degenerate along the boundary. I'll present the well- posedness result of 3D compressible Euler equations for polytropic gases. This is a joint work with Nader Masmoudi.

In this talk, I will explain the correspondence between the Lorenz periodic solution and the topological knot in 3-space.The effect of small random perturbation on the Lorenz flow will lead to a certain nature order developed previously by Chow-Li-Liu-Zhou. This work provides an answer to an puzzle why the Lorenz periodics are only geometrically simple knots.

An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as

The quest for a suitable geometric description of major analyticproperties of sets has largely motivated the development of GeometricMeasure Theory in the XXth theory. In particular, the 1880 Painlev\'eproblem and the closely related conjecture of Vitushkin remained amongthe central open questions in the field. As it turns out, their higherdimensional versions come down to the famous conjecture of G. Davidrelating the boundedness of the Riesz transform and rectifiability.

We discuss a global weighted estimate for a class of divergence form elliptic operators with BMO coefficients on Reifenbergflat domains. Such an estimate implies new global regularity results in Morrey, Lorentz, and H\"older spaces for solutionsof certain nonlinear elliptic equations. Moreover, it can also be used to obtain a capacitary estimate to treat a measuredatum quasilinear Riccati type equations with nonstandard growth in the gradient.

We prove a new Holder estimate for drift-(fractional)diffusion equations similar to the one recently obtained by Caffarelli and Vasseur, but for bounded drifts that are not necessarily divergence free. We use this estimate to study the regularity of solutions to either the Hamilton-Jacobi equation or conservation laws with critical fractional diffusion.

We prove an extension of the Wiener inversion theorem for convolution of summable series, allowing the terms to take values in a space of bounded linear operators. The resulting algebra is no longer commutative due to the composition of operators. Inversion theorems arise naturally in the context of proving dispersive estimates for the Schr\"odinger and wave equation and lead to scale-invariant conditions for the class of admissible potentials. All results are joint work with Marius Beceanu.