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Series: PDE Seminar

We formulate a plasma model in which negative ions tend to a fixed, spatially-homogeneous background of positive charge. Instead of solutions with compact spatial support, we must consider those that tend to the background as x tends to infinity. As opposed to the traditional Vlasov-Poisson system, the total charge and energy are thus infinite, and energy conservation (which is an essential component of global existence for the traditional problem) cannot provide bounds for a priori estimates. Instead, a conserved quantity related to the energy is used to bound particle velocities and prove the existence of a unique, global-in-time, classical solution. The proof combines these energy estimates with a crucial argument which establishes spatial decay of the charge density and electric field.

Series: PDE Seminar

We discuss comparison principle for viscosity solutions of fully nonlinear elliptic PDEs in $\R^n$ which may have superlinear growth in $Du$ with variable coefficients. As an example, we keep the following PDE in mind:$$-\tr (A(x)D^2u)+\langle B(x)Du,Du\rangle +\l u=f(x)\quad \mbox{in }\R^n,$$where $A:\R^n\to S^n$ is nonnegative, $B:\R^n\to S^n$ positive, and $\l >0$. Here $S^n$ is the set of $n\ti n$ symmetric matrices. The comparison principle for viscosity solutions has been one of main issues in viscosity solution theory. However, we notice that we do not know if the comparison principle holds unless $B$ is a constant matrix. Moreover, it is not clear which kind of assumptions for viscosity solutions at $\infty$ is suitable. There seem two choices: (1) one sided boundedness ($i.e.$ bounded from below), (2) growth condition.In this talk, assuming (2), we obtain the comparison principle for viscosity solutions. This is a work in progress jointly with O. Ley.

Series: PDE Seminar

Many problems in Geometry, Physics and Biology are described by nonlinear partial differential equations of second order or four order. In this talk I shall mainly address the blow-up phenomenon in a class of fourth order equations from conformal geometry and some Liouville systems from Physics and Ecology. There are some challenging open problems related to these equations and I will report the recent progress on these problems in my joint works with Gilbert Weinstein and Chang-shou Lin.

Series: PDE Seminar

Multicomponent reactive flows arise in many practical applicationssuch as combustion, atmospheric modelling, astrophysics, chemicalreactions, mathematical biology etc. The objective of this work isto develop a rigorous mathematical theory based on the principles ofcontinuum mechanics. Results on existence, stability, asymptotics as wellas singular limits will be discussed.

Series: PDE Seminar

This seminar concerns the analysis of a PDE, invented by J.M. Lasry
and P.L. Lions
whose applications need not concern us.
Notwithstanding, the focus of the application is the behavior of a
free boundary in a diffusion equation which has dynamically evolving,
non--standard sources. Global existence and uniqueness are
established for this system. The work to be discussed represents a
collaborative effort with
Maria del Mar Gonzalez, Maria Pia Gualdani and Inwon Kim.

Series: PDE Seminar

We prove that solutions of the Navier-Stokes equations of
three-dimensional, compressible flow, restricted to fluid-particle
trajectories, can be extended as analytic functions of complex time. One
important corollary is backwards uniqueness: if two such solutions agree
at a given time, then they must agree at all previous times.
Additionally, analyticity yields sharp estimates for the time
derivatives of arbitrary order of solutions along particle trajectories.
I'm going to integrate into the talk something like a "pretalk" in an
attempt to motivate the more technical material and to make things
accessible to a general analysis audience.

Series: PDE Seminar

Some interesting nonlinear fourth-order parabolic equations, including the "thin-film" equation with linear mobility and the quantum drift-diffusion equation, can be seen as gradient flows of first-order integral functionals in the Wasserstein space of probability measures. We will present some general tools of the metric-variational approach to gradient flows which are useful to study this kind of equations and their asymptotic behavior. (Joint works in collaboration with U.Gianazza, R.J. McCann, D. Matthes, G. Toscani)

Series: PDE Seminar

I will talk about the highlights of a collaborative and multidisciplinary program investigating qualitative features of steady water waves with vorticity in two dimensions. Computational and analytical results together with data from the oceanographic community have resulted in strong evidence that key qualitative features such as amplitude, depth, streamline shape and pressure profile can be fundamentally affected by the presence of vorticity. Systematic studies of constant vorticity and shear vorticity functions will be presented.

Series: PDE Seminar

The Cauchy problem for the Poisson-Nernst-Planck/Navier-Stokes model was investigated by the speaker in [Transport Theory Statist. Phys. 31 (2002), 333-366], where a local existence-uniqueness theory was demonstrated, based upon Kato's framework for examining evolution equations. In this talk, the existence of a global distribution solution is proved to hold for the model, in the case of the initial-boundary value problem. Connection of the above analysis to significant applications is discussed. The solution obtained is quite rudimentary, and further progress would be expected in resolving issues of regularity.

Series: PDE Seminar

We study the problem of constructing systems of hyperbolic conservation laws with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a (typically overdetermined) system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed with techniques from exterior differential systems (Cartan-Kahler theory). The cases of 2x2- and 3x3-systems can be treated in detail, and explicit examples show that already the 3x3-case is fairly complex. We also analyze general rich systems. We also characterize conservative systems with the same eigencurves as compressible gas dynamics. This is joint work with Irina Kogan (North Carolina State University).