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Series: PDE Seminar

In a bounded domain with smooth boundary (which can be considered as a
smooth sub-manifold of R3), we consider the Boltzmann equation with
general Maxwell boundary condition---linear combination of specular
reflection and diffusive absorption. We analyze the kinetic (Knudsen
layer) and fluid (viscous layer) coupled boundary layers in both acoustic
and incompressible regimes, in which the boundary layers behave
significantly different. The existence and damping properties of these
kinetic-fluid layers depends on the relative size of accommodation number
and Kundsen number, and the differential geometric property of the
boundary (the second fundamental form.)
As applications, first we justify the incompressible Navier-Stokes-Fourier
limit of the Boltzmann equation with Dirichlet, Navier, and diffusive
boundary conditions respectively, depending on the relative size of
accommodation number and Kundsen number. Using the damping property of the
boundary layer in acoustic regime, we proved the convergence is strong.
The second application is that we derive and justified the higher order
acoustic approximation of the Boltzmann equation.
This is a joint work with Nader Masmoudi.

Series: PDE Seminar

In this talk, we will discuss the global existence and asymptotic behavior of classical solutions for two dimensional inviscid Rotating Shallow Water system with small initial data subject to the zero-relative-vorticity constraint. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal forms. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity. This is joint work with Bin Cheng.

Series: PDE Seminar

We prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 3. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.

Series: PDE Seminar

We first discuss blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations,
where the sense of convergence and self-similarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. In the second part of the talk we discuss some observations on the Euler equations with symmetries, which shows that the point-wise behavior of the pressure along the flows is closely related to the blow-up of of solutions.

Series: PDE Seminar

Under the classical small-amplitude, long wave-length assumptions in which the
Stokes number is of order one, so featuring a balance between nonlinear and dispersive effects,
the KdV-equation
u_t+ u_x + uu_x + u_xxx = 0 (1)
and the regularized long wave equation, or BBM-equation
u_t + u_x + uu_x-u_xxt = 0 (2)
are formal reductions of the full, two-dimensional Euler equations for free surface flow. This
talk is concerned with the two-point boundary value problem for (1) and (2) wherein the wave
motion is specified at both ends of a finite stretch of length L of the media of propagation.
After ascertaining natural boundary specifications that constitute well posed problems, it is
shown that the solution of the two-point boundary value problem, posed on the interval [0;L],
say, converges as L converges to infinity, to the solution of the quarter-plane boundary value problem in
which a semi-infinite stretch [0;1) of the medium is disturbed at its finite end (the so-called
wavemaker problem). In addition to its intrinsic interest, our results provide justification for the use of the
two-point boundary-value problem in numerical studies of the quarter plane problem for
both the KdV-equation and the BBM-equation.

Series: PDE Seminar

We study the asymptotic behavior of the infinite Darcy-Prandtl number Darcy-Brinkman-Boussinesq model for convection in porous media at small Brinkman-Darcy number. This is a singular limit involving a boundary layer with thickness proportional to the square root of the Brinkman-Darcynumber . This is a joint work with Jim Kelliher and Roger Temam.

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.