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Series: PDE Seminar

Long-time behavior of "generic" 2d Euler solutions is
expected to be governed by conserved quantities and simple variational
principles related to them. Proving or disproving this from the
dynamics is a notoriously difficult problem which remains unsolved.
The variational problems which arise from these conjectures are
interesting by themselves and we will present some results concerning
these problems.

Series: PDE Seminar

In this talk we describe recent results on classification and rigidity
properties of stationary homogeneous solutions to the 3D and 2D Euler
equations. The problem is motivated be recent exclusions of self-similar
blowup for Euler and its relation to Onsager conjecture and
intermittency. In 2D the problem also arises in several other areas such as isometric
immersions
of the 2-sphere, or optimal transport. A full classification of two dimensional
solutions
will be given. In 3D we reveal several new classes of solutions and prove their
rigidity properties. In particular, irrotational solutions are characterized by
vanishing of
the Bernoulli function; and tangential flows are necessarily 2D
axisymmetric pure rotations. In several cases solutions are excluded
altogether. The arguments reveal geodesic features of the Euler equation on
the sphere. We further discuss the case when homogeneity corresponds to
the Onsager-critical state. We will show that anomalous energy flux at
the singularity vanishes, which is suggestive of absence of extreme
$0$-dimensional intermittencies in dissipative flows.

Series: PDE Seminar

The interactions between particles and fluid have received a bulk
of attention due to a number of their applications in the field of, for
example, biotechnology, medicine, and in the study of sedimentation
phenomenon, compressibility of droplets of the spray, cooling tower plumes,
and diesel engines, etc. In this talk, we present coupled hydrodynamic
equations which can formally be derived from Vlasov-Boltzmann/Navier-Stokes
equations. More precisely, our proposed equations consist of the
compressible pressureless Euler equations and the isentropic compressible
Navier-Stokes equations. For the coupled system, we establish the global
existence of classical solutions when the domain is periodic, and its
large-time behavior which shows the exponential alignment between two fluid
velocities. We also remark on blow-up of classical solutions in the whole
space.

Series: PDE Seminar

In this talk, we will discuss a sequence of recent progresses
on the global well-posedness of energy conservative Holder continuous
weak solutions for a class of nonlinear variational wave equations and
the Camassa-Holm equation, etc. A typical feature of solutions in these
equations is the formation of cusp singularity and peaked soliton waves
(peakons), even when initial data are smooth. The lack of Lipschitz
continuity of solutions gives the major difficulty in studying the
well-posedness and behaviors of solutions. Several collaboration works
with Alberto Bressan will be discussed, including the uniqueness by
characteristic method, Lipschitz continuous dependence on a Finsler type
optimal transport metric and a generic regularity result using Thom's
transversality theorem.

Series: PDE Seminar

In this talk, we consider the initial-boundary value problem for
the Fokker-Planck equation in an interval or in a bounded domain with
absorbing boundary conditions. We discuss a theory of well-posedness of
classical solutions for the problem as well as the exponential decay in
time, hypoellipticity away from the singular set, and the Holder continuity
of the solutions up to the singular set. This is a joint work with J.
Jang,J. Jung, and J. Velazquez.

Series: PDE Seminar

We consider a Benney-type system modeling short wave-long wave
interactions in compressible viscous
fluids under the influence of a magnetic
field. Accordingly, this large system now consists of the compressible MHD
equations coupled with a nonlinear Schodinger equation along particle paths.
We study the global existence of smooth solutions to the Cauchy problem in R^3
when the initial data are small smooth perturbations of an equilibrium state.
An important point here is that, instead of the simpler case having zero as
the equilibrium state for the magnetic field, we consider an arbitrary non-zero
equilibrium state B
for the magnetic field. This is motivated by applications,
e.g., Earth's magnetic field, and the lack of invariance of the MHD system
with respect to either translations or rotations of the magnetic field. The usual
time decay investigation through spectral analysis in this non-zero equilibrium
case meets serious difficulties, for the eigenvalues in the frequency space are
no longer spherically symmetric. Instead, we employ a recently developed
technique of energy estimates involving evolution in negative Besov spaces, and
combine it with the particular interplay here between Eulerian and Lagrangian
coordinates. This is a joint work with Junxiong Jia and Ronghua Pan.

Series: PDE Seminar

The study of reflector surfaces in geometric optics necessitates the
analysis of nonlinear equations of Monge-Ampere type. For many important
examples (including the near field reflector problem), the equation no
longer falls within the scope of optimal transport, but within the class of "Generated
Jacobian equations" (GJEs). This class of equations was recently introduced by Trudinger,
motivated by problems in geometric optics, however they appear in many others areas (e.g.
variations of the Minkowski problem in convex geometry). Under natural assumptions, we
prove Holder regularity for the gradient of weak solutions. The results are new in
particular for the near-field point source reflector problem, but are applicable for a
broad class of GJEs: those satisfying an analogue of the A3-weak condition introduced by
Ma, Trudinger and Wang in optimal transport. Joint work with Jun Kitagawa.

Series: PDE Seminar

The smallest eigenvalue of a symmetric matrix A can be
expressed through Rayleigh's formula. Moreover, if the smallest eigenvalue
is simple, it can be approximated by using the inverse iteration method or
by studying the large time behavior of solutions of the ODE x'(t)=-Ax(t).
We discuss surprising analogs of these facts for a nonlinear PDE eigenvalue
problem involving the p-Laplacian.

Series: PDE Seminar

Consider a general linear Hamiltonian system u_t = JLu in a Hilbert
space X, called the energy space. We assume that R(L) is closed, L induces a
bounded and symmetric bi-linear form on X, and the energy functional
has only finitely many negative dimensions n(L). There is no restriction on the
anti-selfadjoint operator J except \ker L \subset D(J), which can be unbounded
and with an infinite dimensional kernel space. Our first result is an index
theorem on the linear instability of the evolution group e^{tJL}. More
specifically, we obtain some relationship between n(L) and the dimensions of
generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the
continuous spectrum. Our second result is the linear exponential trichotomy of the
evolution group e^{tJL}. In particular, we prove the nonexistence of exponential
growth in the finite co-dimensional center subspace and the optimal bounds on the
algebraic growth rate there. This is applied to construct the local invariant
manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state
(standing wave, steady state, traveling waves etc.). For some cases (particularly
ground states), we can prove orbital stability and local uniqueness of center
manifolds. We will discuss applications to examples including dispersive long wave
models such as BBM and KDV equations, Gross-Pitaevskii equation for superfluids,
2D Euler equation for ideal fluids, and 3D Vlasov-Maxwell systems for
collisionless plasmas. This work will be discussed in two talks. In the first
talk, we will motivate the problem by several Hamiltonian PDEs, describe the main
results, and demonstrate how they are applied. In the second talk, some ideas of
the proof will be given.

Series: PDE Seminar