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Series: PDE Seminar

In everyday language, this talk addresses the question about the optimal shape
and location of a thermometer of a given volume to reconstruct the temperature
distribution in an entire room. For random initial conditions, this problem was
considered by Privat, Trelat and Zuazua (ARMA, 2015), and we remove both the randomness
and geometric assumptions in their article. Analytically, we obtain quantitative
estimates for the wellposedness of an inverse problem, in which one determines the
solution in the whole domain from its restriction to a subset of given volume. Using wave
packet decompositions from microlocal analysis, we conclude that there exists a unique
optimal such subset, that it is semi-analytic and can be approximated by solving a
sequence of finite-dimensional optimization problems. This talk will also address future
applications to inverse problems.

Series: PDE Seminar

PDEs (such as Navier-Stokes) are in principle infinite-dimensional
dynamical systems. However, recent studies support conjecture that
the turbulent solutions of spatially extended dissipative systems
evolve within an `inertial' manifold spanned by a finite number of
'entangled' modes, dynamically isolated from the residual set of
isolated, transient degrees of freedom. We provide numerical
evidence that this finite-dimensional manifold on which the
long-time dynamics of a chaotic dissipative dynamical system lives
can be constructed solely from the knowledge of a set of unstable
periodic orbits. In particular, we determine the dimension of the
inertial manifold for Kuramoto-Sivashinsky system, and find it to
be equal to the `'physical dimension' computed previously via the
hyperbolicity properties of covariant Lyapunov vectors.
(with Xiong Ding, H. Chate, E. Siminos and K. A. Takeuchi)

Series: PDE Seminar

Self-organized behaviors are very common in nature and human societies:
flock of birds, school of fishes, colony of bacteria, and even group of people's
opinions. There are many successful mathematical models which capture the large
scale phenomenon under simple interaction rules in small scale. In this talk, I
will present several models on self-organized dynamics, in different scales: from
agent-based models, through kinetic descriptions, to various types of hydrodynamic
systems. I will discuss some recent results on these systems including existence of
solutions, large time behaviors, connections between different scales, and
numerical implementations.

Series: PDE Seminar

Stochastic optimal control problems governed by delay equations
with delay in the control are usually more difficult to study than the ones
when the delay appears only in the state. This is particularly true when we
look at the associated Hamilton-Jacobi-Bellman (HJB) equation. Indeed, even
in the simplified setting (introduced first by Vinter and Kwong for the
deterministic case) the HJB equation is an infinite dimensional second
order semi-linear PDE that does not satisfy the so-called structure
condition which substantially means that "the noise enters the system with
the control". The absence of such condition, together with the lack
of smoothing properties which is a common feature of problems with delay,
prevents the use of known techniques (based on Backward Stochastic
Differential Equations or on the smoothing properties of the linear part)
to prove the existence of regular solutions to this HJB equation and thus
no results in this direction have been proved till now. In this talk we
will discuss results about existence of regular solutions of this kind of
HJB equations and their use in solving the
corresponding control problem by finding optimal feedback controls,
also in the more difficult case of pointwise delay.
This is a joint work with Federica Masiero.

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.