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Series: PDE Seminar

We investigate Lipschitz maps, I, mapping $C^2(D) \to C(D)$, where $D$ is an appropriate domain. The global comparison principle (GCP) simply states that whenever two functions are ordered in D and touch at a point, i.e. $u(x)\leq v(x)$ for all $x$ and $u(z)=v(z)$ for some $z \in D$, then also the mapping I has the same order, i.e. $I(u,z)\leq I(v,z)$. It has been known since the 1960’s, by Courr\`{e}ge, that if I is a linear mapping with the GCP, then I must be represented as a linear drift-jump-diffusion operator that may have both local and integro-differential parts. It has also long been known and utilized that when I is both local and Lipschitz it will be a min-min over linear and local drift-diffusion operators, with zero nonlocal part. In this talk we discuss some recent work that bridges the gap between these situations to cover the nonlinear and nonlocal setting for the map, I. These results open up the possibility to study Dirichlet-to-Neumann mappings for fully nonlinear equations as integro-differential operators on the boundary. This is joint work with Nestor Guillen.

Series: PDE Seminar

The incompressible three-dimensional Euler equations are a basic model of fluid mechanics. Although these equations are more than 200 years old, many fundamental questions remain unanswered, most notably if smooth solutions can form singularities in finite time. In this talk, I discuss recent progress towards proving a finite time blowup for the Euler equations, inspired numerical work by T. Hou and G. Luo and analytical results by A. Kiselev and V. Sverak. My main focus lies on various model equations of fluid mechanics that isolate and capture possible mechanisms for singularity formation. An important theme is to achieve finite-time blowup in a controlled manner using the hyperbolic flow scenario in one and two space dimensions. This talk is based on joint work with B. Orcan-Ekmecki, M. Radosz, and H. Yang.

Series: PDE Seminar

SGSW is a third level specialization of Navier-Stokes (via
Boussinesq, then Semi-Geostrophic), and it accurately describes
large-scale, rotation-dominated atmospheric flow under the extra-assumption
that the horizontal velocity of the fluid is independent of the vertical
coordinate. The Cullen-Purser stability condition establishes a connection
between SGSW and Optimal Transport by imposing semi-convexity on the
pressure; this has led to results of existence of solutions in dual space
(i.e., where the problem is transformed under a non-smooth change of
variables). In this talk I will present recent results on existence and weak
stability of solutions in physical space (i.e., in the original variables)
for general initial data, the very first of their kind. This is based on
joint work with M. Feldman (UW-Madison).

Series: PDE Seminar

We establish global well-posedness and scattering for the cubic
Dirac equation for small data in the critical space. The theory we
develop is the Klein-Gordon counterpart of the Wave Maps / Schroedinger
Maps theory. This is joint work with Sebastian Herr.

Series: PDE Seminar

We will discuss a distance between shapes defined by minimizing the integral of
kinetic energy along transport paths constrained to measures with characteristic-function
densities. The formal geodesic equations for this shape distance are Euler equations for
incompressible, inviscid flow of fluid with zero pressure and surface tension on the free
boundary. We will discuss the instability that the minimization problem develops and the
resulting connections to optimal transportation. The talk is based on joint work with
Jian-Guo Liu (Duke) and Bob Pego (CMU).

Series: PDE Seminar

The subject of this talk is wave equations that arise from
geometric considerations. Prime examples include the wave map equation and
the Yang-Mills equation on the Minkowski space.
On one hand, these are fundamental field theories arising in physics; on the
other hand, they may be thought of as the hyperbolic analogues of the
harmonic map and the elliptic Yang-Mills equations, which are interesting
geometric PDEs on their own. I will discuss the recent progress on the
problem of global regularity and asymptotic behavior of solutions to these
PDEs.

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.