Friday, April 29, 2016 - 14:05 , Location: Skiles 005 , Adrian Tudorascu , West Virginia University , Organizer: Wilfrid Gangbo
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).
Wednesday, April 27, 2016 - 14:05 , Location: Skiles 270 , Ioan Bejenaru , University of California, San Diego , Organizer: Wilfrid Gangbo
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.
Tuesday, April 19, 2016 - 15:05 , Location: Skiles 006 , Dejan Slepcev , Carnegie Mellon University , Organizer: Wilfrid Gangbo
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).
Wednesday, April 6, 2016 - 14:05 , Location: Skiles 270 , Sung-Jin Oh , University of California, Berkeley , Organizer: Wilfrid Gangbo
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.
Wednesday, March 30, 2016 - 14:05 , Location: Skiles 270 , Alden Waters , CNRS Ecole Normale Superieure , Organizer: Wilfrid Gangbo
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.
Wednesday, March 16, 2016 - 14:05 , Location: Skiles 270 , Predrag Cvitanovic , School of Physics, Georgia Tech , Organizer: Wilfrid Gangbo
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)
Friday, March 4, 2016 - 16:05 , Location: Skiles 005 , Changhui Tan , Rice University , Organizer: Wilfrid Gangbo
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.
HJB equations for stochastic control problems with delay in the control: regularity and feedback controlsTuesday, February 23, 2016 - 15:05 , Location: Skiles 006 , Fausto Gozzi , LUISS University, Rome, Italy , Organizer: Wilfrid Gangbo
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.
Tuesday, February 16, 2016 - 18:05 , Location: Skiles 006 , Vladimir Sverak , University of Minneapolis, Minnesota , Organizer: Wilfrid Gangbo
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.
Wednesday, February 3, 2016 - 16:05 , Location: Skiles 006 , Roman Shvydkoy , University of Illinois, Chicago , Organizer: Wilfrid Gangbo
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.