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.
Tuesday, December 1, 2015 - 15:05 , Location: Skiles 006 , Young-Pil Choi , Imperial College London , Organizer: Wilfrid Gangbo
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.
Tuesday, November 17, 2015 - 15:05 , Location: Skiles 006 , Geng Chen , School of Mathematics, Georgia Tech , Organizer: Wilfrid Gangbo
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.
Tuesday, November 10, 2015 - 15:05 , Location: Skiles 006 , Hyung Ju Hwang , POSTECH, Korea , Organizer: Wilfrid Gangbo
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.
Thursday, October 22, 2015 - 15:05 , Location: Skiles 005 , Hermano Frid , Institute de Matematica Pura e Aplicada (IMPA) , Organizer: Ronghua Pan
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.