Seminars and Colloquia by Series

Turbulence, shmurbulence: how fat is it?

Series
PDE Seminar
Time
Wednesday, March 16, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
Predrag CvitanovicSchool of Physics, Georgia Tech
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)

Self-organized dynamics: aggregation and flocking

Series
PDE Seminar
Time
Friday, March 4, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Changhui TanRice University
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 controls

Series
PDE Seminar
Time
Tuesday, February 23, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Fausto GozziLUISS University, Rome, Italy
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.

On long -time behavior of solutions of 2d Euler equations

Series
PDE Seminar
Time
Tuesday, February 16, 2016 - 18:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vladimir SverakUniversity of Minneapolis, Minnesota
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.

Homogeneous solutions to the incompressible Euler equation

Series
PDE Seminar
Time
Wednesday, February 3, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roman ShvydkoyUniversity of Illinois, Chicago
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.

The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations

Series
PDE Seminar
Time
Tuesday, December 1, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Young-Pil ChoiImperial College London
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.

Uniqueness and Finsler type optimal transport metric for nonlinear wave equations

Series
PDE Seminar
Time
Tuesday, November 17, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Geng ChenSchool of Mathematics, Georgia Tech
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.

The Fokker-Planck equation in bounded domains

Series
PDE Seminar
Time
Tuesday, November 10, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyung Ju HwangPOSTECH, Korea
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.

Global Smooth Solutions in R^3 to Short Wave-Long Wave Interactions in Magnetohydrodynamics

Series
PDE Seminar
Time
Thursday, October 22, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hermano FridInstitute de Matematica Pura e Aplicada (IMPA)
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.

Regularity theory for surfaces in geometric optics and other Generated Jacobian Equations

Series
PDE Seminar
Time
Tuesday, October 20, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nestor GuillenUniversity of Massachusetts at Amherst
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.

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