Seminars and Colloquia by Series

Shapes of local minimizers for a model of capillary energy in periodic media

Series
PDE Seminar
Time
Tuesday, September 25, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
William FeldmanUniversity of Chicago
I will discuss the limit shapes for local minimizers of the Alt-Caffarelli energy. Fine properties of the associated pinning intervals, continuity/discontinuity in the normal direction, determine the formation of facets in an associated quasi-static motion. The talk is partially based on joint work with Charles Smart.

Regularity properties of degenerate diffusion equations with drifts

Series
PDE Seminar
Time
Tuesday, September 4, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yuming Paul ZhangUCLA
We consider a class of nonlinear, degenerate drift-diffusion equations in R^d. By a scaling argument, it is widely believed that solutions are uniformly Holder continuous given L^p-bound on the drift vector field for p>d. We show the loss of such regularity in finite time for p≤d, by a series of examples with divergence free vector fields. We use a barriers argument.

Maxmimal regularity properties of local and nonlocal problems for regular and singular degenerate PDEs

Series
PDE Seminar
Time
Tuesday, August 21, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Veli ShakhmurovOkan University
The boundary value and mixed value problems for linear and nonlinear degenerate abstract elliptic and parabolic equations are studied. Linear problems involve some parameters. The uniform L_{p}-separability properties of linear problems and the optimal regularity results for nonlinear problems are obtained. The equations include linear operators defined in Banach spaces, in which by choosing the spaces and operators we can obtain numerous classes of problems for singular degenerate differential equations which occur in a wide variety of physical systems. In this talk, the classes of boundary value problems (BVPs) and mixed value problems (MVPs) for regular and singular degenerate differential operator equations (DOEs) are considered. The main objective of the present talk is to discuss the maximal regularity properties of the BVP for the degenerate abstract elliptic and parabolic equation We prove that for f∈L_{p} the elliptic problem has a unique solution u∈ W_{p,α}² satisfying the uniform coercive estimate ∑_{k=1}ⁿ∑_{i=0}²|λ|^{1-(i/2)}‖((∂^{[i]}u)/(∂x_{k}^{i}))‖_{L_{p}(G;E)}+‖Au‖_{L_{p}(G;E)}≤C‖f‖_{L_{p}(G;E)} where L_{p}=L_{p}(G;E) denote E-valued Lebesque spaces for p∈(1,∞) and W_{p,α}² is an E-valued Sobolev-Lions type weighted space that to be defined later. We also prove that the differential operator generated by this elliptic problem is R-positive and also is a generator of an analytic semigroup in L_{p}. Then we show the L_{p}-well-posedness with p=(p, p₁) and uniform Strichartz type estimate for solution of MVP for the corresponding degenerate parabolic problem. This fact is used to obtain the existence and uniqueness of maximal regular solution of the MVP for the nonlinear parabolic equation.

Scattering below the ground state for nonlinear Schrödinger equations

Series
PDE Seminar
Time
Thursday, May 3, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jason MurphyMissouri University of Science and Technology
The ground state solution to the nonlinear Schrödinger equation (NLS) is a global, non-scattering solution that often provides a threshold between scattering and blowup. In this talk, we will discuss new, simplified proofs of scattering below the ground state threshold (joint with B. Dodson) in both the radial and non-radial settings.

Growth of Sobolev norms for abstract linear Schrödinger Equations

Series
PDE Seminar
Time
Thursday, April 26, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 257
Speaker
Alberto MasperoSISSA
We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.

[MOVED TO THURSDAY] Growth of Sobolev norms for abstract linear Schrödinger Equations

Series
PDE Seminar
Time
Tuesday, April 24, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alberto MasperoSISSA
(Due to a flight cancellation, this talk will be moved to Thursday (Apr 26) 3pm at Skiles 257). We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.

Dynamics of a degenerate PDE model of epitaxial crystal growth

Series
PDE Seminar
Time
Tuesday, April 17, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jian-Guo LiuDuke University
Epitaxial growth is an important physical process for forming solid films or other nano-structures. It occurs as atoms, deposited from above, adsorb and diffuse on a crystal surface. Modeling the rates that atoms hop and break bonds leads in the continuum limit to degenerate 4th-order PDE that involve exponential nonlinearity and the p-Laplacian with p=1, for example. We discuss a number of analytical results for such models, some of which involve subgradient dynamics for Radon measure solutions.

L-infinity instability of Prandtl's layers

Series
PDE Seminar
Time
Tuesday, April 10, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Toan NguyenPenn State University
In 1904, Prandtl introduced his famous boundary layer theory to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary in the inviscid limit. His Ansatz was that the solution of Navier Stokes can be described as a solution of Euler, plus a boundary layer corrector, plus a vanishing error term in $L^\infty$. In this talk, I will present a recent joint work with E. Grenier (ENS Lyon), proving that, for a class of regular solutions of Navier Stokes equations, namely for shear profiles that are unstable to Rayleigh equations, this Prandtl's Ansatz is false. In addition, for shear profiles that are monotone and stable to Rayleigh equations, the Prandtl's asymptotic expansions are invalid.

How to Make a Black Hole

Series
PDE Seminar
Time
Tuesday, February 27, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xinliang AnUniversity of Toronto
Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations, geometric analysis and dynamical systems, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a black hole region in our universe. This result extends the 1965 Penrose’s singularity theorem and it also proves a conjecture of Ashtekar on black-hole thermodynamics. Open problems and new directions will also be discussed.

Time quasi-periodic gravity water waves in finite depth

Series
PDE Seminar
Time
Tuesday, February 20, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Emanuele HausUniversità degli Studi di Napoli Federico II
We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - i.e. periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. To overcome these problems, we first reduce the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme that requires very weak Melnikov non-resonance conditions (which lose derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments. This is a joint work with P. Baldi, M. Berti and R. Montalto.

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