Seminars and Colloquia by Series

Derivative estimates for elliptic systems from composite material

Series
PDE Seminar
Time
Tuesday, October 16, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Yanyan LiRutgers University
We give derivative estimates for solutions to divergence form elliptic equations with piecewise smooth coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces.

Hyperbolic conservation laws arising in the study of the forward-forward Mean-Field Games

Series
PDE Seminar
Time
Thursday, October 11, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marc SedjroAfrican Institute for Mathematical Sciences, Tanzania
In this talk, we introduce several models of the so-called forward-forward Mean-Field Games (MFGs). The forward-forward models arise in the study of numerical schemes to approximate stationary MFGs. We establish a link between these models and a class of hyperbolic conservation laws. Furthermore, we investigate the existence of solutions and examine long-time limit properties. Joint work with Diogo Gomes and Levon Nurbekyan.

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.

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