- You are here:
- GT Home
- Home
- News & Events

Series: PDE Seminar

TBA

Series: PDE Seminar

TBA

Series: PDE Seminar

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.

Series: PDE Seminar

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.

Series: PDE Seminar

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.

Series: PDE Seminar

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.

Series: PDE Seminar

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.

Series: PDE Seminar

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.

Series: PDE Seminar

(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.

Series: PDE Seminar

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.