Seminars and Colloquia by Series

Infinite volume limit for the Nonlinear Schrodinger Equation and Weak Turbulence

Series
PDE Seminar
Time
Tuesday, December 2, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pierre GermainCourant Institute
Abstract: the theory of weak turbulence has been put forward by appliedmathematicians to describe the asymptotic behavior of NLS set on a compactdomain - as well as many other infinite dimensional Hamiltonian systems.It is believed to be valid in a statistical sense, in the weaklynonlinear, infinite volume limit. I will present how these limits can betaken rigorously, and give rise to new equations.

Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

Series
PDE Seminar
Time
Tuesday, November 25, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Changyou WangPurdue University
For a $C^{1,1}$-uniformly elliptic matrix $A$, let $H(x,p)=$ be the corresponding Hamiltonian function. Consider the Aronsson equation associated with $H$: $$(H(x,Du))x H_p(x,Du)=0.$$ In this talk, I will indicate everywhere differentiability of any viscosity solution of the above Aronsson's equation. This extends an important theorem by Evans and Smart on the infinity harmonic functions (i.e. $A$ is the identity matrix).

Shock wave solutions of conservation laws and their regularization by dissipation and dispersion.

Series
PDE Seminar
Time
Tuesday, November 4, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael ShearerNorth Carolina State University
Shock waves are idealizations of steep spatial gradients of physical quantities such as pressure and density in a gas, or stress in an elastic solid. In this talk, I outline the mathematics of shock waves for nonlinear partial differential equations that are simple models of physical systems. I will focus on non-classical shocks and smooth waves that they approximate. Of particular interest are comparisons between nonlinear traveling waves influenced strongly by dissipative effects such as viscosity or surface tension, and spreading waves generated by the balance between dispersion and nonlinearity, when the nonlinearity is non-convex.

Regularity of Solutions of Hamilton-Jacobi Equation on a Domain

Series
PDE Seminar
Time
Tuesday, October 28, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Albert FathiÉcole Normale Supérieure de Lyon, France
In this lecture, we will explain a new method to show that regularity on the boundary of a domain implies regularity in the inside for PDE's of the Hamilton-Jacobi type. The method can be applied in different settings. One of these settings concerns continuous viscosity solutions $U : T^N\times [0,+\infty[ \rightarrow R$ of the evolutionary equation $\partial_t U(x, t) + H(x, \partial_x U(x, t) ) = 0,$ where $T^N = R^N / Z^N$, and $H: T^N \times R^N$ is a Tonelli Hamiltonian, i.e. H(x, p) is $C^2$, strictly convex superlinear in p. Let D be a compact smooth domain with boundary $\partial D$ contained in $T^N \times ]0,+\infty[$ . We show that if U is differentiable at each point of $\partial D$, then this is also the case on the interior of D. There are several variants of this result in different settings. To make the result accessible to the layman, we will explain the method on the function distance to a closed subset of an Euclidean space. This example contains all the ideas of the general case.

The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution

Series
PDE Seminar
Time
Tuesday, October 7, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xuwen ChenBrown University
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to infinity. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.

Hydrodynamic limit of vortices in Ginzburg-Landau theory

Series
PDE Seminar
Time
Tuesday, September 30, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel SpirnUniversity of Minnesota
Vortices arise in many problems in condensed matter physics, including superconductivity, superfluids, and Bose-Einstein condensates. I will discuss some results on the behavior of two of these systems when there are asymptotically large numbers of vortices. The methods involve suitable renormalization of the energies both at the vortex cores and at infinity, along with a renormalization of the vortex density function.

Existence of strong solutions to Compressible Navier-Stokes equations with degenerate viscosities and vacuum

Series
PDE Seminar
Time
Tuesday, September 9, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shengguo ZhuGeorgia Tech
We identify sufficient conditions on initial data to ensure the existence of a unique strong solution to the Cauchy problem for the Compressible Navier-Stokes equations with degenerate viscosities and vacuum (such as viscous Saint-Venants model in $\mathbb{R}^2$). This is a recent work joint with Yachun Li and Ronghua Pan.

Well posedness and decay for full Navier Stokes equations with temperature dependent coefficient

Series
PDE Seminar
Time
Tuesday, August 26, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Junxiong JiaGeorgia Tech
In this talk, firstly, we study the local and global well-posedness for full Navier-Stokes equations with temperature dependent coefficients in the framework of Besov space. We generalized R. Danchin's results for constant transport coefficients to obtain the local and global well-posedness for the initial with low regularity in Besov space framework. Secondly, we give a time decay rate results of the global solution in the Besov space framework which is not investigated before. Due to the low regularity assumption, we find that the high frequency part is also important for us to get the time decay.

Nonlinear, nondispersive surface waves

Series
PDE Seminar
Time
Tuesday, April 22, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
John HunterUniversity California, Davis
Surface waves are waves that propagate along a boundary or interface, with energy that is localized near the surface. Physical examples are water waves on the free surface of a fluid, Rayleigh waves on an elastic half-space, and surface plasmon polaritons (SPPs) on a metal-dielectric interface. We will describe some of the history of surface waves and explain a general Hamiltonian framework for their analysis. The weakly nonlinear evolution of dispersive surface waves is described by well-known PDEs like the KdV or nonlinear Schrodinger equations. The nonlinear evolution of nondispersive surface waves, such as Rayleigh waves or quasi-static SPPs, is described by nonlocal, quasi-linear, singular integro-differential equations, and we will discuss some of the properties of these waves, including the formation of singularities on the boundary.

Infinite energy cascades and modified scattering for the cubic Schr\"odinger on product spaces

Series
PDE Seminar
Time
Thursday, April 17, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zaher HaniNew York University
We consider the cubic nonlinear Schr\"odinger equation posed on the product spaces \R\times \T^d. We prove the existence of global solutions exhibiting infinite growth of high Sobolev norms. This is a manifestation of the "direct energy cascade" phenomenon, in which the energy of the system escapes from low frequency concentration zones to arbitrarily high frequency ones (small scales). One main ingredient in the proof is a precise description of the asymptotic dynamics of the cubic NLS equation when 1\leq d \leq 4. More precisely, we prove modified scattering to the resonant dynamics in the following sense: Solutions to the cubic NLS equation converge (as time goes to infinity) to solutions of the corresponding resonant system (aka first Birkhoff normal form). This is joint work with Benoit Pausader (Princeton), Nikolay Tzvetkov (Cergy-Pontoise), and Nicola Visciglia (Pisa).

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