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Analysis/PDE Seminar

Time: 11:50-12:40 pm Fridays
Room: PSA 302
(unless otherwise specified)

Contact: svetlana[at]math.asu.edu 

Date

Speaker

Topic/Abstract

October 16

Luiz Gustavo Farah
University of California, Santa Barbara

Title: Global rough solutions to the critical generalized KdV equation.

Abstract: Following the $ I $-method scheme, we prove that the initial value problem(IVP) for the critical generalized KdV equation $ u_t + u_{xxx} + (u^5 )_x= 0 $ on the real line is globally well-posed in $ H^s (R) $, $ s > 3/5 $,with the appropriate smallness assumption on the initial data.

Short Bio: Luiz Gustavo Farah was an undergraduate at UFMG, Brazil. At 2008 he obtainedhis Ph.D. from IMPA, Brazil. Now he is working as a Postdoc in the Departmentof Mathematics of the University of California, Santa Barbara. His researchareas are Harmonic Analysis and Partial Differential Equations (PDE) and hehas published some papers concerning the local/global well-posedness andasymptotic behavior of the Cauchy problem for dispersive nonlinear PDE. His Ph.D. thesis was selected as the best Ph.D. thesis at IMPA in 2008 andreceived the Prize "Professor Carlos Teobaldo Gutierrez Vidalon (ICMC-USP)"as the best brazilian mathematical Ph.D thesis in 2008. His home page is http://sites.google.com/site/lgfarah/Home.

October 23

Erwin Suazo
Arizona State University

Title: Uniqueness in the cauchy problem for a family of parabolic equations in an unbounded domain.

Abstract: This will be an informal discussion. We will prove the uniqueness of the solution for the Cauchy initial value problem of a parabolic equation in an unbounded domain. Previously with S. K. Suslov and J. Vega we constructed an explicit solution of this Cauchy initial value problem.

October 30

Cristi Guevara
Arizona State University

Title: Concentration compactness for the focusing 2d quintic NLS.

Abstract:Using the concentration-compactness method and the rigidity arguments, wecharacterize the long time behavior of $ H^1 $ solutions to the focusing quinticNLS in $ \R^2 $. In particular, under a Mass-Energy threshold  we provideconditions for global existence, scattering, finite blowup and "weak" blowup.

November 6

Pavel Lushnikov
University of New Mexico

Title: Critical collapse in Keller-Segel equation

Abstract: Keller-Segel model describes macroscopic dynamics of bacterial colonies and biological cells. Keller-Segel model is a parabolic-elliptic system of partial differential equations for bacterial density and chemical concentration. Bacteria secret chemical which attract other bacteria so that they move along chemical gradient. If bacterial density exceed critical value then the density blows up in a finite time which corresponds to bacterial aggregation. Keller-Segel model has striking qualitative similarities with nonlinear Schrodinger equation including critical collapse in two dimensions and supercritical in three dimensions. The self-similar solutions near blow up point are studied in critical case together with time dependence of these solutions and their stability. Critical collapse has logarithmic corrections to $ (t_0-t)^{1/2} $ scaling law of self-similar solution. Regularization of collapse occurs if we take into account finite size of bacteria. We derive a nonlinear diffusion equation coupled with chemoattractant which depends on cellular volume fraction and provides regularization (prevention) of cellular density collapse.

November 13
special event
 Event Flyer 1:45pm

Vladimir Zakharov
Regent's Professor, Univ of Arizona

Distinguished Lecture
Title: Free Surface Hydrodynamics in Conformal Variables

Abstract: We study the potential flow of deep 2-D fluid with free surface in the presence of gravity. The area $ -\infty<y>&lt;\eta(x, t) $ filled with fluid is conformally mapped onto the lower half-plane $ v $<$ 0 $ on "mathematical" plane $ w=u+i v $. If $ \Phi $ is the hydrodynamic potential, in Dyachenko variables $ R(w, t)=\frac{1}{z'},\,\,V(w, t)=i\frac{\partial\Phi}{\partial z} $, the Euler equations take an unusual but very elegant form. These equations are suitable both for analytical and numerical study. Analytical in the lower half-plane functions $ R(w,t) $ and $ V(w,t) $ have moving singularities in the upper half-plane. Zeros of $ R(w,t) $ in the upper half-plane are also important. We are interested in the "robust" singularities only, which preserve their type with time, and can formulate the following rigorous analytical results:

$ \quad $ 1. Poles of $ R,\,V $ are not robust; they immediately turn to cuts.

$ \quad $ 2. Cuts are robust if they end with simple root-square type branch points

 $ R=a(t)\,\left(w -\lambda (t)\right)^{1/2} $, $ V=V_0(t)+b(t)\left(w -\lambda(t)\right)^{1/2} $.

$ \quad $ 3. Zeros of $ R(w, t) $ are robust; they generate motion constants. In the vicinity of the moving zero $ w=a_n(t) $,

$$ R=R_n(w-a_n(t)),\quad V=V_n(t), $$

where

$$ \frac{d R_n}{dt}=0,\quad \frac{dV_n}{dt}=-g,\quad V=-gt+V_n^{(0)}. $$

Each moving root $ a_n(t) $ generates two complex motion constants $ R_n ,\,V_n^{(0)} $.

$ \quad $ 4. If the cuts are narrow, initial integro-differential equations turn to a system of PDE's. In absence of gravity this system is integrable.

In spite of the progress, central questions of the theory are still unanswered:

$ \quad $ A. Can singularity of the surface occur in a framework of exact equations in a finite time?

$ \quad $ B. Are Euler equations for potential flow of deep fluid with free surface an integrable system?

The most plausible answer on both questions is positive.

November 13
special event

Justin Holmer
Brown University

Title: Various blow-up geometries for the nonlinear Schroedinger equation

Abstract: We begin by surveying basic structural properties of the nonlinear Schroedinger equation in dimensions one, two, and three. Specifically, we mention ground state solitons, complete integrability and multisolitons in one dimension (discovered by Zakharov-Shabat (1972)), and sufficient conditions for finite-time blow-up in two and three dimensions. We describe the Hamiltonian formulation and discuss the relation between symmetries and conservation laws. In the last part of the talk, we apply these concepts to give an interpretation of the derivation of blow-up dynamics in two dimensions obtained recently by Merle-Raphael (2001-2005) following the heuristic and numerical studies by Landman-Papanicolaou-Sulem-Sulem (1988) and Dyachenko-Newell-Pushkarev-Zakharov (1992). We also discuss our construction of a solution to the three dimensional equation blowing up on a circle. This is joint work with Svetlana Roudenko and is based on a result of Raphael (2006).

November 13
special event

Pavel Lushnikov
Brown University

Title: Finite time singularities: from individual collapses to collapse turbulence

Abstract: Many nonlinear systems have a striking phenomenon of spontaneous formation of singularities in a finite time (blow up). Blow up is often accompanied by a dramatic contraction of the spatial extent of solution, which has been called collapse since the pioneering work of Vladimir Zakharov in 1972. Near singularity point there is usually a qualitative change in underlying nonlinear phenomena, reduced models loose their applicability and other mechanisms become important such as inelastic collisions in the Bose-Einstein condensate; optical breakdown and dissipation in nonlinear optical media and plasma, wave breaking in hydrodynamics. Collapses occur in many physical and biological systems including a nonlinear Schroedinger equation (NLS), a Kadomtsev-Petviashvili equation with higher order nonlinearity, a Keller-Segel equation and many others. We will focus on NLS with dissipation and forcing in critical dimension. Without both linear and nonlinear dissipation NLS results in a finite-time singularity (collapse) for any initial conditions. Dissipation ensures collapse regularization. If dissipation is small then multiple near-singular collapses are randomly distributed in space and time forming collapse turbulence. Collapses are responsible for non-Gaussian tails in the probability distribution function of amplitude fluctuations which makes turbulence strong. Power law of non-Gaussian tails is obtained for strong NLS turbulence.

December 4

Erwin Suazo
ASU

Title: The Schroedinger Equation with time dependent potential (cont.)
 

 

 

 
 

 

 

 
 

 

 

 
 

 

 

 
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