BU/Brown PDE Seminar

Each month, a joint PDE seminar between the Departments of Mathematics at Boston University and Brown University and the Division of Applied Mathematics at Brown University will be held. The schedule and location for these events can be found below. To view abstracts, move your mouse over a title. Please visit the BU/Brown PDE Seminar archive page for past events in this seminar series.

Spring 2011

Wednesday, 9 February 2011, 3:30-5:30pm in Room 110 (182 George Street) at Brown

Speakers	Titles
Govind Menon (Brown University)	Complete integrability of shock clusteringRavi Srinivasan and I found a Lax pair that described a kinetic theory for shock statistics in scalar conservation laws with random initial data. More recent work shows that the kinetic equations are completely integrable. Precisely: they have a Hamiltonian structure, can be linearized via a matrix factorization, and solved by an inverse scattering theory.
Margaret Beck (Boston University)	A mechanism for creating strange attractors in infinite-dimensional dynamical systemsIn systems that posses stable limit cycles and a property known as "shear," it is possible to apply time-periodic forcing and create a strange attractor, on which the dynamics can be characterized by an SRB measure. The resulting chaos is sustained (ie is not transient) and is phyically observable. This has been previously analyzed in finite dimensions, and in infinite dimensions for limit cycles that are created by Hopf bifurcations. We show how to extend this to more general limit cycles in infinite dimensions. This is join work with Will Ott.

Speakers

Titles

Govind Menon (Brown University)

Complete integrability of shock clusteringRavi Srinivasan and I found a Lax pair that described a kinetic theory for shock statistics in scalar conservation laws with random initial data. More recent work shows that the kinetic equations are completely integrable. Precisely: they have a Hamiltonian structure, can be linearized via a matrix factorization, and solved by an inverse scattering theory.

Margaret Beck (Boston University)

A mechanism for creating strange attractors in infinite-dimensional dynamical systemsIn systems that posses stable limit cycles and a property known as "shear," it is possible to apply time-periodic forcing and create a strange attractor, on which the dynamics can be characterized by an SRB measure. The resulting chaos is sustained (ie is not transient) and is phyically observable. This has been previously analyzed in finite dimensions, and in infinite dimensions for limit cycles that are created by Hopf bifurcations. We show how to extend this to more general limit cycles in infinite dimensions. This is join work with Will Ott.

Wednesday, 2 March 2011, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University

Speakers	Titles
David Ambrose (Drexel University)	Two problems in interfacial fluid dynamicsFor many problems in interfacial fluid dynamics, such as the water wave or the vortex sheet with surface tension, existence of solutions for initial value problems and existence of traveling solutions has been established in recent years. In this talk, we will look at two other existence problems. First, we will explore the question of existence of weak solutions for interfacial Navier-Stokes flows, in the presence of surface tension. One signicant challenge in this problem is finding a weak formulation of the surface tension force, which for strong solutions is supported only at the interface and is given in terms of the curvature of the interface. Second, we will discuss existence of time-periodic interfacial flows. This includes joint work with Milton Lopes Filho, Helena Nussenzveig Lopes, Walter Strauss, and Jon Wilkening.
Peter Lax (Courant Institute / NYU)	Hyperbolic partial differential equations and degenerate matricesThe occurrence of multiple characteristics for hyperbolic systems gives rise to singularities in solutions. This leads to topological and algebraic questions about degenerate symmetric matrices and their discriminants.

Speakers

Titles

David Ambrose (Drexel University)

Two problems in interfacial fluid dynamicsFor many problems in interfacial fluid dynamics, such as the water wave or the vortex sheet with surface tension, existence of solutions for initial value problems and existence of traveling solutions has been established in recent years. In this talk, we will look at two other existence problems. First, we will explore the question of existence of weak solutions for interfacial Navier-Stokes flows, in the presence of surface tension. One signicant challenge in this problem is finding a weak formulation of the surface tension force, which for strong solutions is supported only at the interface and is given in terms of the curvature of the interface. Second, we will discuss existence of time-periodic interfacial flows. This includes joint work with Milton Lopes Filho, Helena Nussenzveig Lopes, Walter Strauss, and Jon Wilkening.

Peter Lax (Courant Institute / NYU)

Hyperbolic partial differential equations and degenerate matricesThe occurrence of multiple characteristics for hyperbolic systems gives rise to singularities in solutions. This leads to topological and algebraic questions about degenerate symmetric matrices and their discriminants.

Wednesday, 6 April 2011, 3:30-5:30pm in Room 110 (182 George Street) at Brown

Speakers	Titles
Benoit Pausader (Brown University)	The energy-critical nonlinear Schrödinger equation in different geometries.We prove global existence of the solutions of the energy-critical nonlinear Schrödinger equation in the hyperbolic space and in the three-dimensional Torus. The key, besides developping the necessary harmonic analysis is to consider the (Euclidean) scaling limits. This is a joint work with G. Staffilani and A. Ionescu.
Sanjeeva Balasuriya (Connecticut College)	Abnormal Melnikov theoryMelnikov theory usually relates to the normal displacement of a homoclinic manifold under a perturbation. Here, I introduce a method of determining the tangential displacement of a manifold under a time-aperiodic perturbation in a non-area-preserving instance, while improving on existing normal displacement results. I will also present a class of non-standard applications of Melnikov methods, viz., determining the wavespeed and waveprofile perturbation to travelling waves in reaction-diffusion systems arising in ecology and combustion.

Speakers

Titles

Benoit Pausader (Brown University)

The energy-critical nonlinear Schrödinger equation in different geometries.We prove global existence of the solutions of the energy-critical nonlinear Schrödinger equation in the hyperbolic space and in the three-dimensional Torus. The key, besides developping the necessary harmonic analysis is to consider the (Euclidean) scaling limits. This is a joint work with G. Staffilani and A. Ionescu.

Sanjeeva Balasuriya (Connecticut College)

Abnormal Melnikov theoryMelnikov theory usually relates to the normal displacement of a homoclinic manifold under a perturbation. Here, I introduce a method of determining the tangential displacement of a manifold under a time-aperiodic perturbation in a non-area-preserving instance, while improving on existing normal displacement results. I will also present a class of non-standard applications of Melnikov methods, viz., determining the wavespeed and waveprofile perturbation to travelling waves in reaction-diffusion systems arising in ecology and combustion.

Wednesday, 27 April 2011, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University

Speakers	Titles
Irving Epstein (Brandeis University)	Reaction-Diffusion Patterns in Structured MediaI will look at two- and three-dimensional pattern formation in a reverse microemulsion consisting of nanometer diameter droplets of water containing the reactants of the Belousov-Zhabotinsky oscillating chemical reaction dispersed in oil. This system allows the physical structure (size, spacing) of the droplets and their chemical composition to be controlled independently, enabling one to generate a remarkable variety of stationary and moving patterns, including Turing structures, ordinary and antispirals, packet waves and spatiotemporal chaos. I will show examples of patterns and discuss two alternative mechanisms by which they may arise, one involving diffusion of different species at very different rates, the other via cross diffusion, whereby gradients in the concentration of one species influence the rate of diffusion of another species. I will also discuss pattern formation in one and two-dimensional arrays of micrometer diameter aqueous BZ droplets in oil prepared using microfludic techniques.
Simon Tavener (Colorado State University)	A posteriori analysis and adaptive error control for operator decomposition approaches to coupled physics problemsA posteriori error estimation and adaptive error control based on the formulation and solution of an adjoint problem are well established for problems involving a single type of physics. For systems that exhibit sufficiently complicated physics or a range of scales so that they severely challenge standard solution techniques, operator decomposition provides an attractive way to decompose the problem into components with relatively simple physics or into behaviors that occur over a modest range of scales. Operator decomposition creates additional challenges for adaptive error control since errors in one component may limit the accuracy in another component, yet a global adjoint solution is typically not available. In this talk, I will describe an a posteriori analysis of operator decomposition methods for coupled elliptic systems. This analysis takes into account the accuracy with which individual components are solved as well as the global effects of operator decomposition. The estimates obtained provide the means for adaptive error control. Extensions of these ideas that are currently under investigation include coupled elliptic-parabolic and coupled parabolic systems arising in studies of cardiac physiology. Time-dependent problems present additional challenges and I will describe a one approach we have developed which we call "block adaptivity".

Speakers

Titles

Irving Epstein (Brandeis University)

Reaction-Diffusion Patterns in Structured MediaI will look at two- and three-dimensional pattern formation in a reverse microemulsion consisting of nanometer diameter droplets of water containing the reactants of the Belousov-Zhabotinsky oscillating chemical reaction dispersed in oil. This system allows the physical structure (size, spacing) of the droplets and their chemical composition to be controlled independently, enabling one to generate a remarkable variety of stationary and moving patterns, including Turing structures, ordinary and antispirals, packet waves and spatiotemporal chaos. I will show examples of patterns and discuss two alternative mechanisms by which they may arise, one involving diffusion of different species at very different rates, the other via cross diffusion, whereby gradients in the concentration of one species influence the rate of diffusion of another species. I will also discuss pattern formation in one and two-dimensional arrays of micrometer diameter aqueous BZ droplets in oil prepared using microfludic techniques.

Simon Tavener (Colorado State University)

A posteriori analysis and adaptive error control for operator decomposition approaches to coupled physics problemsA posteriori error estimation and adaptive error control based on the formulation and solution of an adjoint problem are well established for problems involving a single type of physics. For systems that exhibit sufficiently complicated physics or a range of scales so that they severely challenge standard solution techniques, operator decomposition provides an attractive way to decompose the problem into components with relatively simple physics or into behaviors that occur over a modest range of scales. Operator decomposition creates additional challenges for adaptive error control since errors in one component may limit the accuracy in another component, yet a global adjoint solution is typically not available. In this talk, I will describe an a posteriori analysis of operator decomposition methods for coupled elliptic systems. This analysis takes into account the accuracy with which individual components are solved as well as the global effects of operator decomposition. The estimates obtained provide the means for adaptive error control. Extensions of these ideas that are currently under investigation include coupled elliptic-parabolic and coupled parabolic systems arising in studies of cardiac physiology. Time-dependent problems present additional challenges and I will describe a one approach we have developed which we call "block adaptivity".