DEPARTMENT OF MATHEMATICS


Math Home > Research > Seminars > PDE Seminar >	[ Search \| Contact \| Help! ]

PDE/Appl. Math. Seminar, Abstracts Spring 2005

(Feb. 3) Konstantina Trivisa On a Multidimensional Model for the Dynamic Combustion of Compresssible Reacting Fluids

Abstract: In this work we present a multidimensional model for the dynamic combustion of compressible reacting fluids formulated by the Navier Stokes equations in Euler coordinates. For the chemical model we consider a one way irreversible chemical reaction governed by the Arrhenius kinetics. The existence of globally defined weak solutions of the Navier-Stokes equations for compressible reacting fluids is established by using weak convergence methods, compactness and interpolation arguments in the spirit of Feireisl and P.L. Lions.

(Feb. 10) Georg Dolzmann A 2D compressible membrane theory as a Gamma-limit of a nonlinear elasticity model for incompressible membranes in 3D

We derive a two-dimensional compressible elasticity model for thin elastic sheets as a Gamma-limit of a fully three-dimensional incompressible theory. The energy density of the reduced problem is obtained in two steps: first one optimizes locally over out-of-plane deformations, then one passes to the quasiconvex envelope of the resulting energy density. This work extends the results by LeDret and Raoult on smooth and finite-valued energies to the case incompressible materials. The main difficulty in this extension is the construction of a recovery sequence which satisfies the nonlinear constraint of incompressibility pointwise everywhere.
This is joint work with Sergio Conti.

(Feb. 24) Cleopatra Christoforou Hyperbolic Systems of Balance Laws via Vanishing Viscosity

The aim is to construct solutions of hyperbolic systems of balance laws with dissipative source terms as limits of solutions of parabolic systems with viscosity tending to zero. The analysis of the vanishing viscosity method of Bianchini and Bressan [BiB] is extended to this class of systems. Because of the presence of the dissipative source terms, supplementary Lyapunov functionals are constructed and additional techniques are employed to those already devised in [BiB]. Moreover, an exponential decay of the total variation of vanishing viscocity approximations is established.

(March 3) Fumioki Asakura Steady Flows in the Laval Nozzle

A flow through the nozzle is modeled as a 1-d steady, isentropic flow. The Laval nozzle consists of a converging entry section, a throat and a diverging exhaust section, and used to accelerate subsonic flow into supersonic flow. We may assume the pressure at the entrance is kept constant, say p_0, which is realized by attaching a sufficiently large chamber at the entrance. The pressure p_1 at the exit will be varied. If p_1 = p_0, the flow in the nozzle is at rest. If p_1 is made slightly lower than p_0, then a flow with low speed arises. This subsonic flow accelerate in the converging section and decelerate in the diverging section. As p_1 reduces more and more, finally, the subsonic flow accelerates into the sonic speed at the throat but still subsonic elsewhere; this is called the choking. We can show that the mass of flow passing through the throat is maximum in this flow. If p_1 reduces still more, the flow accelerates into supersonic flow in the diverging section and a standing shock wave appears there. However the mass of flow is unchanged. Finally, the flow is smooth with steadily decreasing pressure and increasing speed, and sonic at the throat; this is considered the ideal nozzle flow. In this talk, we provide mathematical descriptions of the above phenomena for general flows which do not necessarily obey the gamma law. Moreover, we study the bifurcation of the solution at the throat and the geometry of the Hugoniot curve for the standing shock waves.

(March 10) Min Kang Interface growth models in max-plus perspective

We present a general class of growth models generated by linear operators in max-plus/min-plus algebra. The property enables the immediate derivation of the microscopic Hopf-Lax formula which leads to a law of large numbers for the interface under Euler scaling. Several well-known examples will be given such as ballistic deposition and interface associated with totally asymmetric exclusion processes.

(March 31) Giovanni Leoni Necessary and Sufficient Conditions for the Chain Rule in Sobolev Spaces

In this talk we will present necessary and sufficient conditions for the validity of the classical chain rule in Sobolev spaces and in the space of functions of bounded variation.

(April 7) Jon Wilkening Stress-driven grain boundary diffusion: modeling, analysis and numerical methods

Microchips often fail when the metallic interconnects between transistors and diodes on the chip degrade due to extremely high current densities. The physics of this process is quite interesting; it is a non-local moving interface problem involving elastic deformation and diffusion. Stress singularities can develop which make boundary conditions difficult to understand and numerical simulation difficult to implement reliably. After describing the model, I will outline our recent proof of well-posedness, which uses techniques from semigroup theory and requires an analysis of a type of Dirichlet to Neumann map involving the equations of elasticity. I will also briefly describe my recent work on computing stable asymptotics for singularities of Agmon-Douglis-Nirenberg elliptic systems near corners and interface junctions, and show how to adjoin these singular functions to the finite element basis to accurately and efficiently resolve stress singularities without mesh refinement.

(April 21) Donatella Donatelli Hyperbolic to Parabolic Relaxation and 3D Navier Stokes Equations.

We present some result concerning the relaxation of semilinear hyperbolic systems to parabolic systems. Some applications to fluid dynamics and chemotaxis will be shown.

(April 28) Eun Heui Kim Subsonic solutions for self-similar transonic problems in 2-dimensional conservation laws

We discuss existence and regularity results for subsonic solutions for transonic problems in 2-dimensional conservation laws. We also discuss model problems such as the Unsteady transonic small disturbance system, the nonlinear wave system and the potential flow.

(May 5) Yann Brenier Derivation of particle, string and membrane motions from Born-Infeld electromagnetism

We derive classical particle, string and membrane motion equations from a rigorous asymptotic analysis of the Born-Infeld nonlinear electromagnetic theory. The Born-Infeld model is a modification of the classical Maxwell equations, designed in 1934 to cutoff, in a nonlinear fashion, the infinite electrostatic field generated by point charged particles, at a scale of order 10 to the -15 meters. Quickly shadowed by the success of Quantum Electrodynamics in the 40', this model has known a recent revival through the concept of Dirichlet-Branes in String Theory. Our asymptotic analysis starts by adding to the Born-Infeld equations the corresponding energy-momentum conservation laws and write the resulting system as a non-conservative symmetric system of ten first-order evolution PDEs. (Surprisingly enough, the extended system enjoys Galilean invariance, just like classical Fluid Mechanics.) Then, we show that four rescaled versions of the system have smooth solutions existing in the time interval where the corresponding limit problems have smooth solutions. These limits respectively describe the classical linear Maxwell equations (for weak fields), string motions (for large magnetic fields and moderate electric fields), particle motions (for large electromagnetic fields) and membrane motions. This is partly a joint work with Wen-An Yong, from Heidelberg.

(May 6) Yann Brenier String integration of some MHD equations

We first review the link between strings and some Magnetohydrodynamics equations. Typical examples are the Born-Infeld system, the Chaplygin gas equations and the shallow water MHD model. They arise in Physics at very different (from subatomic to cosmologic) scales. These models can be exactly integrated in one space dimension by solving the 1D wave equation and using the d'Alembert formula. We show how an elementary "string integrator" can be used to solve these MHD equations through dimensional splitting. A good control of the energy conservation is needed due to the repeted use of Lagrangian to Eulerian grid projections. Numerical simulations A good control of the energy conservation is needed due to the repeted use of Lagrangian to Eulerian grid projections. Numerical simulations in 1 and 2 dimensions will be shown.

Last modified: Fri Apr 22 16:07:56 EDT 2005