Hyperbolic Conservation Laws and Kinetic Equations: Theory, Computation, and Applications - IMS

Hyperbolic Conservation Laws and Kinetic Equations: Theory, Computation, and Applications
(01 Nov - 19 Dec 2010)

~ Abstracts ~

On the speed of approach to equilibrium for a collision less gas
Kazuo Aoki, Kyoto University, Japan

We consider a collision less (or highly rarefied) gas enclosed in a closed domain bounded by a diffusely reflecting wall with a uniform temperature. From an initial condition, the gas evolves and approaches the equilibrium state at rest as time goes on, because of the thermalizing effect of the diffuse reflection condition.
(i) We investigate the rate of approach to equilibrium numerically and demonstrate that the approach is slow and proportional to an inverse power of time. This is a joint work with T. Tsuji and F. Golse.
(ii) We give some theoretical considerations on the result obtained numerically. This is a joint work with F. Golse.

Entropy and entropy dissipation for the critical mass Keller-Segel model
Eric Carlen, Rutgers University, USA

We investigate the long time behavior of the critical mass Keller-Segel equation, which has a one parameter family of steady-state solutions whose second moment is not bounded. The equation is gradient flow in the 2-Wasserstein metric for a non-displacement convex functional. However, it has a second Lyapunov function that is displacement convex. Using the interplay between the two Lyapunov functionals, we determine basins of attraction for the steady state solutions. The proof relies heavily on mass transportation techniques, and has a number of novel features. For instance, the second Lyapunov functional, has a dissipation functional that is a difference of two terms, neither of which need to be small when the dissipation is small, and the level sets of this dissipation functional are not compact. We introduce a strategy of controlled concentration to deal with these issue, and then use the regularity obtained from the entropy-entropy dissipation inequality, and then an interplay between the two Lyapunov functionals to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards the steady states. This is joint work with A. Blanchet and J.a. Carrillo.

Boundary singularity for thermal transpiration problem of the linearized Boltzmann equation
I-Kun Chen, Academia Sinica, Taiwan

We consider the thermal transpiration problem in the kinetic theory, which can be modeled by the linearized Boltzmann equation. It is well-known through asymptotic expansions and computations that there is a logarithmic singularity for the fluid velocity around the solid boundary. The goal of this paper is to confirm this basic phenomenon in the kinetic theory through analysis for sufficiently large Knudsen number. We use an iterated scheme, with the "gain" part of the collision operator as a source. The scheme yields an explicit leading term. The remaining converging terms are estimated through a refined pointwise estimate and Maxwellian upper bound for the gain part. Our analysis is motivated by the previous studies of asymptotic and computational analysis. Numerical data supporting the analysis are also provided.

Traveling wave solutions to the 3-species Lotka-Volterra competition-diffusion system
Chiun-Chuan Chen, National Taiwan University, Taiwan

In general it is difficult to find traveling wave solutions for 3-species competition system. Under suitable assumptions on the parameters of the 3-species Lotka-Volterra competition-diffusion system, we show that a 3-species wave can bifurcate from two 2-species waves which connect to a common equilibrium. The three components of the 3-species wave obtained are positive and have the pro files that one is a front, one is a back, and the third one is a pulse in the middle of the previous two with a large part near the equilibrium state.

Vanishing viscosity limit for nonlinear conservation laws
Gui-Qiang Chen, Oxford University, UK

The vanishing viscosity limit is one of the most classical, fundamental problems in the theory of nonlinear conservation laws. In particular, the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow has been a longstanding open problem. In this talk, we will discuss some of old and recent developments in the study of this issue.

Admissibility of shocks revisited
Constantine Dafermos, Brown University, USA

Clarifying the question of shock admissibility was the earliest important contribution of Tai-Ping Liu to the theory of hyperbolic conservation laws. The lecture will discuss this issue in the light of old and recent developments.

Existence of algebraic vortex spirals
Volker Elling, University of Michigan, USA

In fluid dynamics, vortex sheets -- curves across which the tangential flow velocity jumps, while pressure, density and normal velocity are continuous -- occur naturally in many important problems, for example from the trailing edge of wings in accelerating aircraft, when several interacting shock waves meet in some types of Mach reflection, or in buoyant plumes.

Vortex sheets have a tendency to roll up into spirals, whose asymptotic (and sometimes exact) behaviour has an x ~ t^m self-similar scaling. While logarithmic spiral solutions of the corresponding Birkhoff-Rott equation have been known since the 1920s, no existence proof was previously known for the more relevant algebraic spirals. I will discuss a proof in the case of sufficiently many symmetric branches and outline ongoing generalization to single-branched spirals.

Recently, we found several intereting systems with non-symmetric relaxation which have different decay structure. In this talk, we discuss these examples and report the recent progress on the stability theory for a class of symmetric hyperbolic systems.

A part of this talk is based on a joint work with Y. Ueda and R. Duan.

2-D Riemann problems in gas dynamics and their simulation by the GRP scheme
Jiequan Li, Beijing Normal University, China

Two-dimensional (2-D) Riemann problems for compressible fluid flows assume the simplest initial state but provide the most fundamental wave configurations, including the reflection of oblique shocks and vortex-shock interaction etc. In this talk I will show many fascinating pictures, based on 2-D Riemann solutions, to disclose the mysteries of compressible fluid world both through analytical tools (in the form of mathematical theorems) and computational techniques (in the form of simulations). The simulations are obtained using the generalized Riemann problem (GRP) scheme we developed in recent years. The scheme is equipped with the most accurate solver in the construction of numerical fluxes by a way of tracking singularities analytically and keeping entropy exactly computed.

Invariant manifolds for stationary Boltzmann equation
Tai-Ping Liu, Academia Sinica, Taiwan

With Shih-Hsien Yu, we have introduced the Green's function approach for the study of Boltzmann equation in the kinetic theory for gases. The purpose is to build up a more quantitative theory so as to be able to study some physical phenomena. Of particular importance is the boundary effects, a subject the kinetic theory can model much better than the classical fluid dynamics. In this talk we will describe the bifurcation phenomena for the Boltzmann boundary layers. This is an application of the newly developed theory for the invariant manifolds of the stationary Boltzmann equation. The construction of the invariant manifolds is done through time-asymptotic analysis, making essential use of the pointwise estimates of the Green's functions.

Analysis of Ginzburg Landau type approximations for nematic fluid systems
Pierangelo Marcati, University of L'Aquila, Italy

We present the analysis via the cancellation concentration argument. We show the convergence and the existence of at most finitely many singularities in the limit. (joint with D. Donatelli and S. Spirito)

Ghost effect by curvature
Rossana Marra, University of Rome Tor Vergata, Italy

I consider the Boltzmann equation for a gas between two rotating coaxial cylinders, and we study the behavior of the gas in the limit in which the Knudsen number and the inverse of radius (the curvature) of the inner cylinder tend to zero simultaneously, keeping the difference of the radii of the two cylinders fixed. The Mach number goes to zero as α, α < 1. The limiting behavior is described by a modification of the equations for the plane Couette flow. There is a bifurcation phenomenon from a linear profile of the velocity field due to the new terms in the equations. I will discuss in particular the 1d stationary case.

Quantitative uniform in time mean-field limit and a solution to a problem of Kac
Clement Mouhot, University of Cambridge, UK

This talk is devoted to a joint work with Stephane Mischler about the mean-field limit for systems of indistinguable particles undergoing collision processes. As proposed by Kac (1956) this limit can be reduced to a property of chaos propagation. We (1) prove and quantify this property for Boltzmann collision processes with unbounded collision rates (hard spheres or long-range interactions), (2) prove and quantify this property uniformly in time. Point (1) yields quantitative mean-field limit for long-range interactions and hard spheres (the latter case improves on the qualitative results of Sznitman (1984)). Point (2) yields the first uniform in time chaos propagation results for Boltzmann collision processes. With additional work this allows to give a solution to the question raised by Kac of relating the long-time behavior of a particle system with the one of its mean-field limit. Our results are based on a new method which reduces the question of chaos propagation to the one of proving a purely functional estimate on some generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting non-linear equation (stability estimates).

Asymptotic stability of boundary layer to the Euler-Poisson equation in plasma physics
Shinya Nishibata, Tokyo Institute of Technology, Japan

The main concern of the present talk is mathematical analyses on a boundary layer around a surface of a material with which plasma contacts. The layer, called a sheath in plasma physics, has a larger density of positive ions than that of electrons. The Bohm criterion for formation of the sheath requires that ion velocity should be much faster than sound speed. This physical phenomena is studied in the Euler-Poisson equations, describing behavior of ionized gas. We show that the sheath is regarded as a planar stationary solution in multi-dimensional half space. Precisely, under the Bohm sheath criterion, we show that the existence of the stationary solution and it is time asymptotically stable. Moreover we obtain a convergence rate of the solution towards the stationary solution.

The present result is obtained through the joint research with Mr. Masasi Ohnawa and Dr. Masahiro Suzuki.

Heat convection problems of compressible fluids
Takaaki Nishida, Waseda University, Japan

Heat convection problems are naturally formulated by the full system of equations for the compressible, viscous and heat conducting fluids. However the analysis of the system is rather difficult for the pattern formations or bifurcation problems. Thus Oberbeck-Boussinesq equation is used and investigated as an approximation to explain pattern formations and bifurcations. We clarify difficulties of the full system to explain pattern formations or bifurcations. We give some ideas to overcome those difficulties and propose a justification procedure to obtain the Oberbeck-Boussinesq system from the full system.

Large time behavior of collisionless plasma
Stephen Pankavich, United States Naval Academy, USA

The motion of a collisionless plasma, a high-temperature, low-density, ionized gas, is described by the Vlasov-Maxwell equations. In the presence of large velocities, relativistic corrections are meaningful, and when magnetic effects are neglected this formally becomes the relativistic Vlasov-Poisson system. Similarly, if one takes the classical limit as the speed of light tends to infinity, one obtains the classical Vlasov-Poisson system. We study the long-time dynamics of these systems of PDE and contrast the behavior when considering the cases of classical versus relativistic velocities and the monocharged (i.e., single species of ion) versus neutral plasma situations.

Kinetic models of chemotaxis and traveling bands
Benoit Perthame, Pierre and Marie Curie University, France

The question of explaining patterns in cell colonies is old and sometimes called 'socio-biology'. Many PDE models have been proposed in this field, in particular semilinear parabolic and Fokker-Planck equations.

Kinetic models of chemotaxis have been used since the 80s, based on experimental observation or the run and tumble movement of bacteria. They are useful to include informations at the individual level through the tumbling kernel. They also allow to derive macroscopic models (at the population scale) as the famous Keller-Segel system.

We will explain how the microscopic behavior of E. coli and its way to modulate the runs, gives rise to the Flux Limited Keller-Segel equation in the diffusion limit. In opposition to the traditional Keller-Segel system, this new model can sustain robust traveling bands as observed in the famous experiment of Adler.

This work is a collaboration with V. Calvez, N. Bournaveas, A. Buguin, J. Saragosti and P. Silberzan.

The Cauchy problem for the 3-D Vlasov-Poisson system with point charges
Mario Pulvirenti, Sapienza - Universita di Roma, Italy

I discuss the global existence and uniqueness of the solution to the three-dimensional Vlasov-Poisson system in presence of point charges, in case of repulsive interaction. I would also mention recent two-dimensional results in the more physical interesting attractive case.

Conical waves in 3-D Riemann problems. The case of the Chaplygin gas
Denis Serre, Unité de Mathématiques Pures et Appliquées, France

When solving a Riemann problem in space dimension D, one uses the solutions of several Riemann problems in dimensions from 1 to D-1. However, new patterns do occur, which do not exist in lower dimension. In 3-D, an important one is called Conical Wave. It is present in the wake of three interacting planar shocks. We describe the mathematical problem attached to the CW, and we solve it in the case of a Chaplygin gas.

Symmetry of the linearized Boltzmann equation: A theory from steady to unsteady problem
Shigeru Takata, Kyoto University, Japan

Relations between different problems described by the linearized Boltzmann equation (LBE) are discussed on the basis of the classical symmetry of LBE and kinetic boundary condition. The theory is developed for the system of arbitrary Knudsen number (Kn) from the view point of the Green function. After showing the main idea of the approach to steady problems for bounded and unbounded domains, its extension to unsteady problems in a fixed bounded domain is discussed. In particular, in the latter, recovery of Kubo formula for the viscosity and thermal conductivity in the fluctuation-dissipation theorem and its extension to the systems of arbitrary Kn will be presented.

Mixed-type and degenerate equations and steady continuous transonic flows in nozzles
Zhouping Xin, The Chinese University of Hong Kong, Hong Kong

In this talk, I will report some of the recent progress on the studies of steady compressible irrotational flows in a class of two-dimensional nozzles with variable sections. We first present a existence result on the continuous subsonic-sonic flows in a converging nozzle with given mass flux and flow angle on the inlet of the nozzle whose sonic curve is a free boundary where the acceleration blows up. Then we will discuss the existence of smooth transonic flows in a class of smooth de Laval nozzles.

Well-posedness and qualitative properties for Boltzmann equation without angular cutoff
Tong Yang, City University of Hong Kong, Hong Kong

It is known that the singularity in the non-cutoff cross-section of the Boltzmann collision operator leads to the gain of regularity in the velocity variable. By defining and analyzing a new non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a precise coercive estimate for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state where the global existence of classical solution is established in a general setting. With some essential estimates on the collision operators, the proof is based on the energy method through macro-micro decomposition. Furthermore, we study the qualitative properties of solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. The key step to obtain the regularizing effect is a generalized version of the uncertainty principle together with a theory of pseudo-differential calculus on non-linear collision operators.

In this talk, the part on existence and convergence rates will be given and other part will be given by R. Alexandre's talk.

The results of this talk are from a series of joint works with R. Alexandre, Y. Morimoto, S. Ukai and C.J. Xu. And the research was partially supported by GRF of Hong Kong 103109.

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