「流体力学における現代的アプローチ」
Modern Approaches in Fluid Dynamics
平成25年度第7回講演会（通算第22回）
日時：  平成25年12月12日（木） 16:1018:10 
場所：  京都大学 桂キャンパスCクラスタ総合研究棟III（C3棟） 3階 b3n03室（航空宇宙工学専攻会議室） 
講演1：  Construction of BGK models from an entropy minimization principle 
Dr. Stéphane Brull (Institut de Mathématiques de Bordeaux, Université Bordeaux 1, France) 

要旨：  The aim of this talk is to present a new methodology to construct BGK models that are able to give good hydrodynamic coefficients up to NavierStokes. This derivation is based on a systematic procedure minimizing Boltzmann entropy under suitable moments constraints. We firstly obtain a new construction of the Ellipsoidal Statistical Model in the monoatomic and polyatomic setting. In a last part, I will explain how this technique can be applied to gas mixtures. 
講演2： 
Development of a GeneralPurpose Parallel FluxConserved Direct Simulation Monte Carlo Code (PDSC^{++}) Using an Unstructured Grid. 
Prof. JongShinn Wu (Department of Mechanical Engineering, National Chiao Tung University, Taiwan) 

要旨：  A new generalpurpose parallel 2D/3D DSMC (named PDSC^{++}, hereafter) based on the C++ language using a 2D, 2Daxisymmetric or 3D hybrid unstructured grid has been developed and validated. Several key features of the PDSC^{++} code are presented and discussed in the talk, including a variable timestep (VTS) scheme, a transient adaptive subcell (TAS) method, and parallel processing of the DSMC method. For the VTS scheme, the simulation time step, which is proportional to the weight of simulation particles, varies in each cell based on local mean free path. This leads to an efficient particle tracing algorithm on an unstructured grid, which enforces conservation of mass, momentum and energy. This results in great reduction of total simulation particles as compared to the constant timestep scheme. For the TAS method, dynamically adaptive numbers of subcells, based on the local mean free path or number of simulation particles, are imposed in each cell to ensure the average collision distance is less than the local mean free path. The results show that this TAS method coupled with the VTS scheme results in great reduction of computational time of DSMC while maintaining very high quality of collision between particles. For the parallel processing of DSMC method, a simple and efficient method which is termed as domain redecomposition (DRD) method is presented to improve the parallel performance of parallel DSMC simulation without resorting to dynamic domain decomposition. The results indicate that up to 123135 times of speedup can be reached using 192 processors for the large scale problems which is performed at the ALPS cluster of the National Center for HighPerformance Computing (NCHC), Taiwan. In addition, we also have demonstrated the powerful capability of the PDSC^{++} code by simulating a threedimensional problem with more than one billion simulation particles using 768 cores of ALPS. 
平成25年度第6回講演会（通算第21回）
日時：  平成25年10月4日（金） 16:1018:10 
場所：  京都大学 桂キャンパスCクラスタ総合研究棟III（C3棟） 3階 b3n03室（航空宇宙工学専攻会議室） 
講演1：  A CutCell Method for the Simulation of Gas Microflows 
Mr. Guillaume Dechriste (Université Bordeaux 1, France) 

要旨：  Current engineering developments make possible the conception of more and more complex Microelectromechanical systems (MEMS) and the simulation of rarefied gas micro flows in these devices is becoming a challenging issue. In this context, we propose a numerical method for the computation of BGK model of Boltzmann equation in a moving domain. The account of boundary motion within this framework has been treated by several ways for 1D problems, by the use of Lagrangian and semiLagrangian techniques, but also for 2D computations, thanks to immersed boundary method. Another standard approach for the simulation of incompressible viscous flows with moving boundaries is the cutcell technique. This talk is devoted to the presentation of an extension of this approach to deterministic simulation of rarefied gas flows. The cutcell technique consists in building an unstructured mesh based on a primary Cartesian mesh by cutting the cells that are travelled through by the boundary. Unlike immersed boundary method, the numerical scheme is conservative. Moreover this strategy avoids the use of remeshing approaches. Finally, the Cartesian mesh make the parallelization easy to implement. The method has been tested with both specular and diffuse boundary conditions and has been validated on several 1D and 2D problems. 
講演2：  Comparison of DSMC Chemistry and Vibrational Models Applied to Oxygen Shock Measurements 
Dr. Ingrid Wysong (Asian Office Aerospace Research and Development, Tokyo, Japan) 

要旨：  A validation study of three DSMC chemistry models, two recent and one standard, is presented. First the 2D geometry and numerical approach used to simulate the oxygen shock experiments is verified. Next, 2 different vibrational relaxation models are tested by comparison with data for the M=9.3 case where dissociation is small in the nonequilibrium region of the shock. Finally, the 3 DSMC chemistry model results are compared for the M=13.4 case where nonequilbrium dissociation (in the region where the vibrational temperature is greatly different from the rotational and translational temperature) is important. It is shown that the peak vibrational temperature is very sensitive to vibrational favoring in the chemistry model and that the vibrationallyfavored KSS model predicts the measured peak reasonably. 
平成25年度第5回講演会（通算第20回）
日時：  平成25年7月1日（月） 14:4015:40 
場所：  京都大学 桂キャンパスCクラスタ総合研究棟III（C3棟） 3階 b3n03室（航空宇宙工学専攻会議室） 
講演：  Geometrical acoustics (ray acoustics) in applications to aeroacoustics 
Dr. Takao Suzuki (Boeing, Seattle, USA) （鈴木崇夫 博士，音響・流体力学技術部門，ボーイング社） 

要旨：  Geometrical acoustics (i.e. ray acoustics) is a powerful tool providing insight into wave propagation in the highfrequency limit, where the acoustic wavelength is much shorter than the length scale of the medium (e.g. mean flow). By expanding the governing wave equation in terms of the frequency (or the wavenumber), the leadingorder terms become the eikonal equation solving the ray trajectories, and the second leadingorder terms the firstorder transport equation solving the amplitude field. In the presence of velocity and temperature gradients in the mean flow together with a nonslip wall, waves of various types are created due to refraction, reflection, diffraction, and other phenomena. In this talk, derivation of the raytracing equations is first reviewed, and then applications of geometrical acoustics in highspeed flows are introduced. In particular, an application to a jet screech problem (shock associated noise) is discussed in addition to other types of waves, such as channeled waves and diffracted waves in the vicinity of a boundary layer. 
平成25年度第4回講演会（通算第19回）
日時：  平成25年5月23日（木） 16:3017:30 
場所：  京都大学 桂キャンパスCクラスタ総合研究棟III（C3棟） 3階 b3n03室（航空宇宙工学専攻会議室） 
講演：  Boundary conditions for the Boltzmann equation on periodic rough walls 
Prof. Pierre Charrier (Université Bordeaux 1, France) 

要旨：  The derivation of boundary conditions for the Boltzmann equation on walls with nanoscale roughness is revisited. The boundary conditions are formally derived from a nanoscale kinetic model by classical tools of kinetic theory such as scaling and systematic asymptotic analysis. The associated scattering kernels satisfy the classical basic physical requirements. It is shown that this approach allows to obtain complex scattering patterns (with more than one maximum), related to the wall morphology. 
平成25年度第3回講演会（通算第18回）
日時：  平成25年4月11日（木） 16:3017:30 
場所：  京都大学 桂キャンパスCクラスタ総合研究棟III（C3棟） 3階 b3n03室（航空宇宙工学専攻会議室） 
講演：  Is there any physics missed by the NavierStokes equations? 
Dr. Demenico Giordano (European Space Agency & The Von Karman Institute for Fluid Dynamics, Belgium) 

要旨：  Since our student days, we fluid dynamicists become accustomed to the idea that the velocity field is the sole kinematic unknown to consider in flow fields. The internal angular momentum is systematically relegated to a rank of lower importance in the vast majority of the applications on the justifying assumption that torques distributed throughout the fluid are usually negligible. There are, of course, noticeable exceptions such as electricallypolarisable and magnetisable fluids; thus the concept of internal angular momentum has become by default associated with the presence of the electromagnetic field in a fluid that reacts to it through polarisation and magnetisation. But, is this truly always the case? The present study reflects on the bydefault assumption of discardable internal angular momentum and tries to unravel the implications of the presence of this physical variable on the fluiddynamics and thermodynamics equations. The (almost provocative) title descends from the fact that, when internal angular momentum is accounted for in the governing equations, linear irreversible thermodynamics produces phenomenological relations different from the traditional ones that close the fluiddynamics balance equations into the form known as NavierStokes’. Are these modified phenomenological relations leading to any new physics that is, therefore, missed by the NaierStokes equations? Are they able to resolve flowfield features that so far have proven outside the reach of the NavierStokes equations? These interesting questions, stimulating scientific curiosity and deserving inquiring minds’ attention, constitute the main themes of this study ! 
平成25年度第2回講演会（通算第17回）
日時：  平成25年3月21日（木） 16:2517:25 
場所：  京都大学 桂キャンパスCクラスタ総合研究棟III（C3棟） 3階 b3n03室（航空宇宙工学専攻会議室） 
講演：  Deterministic Conservative solvers for the Boltzmann equation 
Prof. Irene Gamba (Department of Mathematics, University of Texas at Austin, USA) 

要旨：  We will discuss the deterministic conservative spectral and FEM methods for both the nonlinear and the linearized Boltzmann equation. These methods uses the weak formulation of the collisional form and the Fourier structure of the collision operator to recast the collisional terms as a weighted convolution with precomputed weights, which contain all of the collision mechanism information, and a constrained optimization problem is solved to ensure preservation of the macroscopic moments. For the case of the nonlinear Boltzmann equation, we will present several examples including the calculations of mixtures, sudden heating simulations, as well as the asymptotic regime for grazing collisions limits with CoulombicRutherford potentials and the order of approximation to the solution of the Landau equation in terms of the Debye screening parameter. For the case of the linearized equation by means of FEM methods, we’ll show calculations of the spectral gap depending on the potential and angular cross section. This presentation is based on a series of works in collaboration with Ricardo Alonso, S. Harsha Tharkabhushanam, Jeff Haack, Alessandro Munafo, Thierry Magin and Chenglong Zhang. 
平成25年度第1回講演会（通算第16回）
日時：  平成25年3月8日（金） 15:3017:30 
場所：  京都大学 桂キャンパスCクラスタ総合研究棟III（C3棟 １階 講義室５） 
講演：  Extended Thermodynamics of Dense Gases 
Part I: Macroscopic Approach Prof. Masaru Sugiyama (Department of Mechanical Engineering, Nagoya Institute of Technology, Japan) Part II: Maximum Entropy Principle for Rarefied Polyatomic Gases Prof. Tommaso Ruggeri (Faculty of Engineering, University of Bologna, Italy) 

要旨：  The kinetic theory and Extended Thermodynamics (ET) are important theories for rarefied nonequilibrium gases. Nevertheless the weak point is that the range is limited mainly to monatomic gases. In this talk we want to present recent new approach to deduce hyperbolic system for dense gases not necessarily monatomic. In the first part of the talk we study extended thermodynamics of dense gases by adopting the system of field equations with a different hierarchy structure to that adopted in the previous works. It is the theory of 14 fields of mass density, velocity, temperature, viscous stress, dynamic pressure and heat flux. As a result, all the constitutive equations can be determined explicitly by the caloric and thermal equations of state as in the case of monatomic gases. It is shown that the rarefiedgas limit of the theory is consistent with the kinetic theory of gases.In the second part, we limit the result to the physically interesting case of rarefied polyatomic gases and we show a perfect coincidence between ET and the procedure of Maximum Entropy Principle. The main difference with respect to usual procedure is the existence of two hierarchies of macroscopic equations for moments of suitable distribution function, in which the internal energy of a molecule is taken into account. References: 1) I. Mueller and T. Ruggeri: Rational Extended Thermodynamics, 2^{nd }ed., Springer Tracts in Natural Philosophy 37, (1998) SpringerVerlag (New York). 2) G. Boillat and T. Ruggeri: Moment equations in the kinetic theory of gases and wave velocities, Continuum Mech. Thermodyn. 9 (1997) 205212. 3) T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama: Extended thermodynamics of dense gases, Continuum Mech. Thermodyn. 24 (2012) 271292. 4) T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama: Extended thermodynamics of real gases with dynamic pressure: An extension of Meixner theory, Phys. Lett. A376 (2012) 27992803. 5) T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama: Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics, Continuum Mech. Thermodyn. (2012) DOI 10.1007/s0016101202718. 6) M. Pavic, T. Ruggeri and S. Simic: Maximum entropy principle for rarefied polyatomic gases, Physica A (2012) (to be published). 
平成24年度第7回講演会（通算第15回）
日時：  平成24年12月26日（水） 15:0016:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  Eighteen Strikes 
Prof. ShihHsien Yu (Department of Mathematics, National University of Singapore, Singapore) 

要旨：  The LY Algorithm for half space problem is applied to solve the Lamb’s problem for linear elasticity, which is a fundamental problem in seismology. There are 16 independent Rayleigh surface waves. All of Them are explicitly constructed as classical solutions in the spacetime variable and the lame constants. The solution is so precise to predict how an earth quake behaves in an isotropic media. One simple consequence is that it will strike any point in the surface 18 times. 
平成24年度第6回講演会（通算第14回）
日時：  平成24年1２月2１日（金） 15:0016:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  自己推進運動をするミクロ物体とその流体力学的効果 
吉川研一 教授 (生命医科学部，同志社大学) 
平成24年度第5回講演会（通算第13回）
日時：  平成24年10月26日（金） 15:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演1：  A conservative scheme for solving coupled surfacebulk convectiondiffusion equations with an application to interfacial flows with soluble surfactant 
Prof. MingChih Lai (Department of Applied Mathematics, National Chiao Tung University, Taiwan) 

要旨：  Many physical problems arising in biological or material sciences involve solving partial differential equations in deformable interfaces or complex domains. For instance, the surfactant (an amphiphilic molecular) which usually favors the presence in the fluid interface may couple with the surfactant soluble in one of bulk domains through adsorption and desorption processes. Thus, it is important to accurately solve coupled surfacebulk convectiondiffusion equations especially when the interface is moving. In this talk, we present a new conservative scheme for solving this coupled surfacebulk concentration equations which the total surfactant mass is conserved in discrete sense. As an application, we extend our previous work to the interfacial flows with soluble surfactant. The effects of solubility of surfactant on drop deformations in a quiescent and shear flow are investigated in detail. 
講演2：  The Landau equation as the weakcoupling limit of interaction particle systems 
Prof. Mario Pulvirenti (Department of Mathematics, University of Rome ‘La Sapienza’, Italy) 

要旨：  In this talk we exploit the weakcoupling limit of a Hamiltonian particle system. We show, formally, that the Landau equation is the kinetic equation expected to be valid in this scaling limit. We also show that this is also true rigorously, up to the first order in the perturbative expansion. 
平成24年度第4回講演会（通算第12回）
日時：  平成24年10月2日（火） 16:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  Deviational Methods for Simulation of Phonon Transport 
Prof. Nicolas Hadjiconstantinou (Department of Mechanical Engineering, MIT, USA) 

要旨：  Heat tranfer in crystalline solids can be modeled by considering the transport of phonons, quasiparticles representing quantized energy states associated with lattice vibrations. In close analogy with the kinetic theory of gases, a kinetic description becomes particularly useful for mesoscopic scales that are comparable to the mean phonon mean free path but also sufficiently large to be intractable via molecular simulation. In this talk we will discuss Monte Carlo methods for solving the governing kinetic equation for such systems, namely the Boltzmann transport equation. Application of these methods to systems of current practical interest will also be presented. 
国際会議
Kinetic Theory and Related Fields: Theoretical and Numerical Approaches (2428, Sept. 2012)
プログラム表紙
平成24年度第3回講演会（通算第11回）
日時：  平成24年8月31日（金） 15:0016:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  Stability theory of polytropic gaseous stars 
Dr. Juhi Jang (Department of Mathematics, University of California, Riverside, USA) 

要旨：  I’ll discuss stability theory of LaneEmden equilibrium stars under EulerPoisson or NavierStokesPoisson system. A linear stability can be characterized by the adiabatic exponent. A nonlinear instability will be also discussed. 
平成24年度第2回講演会（通算第10回）
日時：  平成24年5月17日（木） 15:0017:30 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  Boundary Kernel for Dissipative Systems 
Part I: Dissipative Wave Equations Prof. TaiPing Liu (Academia Sinica, Taiwan & Stanford University, USA) Part II: Compressible NavierStokes Equations Prof. ShihHsien Yu (National University of Singapore, Singapore) 

要旨：  We will present a systematic approach to the study of the kernel for the boundary DirichletNeumann relations for dissipative partial differential equations. Our study exhibits interesting relation between the wave propagation along the boundary and the differentiation normal to the boundary. The key notions of LaplaceFourier Path and Boundary Determinant will be illustrated for the dissipative wave equations and the compressible NavierStokes equations. Our approach involves interesting algebraic calculations for the transformed variables and the inversion of the Fourier and Laplace transforms by the subtle complex analytic techniques. 
平成24年度第1回講演会（通算第9回）
日時：  平成24年3月6日（火） 16:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  On Bohmian and Wigner Measures 
Prof. Peter Markowich (DAMTP, Centre for Mathematical Sciences, University of Cambridge, UK) 

要旨：  We compare the Schrödinger, Wigner and Bohm picture of quantum mechanics and draw conclusions about the classical limit in these different formulations. 
平成23年度第８回講演会（通算第８回）
日時：  平成24年2月17日（金） 15:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演1：  Competing interactions in kinetic models for fluids 
Prof. Rossana Marra (Universita di Roma, “Tor Vergata”, Rome, Italy) 

要旨：  I will discuss kinetic models modeling systems of particles interacting through potentials which are repulsive and attractive, and the corresponding possible kinds of phase transition. In particular, I will present a kinetic model for the formation of microemulsion and discuss the microphase separation. 
講演2：  Solution to the Boltzmann equation with non isothermal boundary 
Prof. Raffaele Esposito (M&MOCS – Universita’ dell’Aquila, Cisterna di Latina, Italy) 

要旨：  We consider the Boltzmann equation in an arbitrary container of finite size, on whose walls we prescribe Maxwell diffuse reflection b.c. with non homogeneous temperature. Under the assumption that the variations of the wall temperature are sufficiently small we prove that there is a locally unique stationary solution continuous out of an explicit singular set which is asymptotically stable with exponential rate. A corollary of these results is a proof of non validity of the Fourier law. The proof of such results is based on new constructive and robust coercivity estimates for the linearized problem. 
平成23年度第7回講演会（通算第7回）
日時：  平成24年2月3日（金） 15:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演1：  Stability of Global Maxwellian solutions of the Boltzmann equation 
Prof. Claude Bardos (Laboratoire JacquesLouis Lions and University Denis Diderot, Paris, France) 

要旨：  Initial observation is the fact that global maxwellian provide “eternal ” solutions at the same time of the Boltzmann the compressible Euler and the compressible Navier Stokes equations. These solutions are determined by their macroscopic global moments including besides the classical one the angular momentum, the scalar momentum and the scalar inertial momentum.Then the stability (under perturbation of the initial data) of these solutions is studied continuing a program initiated by Kaniel and Shinbrot.This is a joint work in progress with Irene Gamba and David Levermore. 
講演2：  The meanfield limit for a regularized VlasovMaxwell dynamics 
Prof. François Golse (Centre de mathématiques Laurent Schwartz, Ecole Polytechnique, France) 
平成23年度第6回講演会（通算第6回）
日時：  平成24年1月25日（水） 16:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  Comparison between the IPL and VHS collisions models 
Dr. Robert Rubinstein (NASA Langley Research Center, Hampton, VA USA) 

要旨：  The success of the variable hard sphere (VHS) collision model, as a computationally convenient substitute for the inverse power law (IPL) collision model in DSMC calculations, is indisputable. But what is the reason for this success? We will attempt to quantify the differences between the IPL and VHS models by considering how the corresponding change of the collision integral changes the terms in the ChapmanEnskog and Grad expansions of the Boltzmann equation. Comparison with recent related work by Gallis and Torczynski will be given. 
平成23年度第5回講演会（通算第5回）
日時：  平成23年12月15日（木） 16:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  A new classical approach for linear PDE in half space. 
Prof. ShihHsien Yu (Department of Mathematics, National University of Singapore, Singapore) 

要旨：  In this talk, we will present a simple but effective way to study linear PDE in half space. 
平成23年度第4回講演会（通算第4回）
日時：  平成23年11月24日（木） 15:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演1：  Boundary singularity for thermal transpiration problem of the linearized Boltzmann equation 
Dr. IKun Chen (Institute of Mathematics, Academia Sinica, Taiwan) 

要旨：  We consider the thermal transpiration problem in the kinetic theory, which can be modeled by the linearized Boltzmann equation. It is wellknown 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. 
講演2：  Diffuse reflection boundary condition for kinetic models 
Prof. TaiPing Liu (Institute of Mathematics, Academia Sinica, Taiwan) 

要旨：  With HungWen Kuo and LiCheng Tsai, we study two problems on the equilibrating effects of the diffuse reflection boundary condition for the gas motion in a bounded domain. The first problem is to confirm analytically the conjecture of Kazuo Aoki on the free molecular flows with constant boundary temperature. The second is the study of the full Boltzmann equation with variable boundary temperature. For our analysis, we generalize and refine the stochastic formulation of the boundary effects initiated by ShihHsien Yu. 
平成23年度第3回講演会（通算第3回）
日時：  平成23年8月2日（火） 16:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  Coupling penalization and vortex methods for numerical simulation of incompressible flows 
Dr. Iraj Mortazavi (Département Matmeca, Ecole ENSEIRBMATMECA, Institut Polytechnique de Bordeaux, France) 

要旨：  The aim of this work is to couple vortex methods with the penalization methods in order to take advantage from both of them. This immersed boundary approach maintains the efficiency of vortex methods for high Reynolds numbers focusing the computational task on the rotational zones and avoids their lack on the noslip boundary conditions replacing the vortex sheet method by the penalization of obstacles. This method that is very appropriate for bluffbody flows is validated for the flow around different fitted or moving, two and three dimensional obstacles. 
平成23年度第2回講演会（通算第2回）
日時：  平成23年3月23日（水） 16:0018:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演1：  Construction of Green’s functions for the Boltzmann equations 
Prof. ShihHsien Yu (Department of Mathematics, National University of Singapore, Singapore) 

要旨1：  In this talk, we will review the development on constructing Green’s functions of linearized Boltzmann equation around a global maxwellian and Boltzmann shock profile and its applications to various problems. 
講演2：  Kinetic modeling of economic games with large number of participants 
Prof. Alexander V. Bobylev (Department of Mathematics, Karlstad University, Karlstad, Sweden) 

要旨2：  We study a Maxwell kinetic model of socioeconomic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys., 291 (2009), 599644. The model depends on three nonnegative parameters (g,q,s) where 0<1 is the control parameter. Two other parameters are fixed by market conditions. Selfsimilar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed. 
平成23年度第1回講演会（通算第1回）
日時：  平成23年3月17日（木） 16:0017:00 
場所：  京都大学 工学部11号館 2F 会議室 
講演：  Generalized Burnett Equations and the shock wave structure 
Prof. Alexander V. Bobylev (Department of Mathematics, Karlstad University, Karlstad, Sweden) 

要旨：  We describe a general idea of regularization of Burnett equations by “small” changes of variables and present the resulting set of equations (GBEs). Then we consider briefly a halfspace problem and explain why GBEs give a better qualitative approximation than the NavierStokes equations (NSEs). We also present recent results on the shock wave structure, which show the improvement of NavierStokes results for moderate Mach numbers. The talk is partly based on joint research with scientists from Parma university (G.Spiga and his group). 