Peter Markowich 教授 特別講演会

• ２１世紀ＣＯＥプログラム「動的機能機械システムの数理モデルと設計論」―複雑系の科学による機械工学の新たな展開―
• 日本航空宇宙学会関西支部「複雑流れの現象解明とその応用」研究分科会

At the kinetic level, it is easy to relate mathematical parameter functions with simple physical quantities, but the price to pay is the high dimensionality of the phase space. On the other hand, hydrodynamical equations or parabolic models are in principle simpler to solve, but their direct derivation is far less intuitive. This motivates the study of hydrodynamic or diffusion limits and in our approach, local or global Gibbs states will be considered as basic input for the modeling. This is a very standard assumption for instance in semiconductor theory when one speaks of Fermi-Dirac distributions, or when one considers polytropic distribution functions in stellar dynamics. It is the purpose of this work to provide a justification of nonlinear diffusions as limits of appropriate simple kinetic models.

Let us mention that in astrophysics, power law Gibbs states are well known (see, e.g., [1], and [6] for some mathematical properties of such equilibrium states).

In our approach [2], we are not concerned with the physical phenomena responsible for the relaxation towards the local Gibbs state and, in the long time range, towards the global Gibbs state. We present at the kinetic level a simple model of a collision kernel, which is simply a projection onto the local Gibbs state with the same spatial density, thus introducing a local Lagrange multiplier which will be referred to as the pseudo Fermi level.

We prove existence and uniqueness of solutions of the kinetic model under the assumption of boundedness of the initial datum and prove the convergence to a global equilibrium. With the parabolic scaling we rigorously prove the convergence of the solutions to a macroscopic limit using compensated compactness. Most notably, we are able to reproduce non-linear diffusion equations ∂_tρ = Δ(ρ^m) + ∇・(ρ∇V), ranging from porous medium equation to fast diffusion, 0 < m < 3/5, as macroscopic limits by employing the appropriate energy profiles.

In the mathematical study of diffusion limits for semiconductor physics, more results are known, starting with [3, 4]. Other reference papers are [5] and [7].

1. Binney, J. and Tremaine, S. (1997). Galactic dynamics, Princeton university press
2. Dolbeault, J., Markowich, P., Oelz, D. and Schmeiser, C. (to be submitted) Nonlinear diffusions as diffusion limits of kinetic equations with relaxation collision kernels
3. Golse, F. and Poupaud, F. (1988). Fluid limit of the Vlasov-Poisson-Boltzmann equation of semiconductors. In: BAIL V (Shanghai, 1988), Boole Press Conf. Ser., 12
4. Golse, F. and Poupaud, F. (1992). Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac. In: Asymptotic Anal., 6, 135-160
5. Goudon, Thierry and Poupaud, Frederic (2001). Approximation by homogenization and diffusion of kinetic equations. Comm. Partial Differential Equations 3-4 26, 537-569
6. Guo, Yan and Rein, Gerhard (2003). Stable models of elliptical galaxies. In: on. Not. R. Astronom.
7. Poupaud, Frédéric and Schmeiser, Christian (1991). Charge transport in semiconductors with degeneracy effects. Math. Methods Appl. Sci 14, 301-318