# Rigorous results for conservation equations and trend to equilibrium in space-inhomogeneous kinetic theory

Chapter

## Abstract

The well-posedness of the initial value problem for the Boltzmann equation means that we prove that there is a unique nonnegative solution preserving the energy and satisfying the entropy inequality, from a positive initial datum with finite energy and entropy. However, for general initial data, it is difficult, and until now not known, whether such a well-behaved solution can be constructed globally in time. The difficulty in doing this is obviously related to the nonlinearity of the collision operator and the apparent lack of conservation laws or a priori estimates preventing the solution from becoming singular in finite time.

## Keywords

Weak Solution Boltzmann Equation Mild Solution Collision Operator Entropy Inequality
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## References

- [1]L. Arkeryd, On the strong
*L*^{1}trend to equilibrium for the Boltzmann equation,*Studies in Appl. Math.***87**, 283–288 (1992).MATHMathSciNetGoogle Scholar - [2]M. Bony, Existence globale et diffusion en théorie cinétique discrète. In
*Advances in Kinetic Theory and Continuum Mechanics*, R. Gatignol and Soubbarameyer, Eds., 81–90, Springer-Verlag, Berlin (1991).Google Scholar - [3]C. Cercignani, Global weak solutions of the Boltzmann equation,
*Jour. Stat. Phys.***118**, 333–342 (2005).MATHCrossRefMathSciNetGoogle Scholar - [4]C. Cercignani,
*The Boltzmann Equation and Its Applications*, Springer-Verlag, New York (1988).MATHGoogle Scholar - [5]C. Cercignani, Weak solutions of the Boltzmann equation without angle cutoff, submitted to
*Jour. Stat. Phys.*2005.Google Scholar - [6]C. Cercignani, Estimating the solutions of the Boltzmann equation, submitted to
*Jour. Stat. Phys.*2005.Google Scholar - [7]C. Cercignani and R. Illner, Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions,
*Arch. Rational Mech. Anal.***134**, 1–16 (1996).MATHCrossRefMathSciNetGoogle Scholar - [8]C. Cercignani, R. Illner, and M. Pulvirenti,
*The Mathematical Theory of Dilute Gases*, Springer-Verlag, New York (1994).MATHGoogle Scholar - [9]C. Cercignani, Equilibrium states and trend to equilibrium in a gas according to the Boltzmann equation,
*Rend. Mat. Appl.***10**, 77–95 (1990).MATHMathSciNetGoogle Scholar - [10]C. Cercignani,
*Slow Rarefied Flow Theory and Application to MEMS*, Birkhäuser, Basel (2006).Google Scholar - [11]L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations,
*Arch. Rational Mech. Analysis***110**, 73–91(1990).MATHCrossRefMathSciNetGoogle Scholar - [12]R. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability,
*Ann. of Math.***130**, 321–366 (1989).CrossRefMathSciNetGoogle Scholar - [13]R. DiPerna and P. L. Lions, Global solutions of Boltzmann’s equation and the entropy inequality,
*Arch. Rational Mech. Anal.***114**, 47–55 (1991).MATHCrossRefMathSciNetGoogle Scholar - [14]P. L. Lions, Compactness in Boltzmann’s equation via Fourier integral operators and applications. I,
*Cahiers de Mathématiques de la décision***9301**, CEREMADE (1993).Google Scholar - [15]E. Ikenberry and C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory. I,
*Jour. Rat. Mech. Anal.***5**, 1–54 (1956).MathSciNetGoogle Scholar - [16]R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum,
*Commun. Math. Phys.***105**, 189–203 (1986).MATHCrossRefMathSciNetGoogle Scholar - [17]R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two-and three-dimensional rare gases in vacuum: Erratum and improved result,
*Commun. Math. Phys.***121**, 143–146 (1989).MATHCrossRefMathSciNetGoogle Scholar - [18]O. Lanford III, in
*Time evolution of large classical systems.*Moser, E. J. (ed.). Lecture Notes in Physics**38**, 1–111. Springer-Verlag (1975).Google Scholar - [19]J. C. Maxwell, On the dynamical theory of gases,
*Phil. Trans. Roy Soc. (London)***157**, 49–88 (1866).Google Scholar

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