ISSN:
1078-0947
eISSN:
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Discrete & Continuous Dynamical Systems - A
January 2009 , Volume 24 , Issue 1
A special issue on
Boltzmann Equations and Applications
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2009, 24(1): i-ii
doi: 10.3934/dcds.2009.24.1i
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Abstract:
The famous Boltzmann equation
The famous Boltzmann equation
$frac{\partial f}{\partial t} + v \cdot \nabla_x f = Q(f,f),$
studied in a classic paper of Ludwig Boltzmann (1872), represents the basic example
of a kinetic equation. It describes the evolution of a distribution of particles subject
to binary collisions. Since then, the field has become extremely vast and active,
and in recent years two of the main problems concerning this equation have been
solved: the Cauchy problem for quite general collision kernels and the convergence
towards Navier-Stokes and Euler equation under suitable rescalings.
Ludwig E. Boltzmann ended his life in Duino, near Trieste, on September 5, 1906.
The conference
Boltzmann equations and applications, SISSA, Trieste, June 12-17, 2006
was organized with the double aim of honoring this great mathematician and reviewing the most recent and exciting developments on kinetic equations. Two main
courses, given by L. Saint-Raymond and C. Villani, and a series of lectures held by
leading mathematicians working in this area, allowed participants from all over the
world to become acquainted with the most interesting results and research directions.
For more information please click the "Full Text" above.
2009, 24(1): 1-11
doi: 10.3934/dcds.2009.24.1
+[Abstract](1616)
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Abstract:
We use Littlewood-Paley theory for the analysis of regularization properties of weak solutions of the homogeneous Boltzmann equation. For non cutoff and non Maxwellian molecules, we show that such solutions are smoother than the initial data. In particular, our method applies to any weak solution, though we assume that it belongs to a weighted $L^2$ space.
We use Littlewood-Paley theory for the analysis of regularization properties of weak solutions of the homogeneous Boltzmann equation. For non cutoff and non Maxwellian molecules, we show that such solutions are smoother than the initial data. In particular, our method applies to any weak solution, though we assume that it belongs to a weighted $L^2$ space.
2009, 24(1): 13-33
doi: 10.3934/dcds.2009.24.13
+[Abstract](1639)
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Abstract:
In this work, we prove that the singularities (in a fractional Sobolev space) of the classical solutions of the Vlasov-Poisson-Boltzmann equation are propagated along the characteristics of the Vlasov-Poisson equation, and decay exponentially.
In this work, we prove that the singularities (in a fractional Sobolev space) of the classical solutions of the Vlasov-Poisson-Boltzmann equation are propagated along the characteristics of the Vlasov-Poisson equation, and decay exponentially.
2009, 24(1): 35-57
doi: 10.3934/dcds.2009.24.35
+[Abstract](1419)
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Abstract:
In this paper we consider Lie group symmetries of evolution equations with non-local operators in context of applications to nonlinear kinetic equations. As an illustration we consider the Boltzmann equation and calculate the admitted group of point transformations.
In this paper we consider Lie group symmetries of evolution equations with non-local operators in context of applications to nonlinear kinetic equations. As an illustration we consider the Boltzmann equation and calculate the admitted group of point transformations.
2009, 24(1): 59-81
doi: 10.3934/dcds.2009.24.59
+[Abstract](1543)
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Abstract:
We introduce and discuss spatially homogeneous Maxwell-type models of the nonlinear Boltzmann equation undergoing binary collisions with a random component. The random contribution to collisions is such that the usual collisional invariants of mass, momentum and energy do not hold pointwise, even if they all hold in the mean. Under this assumption it is shown that, while the Boltzmann equation has the usual conserved quantities, it possesses a steady state with power-like tails for certain random variables. A similar situation occurs in kinetic models of economy recently considered by two of the authors [24], which are conservative in the mean but possess a steady distribution with Pareto tails. The convolution-like gain operator is subsequently shown to have good contraction/expansion properties with respect to different metrics in the set of probability measures. Existence and regularity of isotropic stationary states is shown directly by constructing converging iteration sequences as done in [8]. Uniqueness, asymptotic stability and estimates of overpopulated high energy tails of the steady profile are derived from the basic property of contraction/expansion of metrics. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as $t\to\infty$ to the steady solution in these distances, which metricizes the weak convergence of measures. These results show that power-like tails in Maxwell models are obtained when the point-wise conservation of momentum and/or energy holds only globally.
We introduce and discuss spatially homogeneous Maxwell-type models of the nonlinear Boltzmann equation undergoing binary collisions with a random component. The random contribution to collisions is such that the usual collisional invariants of mass, momentum and energy do not hold pointwise, even if they all hold in the mean. Under this assumption it is shown that, while the Boltzmann equation has the usual conserved quantities, it possesses a steady state with power-like tails for certain random variables. A similar situation occurs in kinetic models of economy recently considered by two of the authors [24], which are conservative in the mean but possess a steady distribution with Pareto tails. The convolution-like gain operator is subsequently shown to have good contraction/expansion properties with respect to different metrics in the set of probability measures. Existence and regularity of isotropic stationary states is shown directly by constructing converging iteration sequences as done in [8]. Uniqueness, asymptotic stability and estimates of overpopulated high energy tails of the steady profile are derived from the basic property of contraction/expansion of metrics. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as $t\to\infty$ to the steady solution in these distances, which metricizes the weak convergence of measures. These results show that power-like tails in Maxwell models are obtained when the point-wise conservation of momentum and/or energy holds only globally.
2009, 24(1): 83-94
doi: 10.3934/dcds.2009.24.83
+[Abstract](1526)
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Abstract:
The developments in the use and understanding of the Boltzmann equation during the 20th century are briefly surveyed.
The developments in the use and understanding of the Boltzmann equation during the 20th century are briefly surveyed.
2009, 24(1): 95-114
doi: 10.3934/dcds.2009.24.95
+[Abstract](1294)
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Abstract:
Higher order entropies are kinetic entropy estimators suggested by Enskog expansion of Boltzmann entropy. These quantities are quadratic in the density $\rho$, velocity $v$ and temperature $T$ renormalized derivatives. We investigate asymptotic expansions of higher order entropies for compressible flows in terms of the Knudsen $\epsilon_k$ and Mach $\epsilon_m$ numbers in the natural situation where the volume viscosity, the shear viscosity, and the thermal conductivity depend on temperature, essentially in the form $T^x$. Entropic inequalities are obtained when ||$\log \rho$||BMO,$\quad$ $\epsilon_m$||$v/\sqrt{T}$|| L ∞ ,$\quad$ ||$\log T$||$BMO$,$\quad$ $\epsilon_k$||$h\partial_{x} \rho$/$\rho$|| L ∞ , $\epsilon_k$$\epsilon_m$||$h\partial_{x} v$/$\sqrt{T}$ || L ∞ , $\epsilon_k$||$h\partial_{x}T$/$T$|| L ∞ , and $\epsilon_k^2$||$h^2\partial^2_x T$/$T$|| L ∞ are small enough, where $h = 1/(\rho T^{(1/2) -x)}$ is a weight associated with the dependence on density and temperature of the mean free path.
Higher order entropies are kinetic entropy estimators suggested by Enskog expansion of Boltzmann entropy. These quantities are quadratic in the density $\rho$, velocity $v$ and temperature $T$ renormalized derivatives. We investigate asymptotic expansions of higher order entropies for compressible flows in terms of the Knudsen $\epsilon_k$ and Mach $\epsilon_m$ numbers in the natural situation where the volume viscosity, the shear viscosity, and the thermal conductivity depend on temperature, essentially in the form $T^x$. Entropic inequalities are obtained when ||$\log \rho$||BMO,$\quad$ $\epsilon_m$||$v/\sqrt{T}$|| L ∞ ,$\quad$ ||$\log T$||$BMO$,$\quad$ $\epsilon_k$||$h\partial_{x} \rho$/$\rho$|| L ∞ , $\epsilon_k$$\epsilon_m$||$h\partial_{x} v$/$\sqrt{T}$ || L ∞ , $\epsilon_k$||$h\partial_{x}T$/$T$|| L ∞ , and $\epsilon_k^2$||$h^2\partial^2_x T$/$T$|| L ∞ are small enough, where $h = 1/(\rho T^{(1/2) -x)}$ is a weight associated with the dependence on density and temperature of the mean free path.
2009, 24(1): 115-143
doi: 10.3934/dcds.2009.24.115
+[Abstract](1253)
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Abstract:
In this paper, we present an $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with cut-off and inverse power law potentials, when initial data are sufficiently small and decay fast enough in phase space. For moderately soft potentials, we show that classical solutions depend Lipschitz continuously on the initial data in weighted $L^p$-norm. In contrast for hard potentials, we show that classical solutions depend Hölder continuously on the initial data. Our stability estimates are based on the dispersion estimates due to time-asymptotic linear Vlasov dynamics.
In this paper, we present an $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with cut-off and inverse power law potentials, when initial data are sufficiently small and decay fast enough in phase space. For moderately soft potentials, we show that classical solutions depend Lipschitz continuously on the initial data in weighted $L^p$-norm. In contrast for hard potentials, we show that classical solutions depend Hölder continuously on the initial data. Our stability estimates are based on the dispersion estimates due to time-asymptotic linear Vlasov dynamics.
2009, 24(1): 145-157
doi: 10.3934/dcds.2009.24.145
+[Abstract](1624)
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Abstract:
The Boltzmann equation, [1], offers richer and physically more realistic modelling of the boundary effects than the fluid dynamic equations. Important phenomena such as the thermal transpiration and some of the bifurcations due to curvature of the boundary can only modeled using the kinetic formulation. In this paper we survey the analytical ideas that have been introduced in recent years for the study of the boundary effects. The main point is that more quantitative estimates of the solutions are needed for such a study.
The Boltzmann equation, [1], offers richer and physically more realistic modelling of the boundary effects than the fluid dynamic equations. Important phenomena such as the thermal transpiration and some of the bifurcations due to curvature of the boundary can only modeled using the kinetic formulation. In this paper we survey the analytical ideas that have been introduced in recent years for the study of the boundary effects. The main point is that more quantitative estimates of the solutions are needed for such a study.
2009, 24(1): 159-185
doi: 10.3934/dcds.2009.24.159
+[Abstract](1449)
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Abstract:
We consider a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing (in the framework of so-called constant normal restitution coefficients $\alpha \in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_$∗,1) for some constructive $\alpha_$∗, $\in$ [0,1)) we prove uniqueness of the stationary solution for given values of the restitution coefficient $\alpha \in [\alpha_$∗,1), the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.
We consider a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing (in the framework of so-called constant normal restitution coefficients $\alpha \in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_$∗,1) for some constructive $\alpha_$∗, $\in$ [0,1)) we prove uniqueness of the stationary solution for given values of the restitution coefficient $\alpha \in [\alpha_$∗,1), the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.
2009, 24(1): 187-212
doi: 10.3934/dcds.2009.24.187
+[Abstract](1681)
+[PDF](286.7KB)
Abstract:
Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case. As a further study on the problem in the spatially homogeneous situation, in this paper, we will prove the Gevrey smoothing property of the solutions to the Cauchy problem for Maxwellian molecules without angular cutoff by using pseudo-differential calculus. Furthermore, we apply similar analytic techniques for the Sobolev space regularity to the nonlinear equation, and prove the smoothing property of solutions for the spatially homogeneous nonlinear Boltzmann equation with the Debye-Yukawa potential.
Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case. As a further study on the problem in the spatially homogeneous situation, in this paper, we will prove the Gevrey smoothing property of the solutions to the Cauchy problem for Maxwellian molecules without angular cutoff by using pseudo-differential calculus. Furthermore, we apply similar analytic techniques for the Sobolev space regularity to the nonlinear equation, and prove the smoothing property of solutions for the spatially homogeneous nonlinear Boltzmann equation with the Debye-Yukawa potential.
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