Group analysis of the one-dimensional boltzmann equation: II. Equivalence groups and symmetry groups in the special case

We obtain relations that define the equivalence algebra of the family of one-dimensional Boltzmann equations f _t + cf _x + F(t, x, c)f _c = 0 and show that all equations of that form are locally equivalent. We carry out the group classification of the equation with respect to the function F in the special case where the function F and the transformations of the variables t and x are assumed to be independent of c. We show that, under such constraints for the transformation and the family of equations, the maximum possible symmetry algebra is eight-dimensional, which corresponds to an equation with a linear function F.     

Group analysis of the one-dimensional Boltzmann equation. Invariants and the problem of moment system closure

Theoretical and Mathematical Physics - We complete the investigation of the feasibility in principle to close the moment system for the kinetic equation via invariant relations obtained by group...     

Viscosity analysis on the Boltzmann equation

This paper is devoted to investigating the asymptotic properties of the renormalized solution to the viscosity equation tfε + v ·▽xfε = Q (fε,fε ) + εΔvfε as ε→ 0+ . We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L1 ((0 , T ) × RN × RN ). The proof is based on compactness analysis and velocity averaging theory.     

Symmetries and invariant solutions of the one-dimensional Boltzmann equation for inelastic collisions

We consider the one-dimensional integro-differential Boltzmann equation for Maxwell particles with inelastic collisions. We show that the equation has a five-dimensional algebra of point symmetries for all dissipation parameter values and obtain an optimal system of one-dimensional subalgebras and classes of invariant solutions.     

Asymptotic analysis of the linear Boltzmann equation

The stiff boundary value problem for the time-independent linear Boltzmann equation is considered in the variational formulation in terms of the coercive symmetric bilinear form. It is shown that it is asymptotically equivalent to the related variational diffusion problem when the stiffness parameter tends to zero. Higher order perturbations are also discussed.     

Half-space analysis basic to the linearized Boltzmann equation

The elementary solutions and the half-range completeness and orthogonality theorems concerning the linearized Boltzmann equation are discussed.

---

Zusammenfassung Die elementaren Lösungen und die halbräumigen Vollständigkeits- und Orthogonalitätstheoreme die linearen Boltzmann-Gleichungen betreffend, werden diskutiert.

     

L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

Finite symmetry transformation groups and exact solutions of the cylindrical Korteweg-de Vries equation

Based on the new symmetry group method developed by Lou et al. and symbolic computation, both the Lie point groups and the non-Lie symmetry groups of the cylindrical Korteweg-de Vries (cKdV) equation are obtained. With the transformation groups, a type of group invariant solutions of cKdV equation can be derived from a simple one. Furthermore, some transformations from the cKdV equation to KP equation can also be discovered by this method.     

Perturbation of Yamabe equation on Iwasawa N groups in presence of symmetry

Let G be a simple Lie group of real rank one and N be in the Iwasawa decomposition of G. Under the assumption of some symmetries, we obtain an existent result for the nonlinear equation △NU ＋ （1 ＋ ∈K（x, z））u2＊-1 = 0 on N, which generalizes the result of Malchiodi and Uguzzoni to the Kohn＇s subelliptic context on N in presence of symmetry.     

On measure solutions of the Boltzmann equation,part I: Moment production and stability estimates

The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.     

Nonclassical symmetry analysis for hyperbolic partial differential equation

We consider the nonclassical symmetry of one-dimensional hyperbolic differential equations of the form u_t + M(u)u_x = 0. For the infinitesimal generator $\mathbf{V}=\tau {\mathrm{\partial }}_t+\xi {\mathrm{\partial }}_x+{\sum }_{i=1}^n{\varphi }_i{\mathrm{\partial }}_{u_i}$, it is shown that ξ is an eigenvalue of the matrix M when ϕ_i = 0 [Souichi M. Nonclassical symmetry and Riemann invariants. Int J Nonlinear Mech, [in press]]. In this paper, we prove a sufficient condition of a lemma.     

On the asymptotic analysis of the Boltzmann equation tending towards the Stokes equations

This work deals with the analysis of the asymptotic limit for the Boltzmann equation tending towards the linearized Navier–Stokes equations when the Knudsen number $\epsilon$ tends to zero. Global existence and uniqueness theorems are proven for regular initial fluctuations. As $\epsilon$ tends to zero, the solution converges strongly to the solution of the linearized Navier–Stokes systems.     

Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of(x, t) in the region R3×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.     

Group analysis of the modified generalized Vakhnenko equation

In this paper the Lie symmetry group, the corresponding symmetry reductions and invariant solutions of the modified generalized Vakhnenko equation are determined. Moreover a numerical algorithm that is based on a Lie symmetry group preserving scheme is applied to the ordinary differential equations obtained by symmetry reduction.     

Group Analysis of the One-Dimensional Boltzmann Equation: IV. Complete Group Classification in the General Case

Theoretical and Mathematical Physics - We consider the one-dimensional Boltzmann equation $$f_t+cf_x+(\mathcal{F}f)_c=0$$ with a function $$\mathcal{F}$$ depending on (t,x,c,f) and obtain the...     

Full symmetry groups, Painlevé integrability and exact solutions of the nonisospectral BKP equation

Based on the generalized symmetry group method presented by Lou and Ma [Lou and Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A: Math. Gen. 38 (2005) L129], firstly, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between the new solutions and the old ones of equation. Secondly, the nonisospectral BKP can be proved to be Painlevé integrability by combining the standard WTC approach with the Kruskal’s simplification, some solutions are obtained by using the standard truncated Painlevé expansion. Finally, based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated Painlevé expansion, some interesting solution are constructed.     

Numerical analysis of the weighted particle method applied to the semiconductor Boltzmann equation

Summary The semiconductor Boltzmann equation involves an integral operator, the kernel of which is a measure supported by a surface. This feature introduces some singularities of the exact solution, which makes the numerical approximation of this equation difficult. This paper is devoted to the error analysis of the weighted particle method (introduced by Mas-Gallic and Raviart [14]) applied to the space homogeneous semiconductor Boltzmann equation. The results are commented in view of the practical use of the method. This paper is closely related to [12], where results of numerical simulations on both test and real problems are given.