Boltzmann Equation For Hard Potentials Research Articles

On Exponential Moments of the Homogeneous Boltzmann Equation for Hard Potentials Without Cutoff

We consider the spatially homogeneous Boltzmann equation for hard potentials without cutoff. We prove that an exponential moment of order $\rho=\min\{2\gamma/(2-\nu),2\}$, with the usual notation, is immediately created. This is stronger than what happens in the case with cutoff. We also show that exponential moments of order $\rho\in (0,2]$ are propagated.

Spatial behavior of the solution to the linearized Boltzmann equation with hard potentials

The main goal is to understand the quantitative spatial decay of the solution to the linearized Boltzmann equation for hard potentials with cutoff. We obtain the quantitative space-time behavior of the linearized Boltzmann equation under some slow velocity decay assumption but without any regularity assumption on the initial data. This result extends the classical results of Liu and Yu [Commun. Pure Appl. Math. 57, 1543–1608 (2004); Bull. Inst. Math. Acad. Sin. 1, 1–78 (2006); and ibid. 6, 151–243 (2011)] from hard sphere to hard potential under the same assumption on the initial condition [compactly support in the space variable and polynomial decay in the velocity variable with power β = (3/2)+].

A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution \begin{document}$ (f_t)_{t\geq 0} $\end{document} , once the initial condition \begin{document}$ f_0 $\end{document} with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a \begin{document}$ (0,\infty)\times {\mathbb{R}}^3 $\end{document} random variable \begin{document}$ (M_t,V_t) $\end{document} such that \begin{document}$ \mathbb{E}[M_t {\bf 1}_{\{V_t \in \cdot\}}] = f_t $\end{document} . We also write down a series expansion of \begin{document}$ f_t $\end{document} . Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express \begin{document}$ f_t $\end{document} in terms of \begin{document}$ f_0 $\end{document} , we believe that the algorithm is not very efficient in practice and that the series expansion is rather intractable. This is a tedious extension to non-Maxwellian molecules of Wild's sum [ 18 ] and of its interpretation by McKean [ 10 , 11 ].

Exponential convergence for the linear homogeneous Boltzmann equation for hard potentials

In this paper, we consider the asymptotic behavior of solutions to the linear spatially homogeneous Boltzmann equation for hard potentials without angular cutoff. We obtain an optimal rate of exponential convergence towards equilibrium in a L1-space with a polynomial weight. Our strategy is taking advantage of a spectral gap estimate in the Hilbert space L2(μ−12) and a quantitative spectral mapping theorem developed by Gualdani et al. (2017).

Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region, we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region, we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results largely extend the classical results of Liu-Yu \cite{[LiuYu], [LiuYu2], [LiuYu1]} and Lee-Liu-Yu \cite% {[LeeLiuYu]} to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.

On Mittag-Leffler Moments for the Boltzmann Equation for Hard Potentials Without Cutoff

We establish the $L^1$ weighted propagation properties for solutions of the Boltzmann equation with hard potentials and nonintegrable angular components in the collision kernel. Our method identifi...

Spectrum Analysis of Some Kinetic Equations

We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with \({\gamma\geqq-2}\). As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate \({t^{-5/4}}\)) as \({t\to\infty}\) to that of the compressible Navier–Stokes equations for initial data around an equilibrium state.

Global Well-Posedness in Spatially Critical Besov Space for the Boltzmann Equation

The unique global strong solution in the Chemin–Lerner type space to the Cauchy problem on the Boltzmann equation for hard potentials is constructed in a perturbation framework. Such a solution space is of critical regularity with respect to the spatial variable, and it can capture the intrinsic properties of the Boltzmann equation. For the proof of global well-posedness, we develop some new estimates on the nonlinear collision term through the Littlewood–Paley theory.

On measure solutions of the Boltzmann equation, Part II: Rate of convergence to equilibrium

The paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. We prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy. The proof is based on the regularizing property of the iterated collision operators, exponential moment production estimates, and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data. We also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the Maxwellian equilibrium. The constants in these convergence rates depend only on the collision kernel and conserved quantities (mass, momentum, and energy). We finally use these convergence rates in order to deduce global-in-time strong stability of measure solutions.

Exponential Convergence to Equilibrium for the Homogeneous Boltzmann Equation for Hard Potentials Without Cut-Off

This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off and non mollified) hard potentials. We prove an exponential in time convergence towards the equilibrium, improving results of Villani from \cite{Vill1} where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a $L^1$ space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani and al in \cite{GMM}. We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.

Regularization properties of the 2D homogeneous Boltzmann equation without cutoff

We consider the 2-dimensional spatially homogeneous Boltzmann equation for hard potentials. We assume that the initial condition is a probability measure that has some exponential moments and is not a Dirac mass. We prove some regularization properties: for a class of very hard potentials, the solution instantaneously belongs to H r, for some \({r\in (-1,2)}\) depending on the parameters of the equation. Our proof relies on the use of a well-suited Malliavin calculus for jump processes.

Global Existence of Classical Solutions to the Boltzmann Equation with External Force for Hard Potentials

The Boltzmann equation with external force describes the time evolution of rarefied gas in an external field. In this paper, we construct the global existence of classical solutions to the Boltzmann equation for hard potentials with Grad's angular cutoff in the whole space when the initial datum is a perturbation of the local Maxwellian. The proof relies on the nonlinear energy method where the crucial step is to show the positivity of the linearized collision operator. Further, we introduce new energy functional to obtain the desired differential inequality.

Nonlinear stability of boundary layers of the Boltzmann equation for cutoff hard potentials

Many physical models have boundaries. For the Boltzmann equation, the study on the boundary layer in the region of the width in the order of the Kundsen number along the boundary is important both in mathematics and physics. In this paper, we consider the nonlinear stability of boundary layer solutions to the Boltzmann equation for hard potentials with angular cut-off. The boundary condition is imposed on incoming particles of the Dirichlet type and the solution tends to a global Maxwellian in the far field. For the existence of the boundary layer solutions, it is proved by Chen et al. [Anal. Appl. 2, 337–363 (2004)] by introducing a weight function which is a function of both position and velocity to overcome the difficulty from the sublinear growth in the collision frequency. Unlike the hard sphere model, even for stability in the case when the Mach number of the far field is less than −1, exponential decay in time cannot be expected for the cutoff hard potentials. Instead, an algebraic decay in time to the boundary layer solution is proved in this paper by using some recursive weighted energy estimates and the bootstrap argument.

An Example of Nonuniqueness for Solutions to the Homogeneous Boltzmann Equation

The paper deals with the spatially homogeneous Boltzmann equation for hard potentials. An example is given which shows that, even though it is known that there is only one solution that conserves energy, there may be other solutions for which the energy is increasing; uniqueness holds if and only if energy is assumed to be conserved.

The Navier-Stokes limit of the stationary boltzmann equation for hard potentials

In this paper we extend recent results on the hydrodynamic Navier-Stokes limit of the stationary Boltzmann equation for the flow of a gas of hard spheres in a channel in the presence of an external force to the case of a hard intermolecular potential with Grad angular cutoff. We prove the convergence of the solution, for small Knudsen numbers, to the Maxwellian with parameters solving the corresponding Navier-Stokes equation. In the present case we only get polynomial decay of the solution for large velocities, instead of the exponential decay which holds for hard spheres.