Boltzmann Equation For Soft Potentials Research Articles

Existence and uniqueness of solutions to the Fermi-Dirac Boltzmann equation for soft potentials

In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter δ \delta that can decrease from δ = 1 \delta =1 for the Fermi-Dirac particles to δ = 0 \delta =0 for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space L T ∞ L v , x ∞ ∩ L T ∞ L x ∞ L v 1 L^{\infty }_{T}L^{\infty }_{v,x}\cap L^{\infty }_{T}L^{\infty }_{x}L^1_v with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maximum upper bound F ( t , x , v ) ≤ 1 δ F(t,x,v)\leq \frac {1}{\delta } . The key point is that the obtained estimates are uniform in the quantum parameter 0 > δ ≤ 1 0> \delta \leq 1 .

Asymptotic behavior of linearized Boltzmann equations for soft potentials with cut-off

We consider the asymptotic behavior of the linearized Boltzmann equation for soft potentials with cut-off. By introducing a new decomposition of the linearized Boltzmann operator, we analyze the spectrum of the linearized Boltzmann operator and obtain the asymptotic behaviors of the linearized Boltzmann equation for \(\gamma\in (-3,0)\), extending the result in [12] for \(\gamma\in (-1,0)\).
 For more information see https://ejde.math.txstate.edu/Volumes/2021/46/abstr.html

On the inelastic Boltzmann equation for soft potentials with diffusion

We are concerned with the Cauchy problem of the inelastic Boltzmann equation for soft potentials, with a Laplace term representing the random background forcing. The inelastic interaction here is characterized by the non-constant restitution coefficient. We prove that under the assumption that the initial datum has bounded mass, energy and entropy, there exists a weak solution to this equation. The smoothing effect of weak solutions is also studied. In addition, it is shown the solution is unique and stable with respect to the initial datum provided that the initial datum belongs to $ L^{2}(R^{3}) $.

The relativistic Boltzmann equation for soft potentials

The paper concerns the Cauchy problem on the relativistic Boltzmann equation for soft potentials in a periodic box. We show that the global-in-time solutions around relativistic Maxwellians exist in the weighted L∞ perturbation framework and also approach equilibrium states in large time in the weighted L2 framework at the rate of exp⁡(−λtβ) for some λ>0 and β∈(0,1). The proof is based on the nonlinear L2 energy method and nonlinear L∞ pointwise estimate with appropriate exponential weights in momentum. The results extend those on the classical Boltzmann equation by Caflisch [2,3] and Strain and Guo [31] to the relativistic version, and also improve the recent result on almost exponential time-decay by Strain [28] to the exponential rate. Moreover, we study the propagation of spatial regularity for the obtained solutions and also the large time behavior in the corresponding regular Sobolev space, provided that the spatial derivatives of initial data are bounded, not necessarily small.

Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials

The existence of classical solutions to the Cauchy problem for the Boltzmann equation without angular cutoff has been extensively studied in the framework when the solution has Maxwellian decay in the velocity variable, cf. [6, 8] and the references therein. In this paper, we prove local existence of solutions with polynomial decay in the velocity variable for the Boltzmann equation with soft potential. In the proof, the singular change of variables between post- and pre-collision velocities plays an important role, as well as the regular one introduced in the celebrated cancelation lemma by Alexandre–Desvillettes–Villani–Wennberg [Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000) 327–355].

Distributional solutions of the Boltzmann equation with infinite energy

We prove an existence theorem of distributional solutions to the Boltzmann equation for soft potentials with angular cut-off and initial data close to the local Maxwellian M0=exp(−|x−ξ|22). The solutions constructed in this paper have infinite mass and energy since they are located in a neighborhood of the Maxwellian M(t,x,ξ)=exp(−|x−ξ(1+t)|22) (t⩾0).

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns L 1-convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (−4≤γ<0), with and without angular cutoff. We prove the time-averaged L 1-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than 2+|γ|. For the usual L 1-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with −1≤γ<0, there are algebraic upper and lower bounds on the rate of L 1-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates.