Soft Potential Case Research Articles

About the Use of Entropy Production for the Landau\u2013Fermi\u2013Dirac Equation

In this paper, we present new estimates for the entropy dissipation of the Landau–Fermi–Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.

Shock waves from the inhomogeneous Boltzmann equation.

We revisit the problem on the inner structure of shock waves in simple gases modelized by the Boltzmann kinetic equation. In a paper by Pomeau [Y. Pomeau, Transp. Theory Stat. Phys. 16, 727 (1987)10.1080/00411458708204311], a self-similarity approach was proposed for infinite total cross section resulting from a power-law interaction, but this self-similar form does not have finite energy. Motivated by the work of Pomeau [Y. Pomeau, Transp. Theory Stat. Phys. 16, 727 (1987)10.1080/00411458708204311] and Bobylev and Cercignani [A. V. Bobylev and C. Cercignani, J. Stat. Phys. 106, 1039 (2002)10.1023/A:1014037804043], we started the research on the rigorous study of the solutions of the spatial homogeneous Boltzmann equation, focusing on those which do not have finite energy. However, infinite energy solutions do not have physical meaning in the present framework of kinetic theory of gases with collisions conserving the total kinetic energy. In the present work, we provide a correction to the self-similar form, so that the solutions are more physically sound in the sense that the energy is no longer infinite and that the perturbation brought by the shock does not grow at large distances of it on the cold side in the soft potential case.

Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces

In this paper the Boltzmann equation near global Maxwellians is studied in the $d$ -dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin-Lerner type space with respect to the phase variable $(x,v)$ . Both hard and soft potentials with angular cutoff are considered. The new function space for global well-posedness is introduced to essentially treat the case of soft potentials, and the key point is that the velocity variable is taken in the weighted supremum norm, and the space variable is in the $s$ -order Besov space with $s≥ d/2$ including the spatially critical regularity. The proof is based on the time-decay properties of solutions to the linearized equation together with the bootstrap argument. Particularly, the linear analysis in case of hard potentials is due to the semigroup theory, where the extra time-decay plays a role in coping with initial data in $L^2$ with respect to the space variable. In case of soft potentials, for the time-decay of linear equations we borrow the results based on the pure energy method and further extend them to those in $L^∞$ framework through the technique of $L^2-L^∞$ interplay. In contrast to hard potentials, $L^1$ integrability in $x$ of initial data is necessary for soft potentials in order to obtain global solutions to the nonlinear Cauchy problem.

Asymptotic behaviour of the Boltzmann equation as a cosmological model

As a Newtonian cosmological model the Vlasov-Poisson-Boltzmann system is considered, and a slightly modified Boltzmann equation, which describes the stability of an expanding universe, is derived. Asymptotic behaviour of solutions turns out to depend on the expansion of the universe, and in this paper we consider the soft potential case and will obtain asymptotic behaviour.

$${L^{p}}$$ L p stability for the Boltzmann equation near vacuum

The paper is concerned with the uniform time stability in the Lebesgue space \({L^{p}(\mathbb{R}^{3} \times \mathbb{R}^{3})}\) of solutions to the Boltzmann equation near vacuum. Precisely, for the soft potential case \({-2 1\) such that the nonnegative solution with algebraic decay rate in x, v at infinity is stable with respect to small initial data uniformly in time in \({L^{p}}\) with \({1 \leq p < p_{\gamma}}\).

Gevrey regularity of solutions to the non-cutoff homogeneous Boltzmann equation for soft potential with strong singularity

In this study, we analyze the Cauchy problem for the non-cutoff homogeneous Boltzmann equation with strong singularity. In contrast to our previous work Lin (2014) [8], the present study considers the soft potential case and derives the Gevrey regularity of the C∞ solutions.

Uniform stability of the Boltzmann equation with an external force near vacuum

The temporal uniform $L^1(x,v)$ stability of mild solutions for the Boltzmann equation with an external force is considered. We give a unified proof of the stability for two kinds of the forces. Firstly, we extend the soft potential case in [9] to both soft and hard cases. Secondly, we weaken the condition on the force in [13]. Furthermore, we give some new examples satisfying the constructive conditions on the force in [11].

Global existence and decay estimates of the Boltzmann equation with frictional force: The general results

In this paper, we give the existence theory and the optimal time convergence rates of the solutions to the Boltzmann equation with frictional force near a global Maxwellian. We generalize our previous results on the same problem for hard sphere model into both hard potential and soft potential case. The main method used in this paper is the classic energy method combined with some new time–velocity weight functions to control the large velocity growth in the nonlinear term for the case of interactions with hard potentials and to deal with the singularity of the cross-section at zero relative velocity for the soft potential case.

On the Cauchy Problem for the Inelastic Boltzmann Equation with External Force

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered for near vacuum data. Under the assumptions on the bicharacteristic generated by external force which can be arbitrarily large, we prove the global existence of mild solution for initial data small enough with respect to the sup norm with exponential weight by using the contraction mapping theorem. Furthermore, we prove the uniform L1 stability of the mild solution following from the exponential decay estimate and the Gronwall’s inequality for the case of soft potentials.

The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.

Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential

In this paper, we consider the Cauchy problem for the non-cutoff Boltzmann equation in the soft potential case. By using a singular change of velocity variables before and after collision, we prove the uniqueness of weak solutions to the Cauchy problem in the space of functions with polynomial decay in the velocity variable.

Distributional and Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials with Integrable Angular Cross Section

This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming S n−1 integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel–Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes the large data near local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in (Alonso et al. in Convolution inequalities for the Boltzmann collision operator. arXiv:0902.0507 [math.AP]) , by E. Carneiro and the authors, that allows us to show such propagation without additional conditions on the collision kernel. Finally, an L p -stability result (with 1≤p≤∞) is presented assuming the aforementioned condition.

From the Boltzmann Equation to an Incompressible Navier–Stokes–Fourier System

We establish a Navier–Stokes–Fourier limit for solutions of the Boltzmann equation considered over any periodic spatial domain of dimension two or more. We do this for a broad class of collision kernels that relaxes the Grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials. Appropriately scaled families of DiPerna–Lions renormalized solutions are shown to have fluctuations that are compact. Every limit point is governed by a weak solution of a Navier–Stokes–Fourier system for all time.

Well-posedness of the spatially homogeneous Landau equation for soft potentials

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderately soft potentials, we assume additionally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka [H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Geb. 46 (1) (1978–1979) 67–105].

Uniform stability of solutions of Boltzmann equation for soft potential with external force

The study of Cauchy problem of the Boltzmann equation is important in both theory and applications. Existence of global solutions to the equation and uniform stability of solutions in the absence of external force were introduced in the previous work on the Boltzmann equation. In this paper, we will investigate the uniform stability of solutions in L 1 for the Cauchy problem of the Boltzmann equation when there is an external force for the case of soft potentials.

Phase transition in a continuum classical system with finite interactions

Ruelle's proof of the existence of a phase transition in a classical continuum system composed of two kinds of particles with hard core repulsions between them (the Widom-Rowlinson model) is here extended to the case of soft potentials.