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In this paper, we are concerned with the global existence and time decay rates of the solutions to Boltzmann equation for hard potential in the critical Chemin-Lerner type space. The evolution of the energy in negative Besov space is derived to be preserved. Our results are based on the modified trilinear estimates and some new observations. We use the Lyapunov-type inequality of the energy functionals to show the time decay rates without linear decay analysis. Compared with the previous work, the decay rates of the global solution are established for lower-regularity initial data. [ABSTRACT FROM AUTHOR]

In this paper, we prove the propagation of L p upper bounds for the spatially homogeneous relativistic Boltzmann equation for any 1 < p < ∞. We consider the case of relativistic hard ball with Grad's angular cutoff. Our proof is based on a detailed study of the inter-relationship between the relative momenta, the regularity and the L p estimates for the gain operator, the development of the relativistic Carleman representation, and several estimates on the relativistic hypersurface E v ′ − v v ⁎ . We also derive a Pythagorean theorem for the relative momenta g (v , v ⁎) , g (v , v ′) , and g (v ′ , v ⁎) , which has a crucial role in the reduction of the momentum singularity. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the works in [35] and [37] , where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of Boltzmann equation with large external potential in T 3. For a class of initial data with large oscillations in L x , v ∞ around the local Maxwellian, we prove the existence of a global solution to the Boltzmann equation provided the initial perturbation is suitably small in L 2 -norm. The large time behavior of the Boltzmann solution with exponential decay rate is also obtained. This seems to be the first result on the perturbation theory of large-amplitude non-constant equilibriums for large-amplitude initial data. [ABSTRACT FROM AUTHOR]

Abstract: The Euler equations with frictional force have been extensively studied. Since the Boltzmann equation is closely related to the equations of gas dynamics, we study, in this paper, the Boltzmann equation with frictional force when the external force is proportional to the macroscopic velocity. It is shown that smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution which approaches to the global Maxwellian time asymptotically. The analysis is based on the macro–micro decomposition for the Boltzmann equation introduced in Liu et al. [Energy method for the Boltzmann equation, Physica D 188 (3–4) (2004) 178–192] and Liu and Yu [Boltzmann equation: micro–macro-decompositions and positivity of shock profiles, Comm. Math. Phys. 246(1) (2004) 133–179] through energy estimates. [Copyright &y& Elsevier]