Based on the existence theory on the Boltzmann equation with external forces in infinite vacuum, in this paper, we will study the L1 and BV-type stability of the classical solutions for small initial data. The stability results generalize those for the Boltzmann equation without force to the case with external force. In particular, we show that the stability can be established for the soft potentials directly, while the stability for the hard potentials and hard sphere model is obtained through the construction of some nonlinear functionals. The functionals thus constructed generalize those constructed in [S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations 215 (2005) 178205] for the case without force to capture the effect of the force term on the time evolution of the solutions. 2006 Elsevier Inc. All rights reserved.

In this paper, we study the nonlinear stability of travelling wave solutions with shock profile for a relaxation model with a nonconvex flux, which is proposed by Jin and Xin [<i>Comm. Pure Appl. Math.</i>, 48 (1995), pp. 555--563] to approximate an original hyperbolic system numerically under the subcharacteristic condition introduced by T. P. Liu [<i>Comm. Math. Phys.</i>, 108 (1987), pp. 153--175]. The travelling wave solutions with strong shock profile are shown to be asymptotically stable under small disturbances with integral zero using an elementary but technical energy method. Proofs involve detailed study of the error equation for disturbances using the same weight function introduced in [<i>Comm. Math. Phys.</i>, 165 (1994), pp. 83--96].

This is a continuation of the paper [15] on nonlinear boundary layers of the Boltzmann equation where the existence is established and shown to be strongly dependent on the Mach number <i>M</i> ^<i></i> of the Maxwellian state at far field. In this paper, when <i>M</i> ^<i></i>&lt;1, we will show that the linearized operator has the exponential decay in time property and therefore a bootstrapping argument yields nonlinear stability of the boundary layers.

In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in Lin and Yang (Methods Appl. Anal. 7 (3) (2000) 495) to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using LittlewoodPaley theory and justifies the transformation introduced in Liu and Yang (2000) in a rigorous setting.

In this paper, we are concerned with the one-species VlasovPoissonBoltzmann system with a nonconstant background density in full space. There exists a stationary solution when the background density is a small perturbation of a positive constant state. We prove the nonlinear stability of solutions to the Cauchy problem near the stationary state in some Sobolev space without any time derivatives. This result is nontrivial even when the background density is a constant state. In the proof, the macroscopic balance laws are essentially used to deal with the a priori estimates on both the microscopic and macroscopic parts of the solution. Moreover, some interactive energy functionals are introduced to overcome difficulty stemming from the absence of time derivatives in the energy functional.