We consider a body moving along the x-axis under the action of an external force E and immersed in an infinitely extended perfect gas. We assume the gas to be described by the mean-field approximation and interacting elastically with the body. In this setup, we discuss the following statement: "Let V0 be the initial velocity of the body and V∞ its asymptotic velocity, then for |V0 - V∞| small enough it results |V(t) - V∞| ≈ C t-d-2 for t large, where V(t) is the velocity of the body at time t, d the dimension of the space and C is a positive constant depending on the medium and on the shape of the body". The reason for the power law approach to the stationary state instead of the exponential one (usually assumed in viscous friction problems), is due to the long memory of the dynamical system. In a recent paper by Caprino, Marchioro and Pulvirenti,3 the case of E constant and positive, with 0 < V0 < V∞, for a disk orthogonal to the x-axis has been discussed. Here we complete the analysis in the cases E > 0 with V0 > V∞ and E = 0. We also approach the problem of an x-dependent external force, by choosing E of harmonic type. In this case we obtain the power-like asymptotic time behavior for the body position X(t). The investigation is done in detail for a disk orthogonal to the x-axis and then, by a sketched proof, extended to a body with a general convex shape.
We consider a one-species plasma moving in an infinite cylinder, in which it is confined by means of a magnetic field diverging on the walls of the cylinder. It is assumed that initially the particles have bounded velocities and are distributed according to a density which is bounded, without any decreasing at infinity. The mutual interaction is of Yukawa type, i.e., Coulomb at short distance and exponentially decreasing at infinity. We prove the global in time existence and uniqueness of the time evolution of the plasma and its confinement
Abstract.We consider a body immersed in a perfect gas and moving under the action of a constant force. Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V (t) to the limiting velocity V∞ and prove that, under suitable smallness assumptions, the approach to equilibrium iswhere d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions.Mathematics Subject Classification. 76P05, 82B40, 82C40, 35L45, 35L50.
A thin plate accelerated or decelerated in a free-molecular gas at rest by a constant external force is considered. The force is in the direction perpendicular to the plate. In this situation, the plate velocity approaches its final constant velocity as time goes on. It is shown numerically that, under the diffuse-reflection boundary condition, the difference between the plate velocity and its final value decreases in proportion to an inverse power of time. This agrees with the previous theoretical result obtained under the assumption that the initial plate velocity is sufficiently close to the final one.
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