We consider the Dirichlet problem for a quasilinear degenerate parabolic stochastic partial differential equation with multiplicative noise and nonhomogeneous Dirichlet boundary condition. We introduce the definition of kinetic solution for this problem and prove existence and uniqueness of solutions. For the uniqueness of kinetic solutions we prove a new version of the doubling of variables method and use it to deduce a comparison principle between solutions. The proof requires a delicate analysis of the boundary values of the solutions for which we develop some techniques that enable the usage of the existence of normal weak traces for divergence measure fields in this stochastic setting. The existence of solutions, as usual, is obtained through a two-level approximation scheme consisting of nondegenerate regularizations of the equations which we show to be consistent with our definition of solutions. In particular, the regularity conditions that give meaning to the boundary values of the solutions are shown to be inherited by limits of nondegenerate parabolic approximations provided by the vanishing viscosity method.

We study a relaxation dynamics of the kinetic thermodynamic Cucker-Smale (TCS) model under the effect of potential force field. For this, we present a sufficient framework on the emergence of a periodically rotating one-point cluster to the TCS model with a harmonic potential force. In the presence of external potential force, large-time behavior of the kinetic TCS model is completely different from that of the kinetic TCS model without an external force. For the relaxation toward the periodic motion in thermo-mechanical observables, we use dynamical systems theory such as the Lyapunov functional approach, method of characteristics and continuity argument. First, we derive the exponential decay estimate for the Lyapunov functionals of thermo-mechanical observables for the kinetic density $f$ with a compact support. Second, we show that the supports of $f$ in $(x,v,θ)$ shrink to a periodically rotating one-point cluster exponentially fast. Our sufficient framework is formulated in terms of initial configuration, coupling strengths and communication weight functions.

This paper is devoted to the study of the asymptotic stability of the shock wave of the outflow problem governed by the one-dimensional radiative Euler equations, which are a fundamental system in the radiative hydrodynamics with many practical applications in astrophysical and nuclear phenomena. The outflow problem means that the flow velocity on the boundary is negative. Comparing with our previous work on the asymptotic stability of the rarefaction wave of the outflow problem for the radiative Euler equations in [6], two points should be pointed out. On one hand, boundary condition on velocity is considered instead of boundary condition on temperature, which induces a perfect boundary condition on anti-derivative perturbations so that boundary estimates on perturbed unknowns are trickily and smoothly established. On the other hand, the rarefaction wave is an expansive wave, while the shock wave is a compressive wave. So we need take good advantages of properties of the shock wave instead. Our investigation on the outflow problem provides a good understanding on the radiative effect and boundary effect in the setting of shock wave.

This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potentials with angular cutoff. We construct the smooth steady solutions via a perturbation approach when the deformation strength is sufficiently small. The steady solution is a spatially homogeneous non Maxwellian state and may have the polynomial tail at large velocities. Moreover, we also establish the long time asymptotics toward steady states for the Cauchy problem on the corresponding spatially inhomogeneous equation in torus, which in turn gives the non-negativity of steady solutions.

As a problem that dates back to the end of the seventieth century, the brachistochrone problem is one of the oldest problems in the calculus of variations and as such, has generated a myriad of publications. However, in most classical texts and in most papers, the favored way to solve the problem is to make two a priori assumptions, viz., that the brachistochrone lies in a vertical plane, and that it can be represented as the graph of a function in this plane; besides, with few exceptions, the existence of a solution is not rigorously established: instead it is sometimes even taken for granted that the solution is that of the associated Euler-Lagrange equations, even though these are well known to be only necessary conditions for the existence of a minimizer. The objective of this article is to show how all these shortcomings can be very simply, and rigorously, overcome, by means of arguments that do not need any a priori assumptions and that otherwise require only a modicum of basic notions from calculus, so as to rigorously establish the existence and uniqueness of the brachistochrone in full generality. One originality of our approach is that from the outset we seek the brachistochrone as a parameterized curve in the three-dimensional space, i.e., that can be represented by means of three parametric equations, instead of by means of a single graph in a vertical plane. Contrary to expectations, this increase of generality renders the ongoing analysis much simpler. Our objective is thus to show how the direct method of the calculus of variations based on the Euler-Lagrange equations can be used to solve the brachistochrone problem. Otherwise, there are other ways to solve this problem, for instance by means of convex optimization or optimal control; such methods are briefly described at the end of the paper.