We establish the solvability of second order divergence-type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be merely measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable, the coefficients have locally small mean oscillations. We also obtain the corresponding $W^1_p$-solvability of second order elliptic systems in divergence form. This type of system arises from the problems of linearly elastic laminates and composite materials. Our results are new even for scalar equations, and the proofs differ from and simplify the methods used previously in [H. Dong and D. Kim, <i>Arch. Ration. Mech. Anal.</i>, 196 (2010), pp. 2570]. As an application, we improve a result by Chipot, Kinderlehrer, and Vergara-Caffarelli [<i>Arch. Ration. Mech. Anal.</i>, 96 (1986), pp. 8196] on

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This article is concerned with the time periodic solution to the isentropic compressible Navier-Stokes equations in a periodic domain. Using an approach of parabolic regularization, we first obtain the existence of the time periodic solution to a regularized problem under some smallness and symmetry assumptions on the external force. The result for the original compressible Navier-Stokes equations is then obtained by a limiting process. The uniqueness of the periodic solution is also given.

The purpose of this paper is to study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa–Holm equation. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a particle method to the Camassa–Holm equation and show that the nonlinear interaction among the peakon solutions resembles an elastic collision, i.e., the total energy and momentum of the system before the peakon interaction is equal to the total energy and momentum of the system after the collision. From this result, we provide several numerical illustrations which support the analytical study, as well as showcase the merits of using a particle method to simulate solutions to the Camassa–Holm equation under a wide class of initial data.

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