Following the global strategy introduced recently by Bona, Lannes and Saut in Ref. 7, we derive here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, the presence of surface tension and with uneven bottoms. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on ℝd, d = 1, 2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators and the surface tension term with respect to suitable small parameters that depend variously on the amplitude, wavelengths and depth ratio of the two layers. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established.
We study the first initial-boundary-value problem for the three-dimensional non-autonomous Navier-Stokes-Voigt equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Faedo-Galerkin method. We then show the existence of a unique minimal finite-dimensional pull-back Dσ-attractor for the process associated with the problem, with respect to a large class of non-autonomous forcing terms. We also discuss relationships between the pull-back attractor, the uniform attractor and the global attractor.
We study the existence of nontrivial weak solutions for the following boundary value problemwhere is a bounded domain in R N (N ≥ 2), λ is a strongly degenerate elliptic operator, the nonlinearity f (x, u) is subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.
In this paper we study the decay characterization in the space H K σ (R n ) of solutions to the viscous Camassa-Holm equations (VCHE) in the whole space R n (n = 2, 3, 4), namely,where m + 2p K, r * = r * (v 0 ) is the decay character of the initial datumWe also get the optimal lower bounds for decay rates of solutions to VCHE when −n/2 < r * 1. In particular, when v 0 ∈ H K σ (R n ) ∩ L 1 (R n ) has decay character r * (v 0 ) = 0, then we recover the previous results of Bjorland
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.