We study the backward self-similar solution of Leray’s type for compressible Navier-Stokes equations in dimension two. The existence of weak solutions is established via a compactness argument with the help of an higher integrability of density. Moreover, if the density belongs to $L^∞(\mathbb{R}^2)$ and the velocity belongs to $L^2(\mathbb{R}^2),$ the solution is trivial; that is $(\rho,\mathbf{u})=0.$

In this paper we introduce models of short wave-long wave interactions in the relativistic setting. In this context the nonlinear Schrödinger equation is no longer adequate for describing short waves and is replaced by a nonlinear Dirac equation. Two specific examples are considered: the case where the long waves are governed by a scalar conservation law; and the case where the long waves are governed by the augmented Born-Infeld equations in electromagnetism.

In this paper, we are concerned with the large time behavior of global solutions to the Cauchy problem of one-dimensional compressible Navier-Stokes-Poisson equations with density and/or temperature dependent transport coefficients and large initial data. The initial data are assumed to be without vacuum and mass concentrations, and the same is shown to be hold for the global solution constructed. The proof is based on some detail analysis on uniform positive lower and upper bounds of the specific volume and the absolute temperature.

In this paper, we consider a new type of chemotaxis model with acceleration, which assumes that the advective acceleration, instead of velocity in the classical chemotaxis model, of species is proportional to the chemical signal concentration gradient. This new model has an additional equation governing the velocity field with more delicate boundary conditions. We show that this new type of chemotaxis model with acceleration precludes the blow-up and admits globally bounded solutions in two and three dimensions, in contrast to the classical chemotaxis models which may have blow-up in three dimensions or in two dimensions with a critical mass. Moreover, we numerically illustrate that this new type of chemotaxis model can typically generate aggregation patterns. Some open questions are discussed for further studies.

It has been a fascinating topic in the study of boundary layer theory about the well-posedness of Prandtl equation that was derived in 1904. Recently, new ideas about cancellation to overcome the loss of tangential derivatives were obtained so that Prandtl equation can be shown to be well-posed in Sobolev spaces to avoid the use of Crocco transformation as in the classical work of Oleinik. This short note aims to show that the cancellation mechanism is in fact related to some intrinsic directional derivative that can be used to recover the tangential derivative under some structural assumption on the fluid near the boundary.