In this paper we discuss the localization and the multiplicity of solutions for the stationary Stokes system with variable viscosity and a reaction force term. The results obtained apply to systems with strongly oscillating periodic viscosity and the corresponding homogenized systems.

This paper is concerned with a fully parabolic Patlak-Keller-Segel-Navier-Stokes system

where $Ω ⊂\mathbb{R}^2$ is a smoothly bounded domain and the parameter $χ$ is positive. The main aim of this note is to show that if

then the solution of the above system is global and bounded in time.

In this paper, we investigate the local well-posedness of strong solutions and weak-strong uniqueness property to the incompressible Vlasov-magnetohydrodynamic (Vlasov-MHD) model in $\mathbb{R}^3_x.$ This model consists of a Vlasov equation and the incompressible MHD equations, which interact via the so-called Lorentz force. We first establish the local well-posedness of a strong solution $(f ,u,B)$ by utilizing the delicate energy method for the iteration sequence of approximate solutions, provided that the initial data $(f_0,u_0,B_0)$ are $H^2$-regular and $f_0(x,v)$ has a compact support in the velocity $v.$ We further demonstrate the weak-strong uniqueness property of solutions if $f_0(x,v)∈L^1∩L^∞(\mathbb{R}^3_x×\mathbb{R}^3_v),$ and thereby establish a rigorous connection between the strong and weak solutions to the Vlasov-MHD system. The absence of a dissipation structure in the Vlasov equation and the presence of the strong trilinear coupling term $((u−v)×B)f$ in the model pose significant challenges in deriving our results. To address these issues, we employ the method of characteristics to estimate the size of the support of $f,$ which enables us to overcome the difficulties associated with evaluating the integral $\int_{\mathbb{R}^3} ((u−v)×B)f {\rm d}v.$

In this paper, we investigate a fully parabolic Patlak-Keller-Segel-Navier-Stokes system with a self-consistent mechanism near the Poiseuille flow $A(y^2,0)$ in $\mathbb{T}×\mathbb{R},$ which is more natural than the Couette flow from a biomathematical perspective. We demonstrate that the solution to this system maintains global regularity, provided the amplitude $A$ is suitably large and the non-zero modes of the initial chemical density and vorticity are suitably small. To avoid the complex study of the spectral properties of the linear operator and its resolvent, we prove our result using a straightforward weighted energy method combined with a bootstrap argument.

In this article we introduce a new class of weighted sequence spaces of Sobolev type and prove several compact embedding theorems for them. It is our contention that the chosen class is general enough so as to allow applications in various areas of mathematics and mathematical physics. In particular, our results constitute a generalization of those compact embeddings recently obtained in relation to the spectral analysis of a class of master equations arising in non-equilibrium statistical mechanics. As a byproduct of our considerations, we also prove a theorem of Pitt’s type asserting that under some conditions every linear bounded operator acting between such weighted spaces is compact.

In this paper, we study the Zener-type viscoelastic plate in that heat conduction is determined by the type II Green-Naghdi theory. This model considers the finite propagation of thermal waves, which is compatible with the causality principle. We consider the Cauchy problem of this model and derive the decay estimates of solutions. Furthermore, the optimality of the decay estimate is discussed.