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In this article, we study the Cauchy problem of the chemotaxis-Navier–Stokes system with the consumption and production of chemosignals with a logistic source. The parameters χ ≠ 0 ,   ξ ≠ 0 ,   λ > 0 and μ > 0 . The system is a model that involves double chemosignals; one is an attractant consumed by the cells themselves, and the other is an attractant or a repellent produced by the cells themselves. We prove the global-in-time existence and uniqueness of the weak solution to the system for a large class of initial data on the whole space R 2 . [ABSTRACT FROM AUTHOR]

The paper is concerned with the Navier-Stokes-Nernst-Planck-Poisson system arising from electrohydrodynamics in Rd. By means of the implicit function theorem, we prove the global existence of mild solutions for Cauchy problem of this system with small initial data in critical Besov-Morrey spaces. In comparison to the previous works, our existence result provides a new class of initial data, for which the problem is global solvability. Meanwhile, based on the so-called Gevrey estimates, we verify that the obtained mild solutions are analytic in the spatial variables. As a byproduct, we show the asymptotic stability of solutions as the time goes to infinity. Furthermore, decay estimates of higher-order derivatives of solutions are deduced in Morrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, the three-dimensional rotating Boussinesq equations are considered. By striking new balances between the regularizing effects of the Laplacian dissipation and the dispersive effects of the Coriolis force, we obtain the global existence of mild solutions to Cauchy problem of the three-dimensional rotating Boussinesq equations. Particularly, the size of the norm of initial velocity can be arbitrarily large, provided that the speed of rotation is sufficiently high. [ABSTRACT FROM AUTHOR]

In this article, we consider the Cauchy problem to chemotaxis model coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish the global-in-time existence of the solution when the gravitational potential ϕ and the small initial data (u0,n0,c0) in critical Besov spaces under certain conditions. Moreover, we prove that there exist two positive constants σ 0 and C0 such that if the gravitational potential ϕ ∈Bp3,/1p (R3) and the initial data (u0,n0,c0):=(u01,u02,u03,n0,c0):=(u0h,u03,n0,c0) satisfies . . . for some p; q with …, then the global existence results can be extended to the global solutions without any small conditions imposed on the third component of the initial velocity field u03 in critical Besov spaces with the aid of continuity argument. Our initial data class is larger than that of some known results. Our results are completely new even for three-dimensional chemotaxis-Navier-Stokes system. [ABSTRACT FROM AUTHOR]

In this article, we consider the Cauchy problem to Keller-Segel equations coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let u F:= e tΔ u0; we prove that there exist 2 positive constants σ0 and C0 such that if the gravitational potential [ABSTRACT FROM AUTHOR]

We study global well-posedness for Cauchy problem of the three-dimensional viscous primitive equations of geophysics in the critical functional framework. [ABSTRACT FROM AUTHOR]

In this article, we establish higher-order regularizing rate estimates of solutions to generalized magneto-hydrodynamic equations in Morrey spaces with initial data $(u_{0}, d_{0})$ in Besov-Morrey spaces $\dot {\mathbf{N}}_{r, \lambda,\infty}^{-s}\times\dot{\mathbf{N}}_{r, \lambda , \infty}^{-s}$, where $n\geq2$, $1\leq r<\infty$, $0\leq\lambda< n$, $r>n-\lambda$, $\frac {1}{2}+\frac{n-\lambda}{4r}<\sigma< 1+\frac{n-\lambda}{4r}$, and $s=2\sigma-1-\frac{n-\lambda}{r}$, for which under some smallness condition, the solution of the Cauchy problem is analytic in the spatial variable. Our class of initial data contains strongly singular functions and measures and extends the ones in early work. [ABSTRACT FROM AUTHOR]