The Exact Limits and Improved Decay Estimates for All Order Derivatives of the Global Weak Solutions to a Two-Dimensional Incompressible Dissipative Quasi-Geostrophic Equation

We will accomplish the exact limits for all order derivatives of the global weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will also establish the improved decay estimates with sharp rates for all order derivatives. We will consider two cases for the initial function and the external force and prove the optimal results for both cases. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval''s identity, iteration technique, Lebesgue''s dominated convergence theorem, Gagliardo-Nirenberg-Sobolev interpolation inequality, squeeze theorem, Cauchy-Schwartz''s inequality, etc) existing results (the existence of global weak solutions, the existence of local smooth solution on $(T,\infty)$ and the elementary decay estimate with a sharp rate) and a few novel ideas to obtain the main results.     

THE IMPROVED FOURIER SPLITTING METHOD AND DECAY ESTIMATES OF THE GLOBAL SOLUTIONS OF THE CAUCHY PROBLEMS FOR NONLINEAR SYSTEMS OF FLUID DYNAMICS EQUATIONS

Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall''s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980''s to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for $n$-dimensional incompressible Navier-Stokes equations, for the $n$-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.     

具间断系数的N维拟线性椭圆方程弱解的正则性

讨论了具间断系数的N维拟线性椭圆方程. 利用估计和差分逼近方法,证明了弱解的一阶导数H\"{o}lder连续到方程系数间断的内边界.     

Global regularity for 3D generalized Hall magneto-hydrodynamics equations

For the 3D incompressible Hall magneto-hydrodynamics equations, global regularity of the weak solutions is not established so far. The major difficulty is that the dissipation given by the Laplacian operator is insufficient to control the nonlinearities. Wan obtained the global regularities of the 3D generalized Hall-MHD equations with critical and subcritical hyperdissipation in ({\em Global regularity for generalized Hall-magnetohydrodynamics systems}, Electron. J. Differential Equations, 2015, 2015(179), 1--18). We improve this slightly by making logarithmic reductions in the dissipation and still obtain the global regularity.     

A note on statistical solutions of the three-dimensional Navier–Stokes equations: The stationary case

Stationary statistical solutions of the three-dimensional Navier–Stokes equations for incompressible fluids are considered. They are a mathematical formalization of the notion of ensemble average for turbulent flows in statistical equilibrium in time. They are also a generalization of the notion of invariant measure to the case of the three-dimensional Navier–Stokes equations, for which a global uniqueness result is not known to exist and a semigroup may not be well-defined in the classical sense. The two classical definitions of stationary statistical solutions are considered and compared, one of them being a particular case of the other and possessing a number of useful properties. Furthermore, the so-called time-average stationary statistical solutions, obtained as generalized limits of time averages of weak solutions as the averaging time goes to infinity are shown to belong to this more restrictive class. A recurrent type result is also obtained for statistical solutions satisfying an accretion condition. Finally, the weak global attractor of the three-dimensional Navier–Stokes equations is considered, and in particular it is shown that there exists a topologically large subset of the weak global attractor which is of full measure with respect to that particular class of stationary statistical solutions and which has a certain regularity property.     

The combined inviscid and non-resistive limit for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions

In this paper, we establish the existence of the global weak solutions for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions for the velocity field and the magnetic field in a bounded domain ? ? R3. Furthermore,we prove that as the viscosity and resistivity coefficients go to zero simultaneously, these weak solutions converge to the strong one of the ideal nonhomogeneous incompressible magnetohydrodynamic equations in energy space.     

Attractors for the Long-short Wave Equations

In the present paper, we study the long time behaviour of solutions for the long-short wave equations with zero order dissipation. We first construct the global weak attractor for this system in H²_{per} × H¹_{per}. And then by exact analysis of two energy equations, we show that the global weak at attractor is actually the global strong attractor in H²_{per}.     

Determining nodes of the global attractor for an incompressible non-Newtonian fluid

This paper estimates the finite number of the determining nodes to the equations for an incompressible non-Newtonian fluid with space-periodic or no-slip boundary conditions. The authors prove that, whenever the second order derivatives of two different solutions within the global attractor have the same time-asymptotic behavior at finite number of points in the physical space, then the two solutions possess the same time-asymptotic behavior at almost everywhere points of the physical space.     

Global existence and asymptotic analysis of weak solutions to the equations of Ferrohydrodynamics

In this paper, we consider the three-dimensional equations of Ferrohydrodynamics describing the motion of an incompressible ferrofluid under the action of an external magnetic field. We first consider the system introduced by Rosensweig. We establish the existence of globally defined finite energy weak solutions and also investigate the large-time behavior of such solutions. Then we study the asymptotic behavior of the solutions as some physical parameters in the angular momentum equations tend to zero, we prove that the limit functions will be weak solutions of the equations proposed by Shliomis. This shows rigorously the exact relationship of these two principal systems which describe the motion of ferrofluids.     

Asymptotic behavior of exact solutions for the Cauchy problem to the 3D cylindrically symmetric Navier–Stokes equations

In this paper, we establish exact solutions of the Cauchy problem for the 3D cylindrically symmetric incompressible Navier–Stokes equations and further study the global existence and asymptotic behavior of solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Global Solutions to the Coupled Chemotaxis-Fluid Equations

In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier–Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small.     

Global existence of weak solutions to a diffuse interface model for magnetic fluids

This article is devoted to the derivation and analysis of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier–Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn–Hilliard equations. We show global in time existence of weak solutions to the system using the time discretization method.     

Approximation by an iterative method for regular solutions for incompressible fluids with mass diffusion

We study the approximation by means of an iterative method towards strong (and more regular) solutions for incompressible Navier-Stokes equations with mass diffusion. In addition, some convergence rates for the error between the approximation and the exact solution will be given, for weak, strong and more regular norms.     

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.     

Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.     

Global weak solutions for variable density asymmetric incompressible fluids

We establish the existence of global in time weak solutions for the equations of asymmetric incompressible fluids with variable density, when the initial density is not necessarily strictly positive.     

Global existence of weak and strong solutions to Cucker–Smale–Navier–Stokes equations in R2

We study the existence theory for the Cucker–Smale–Navier–Stokes (in short, CS–NS) equations in two dimensions. The CS–NS equations consist of Cucker–Smale flocking particles described by a Vlasov-type equation and incompressible Navier–Stokes equations. The interaction between the particles and fluid is governed by a drag force. In this study, we show the global existence of weak solutions for this system. We also prove the global existence and uniqueness of strong solutions. In contrast with the results of Bae et al. (2014) on the CS–NS equations considered in three dimensions, we do not require any smallness assumption on the initial data.     

A coupled chemotaxis-fluid model: Global existence

We consider a model arising from biology, consisting of chemotaxis equations coupled to viscous incompressible fluid equations through transport and external forcing. Global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the chemotaxis-Navier-Stokes system in two space dimensions, we obtain global existence for large data. In three space dimensions, we prove global existence of weak solutions for the chemotaxis-Stokes system with nonlinear diffusion for the cell density.     

Asymptotic expansions in time for rotating incompressible viscous fluids

We study the three-dimensional Navier–Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray–Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and without the zero spatial average of the solutions. Our method makes use of the Poincaré waves to rewrite the equations, and then implements the Gevrey norm techniques to deal with the resulting time-dependent bi-linear form. Special solutions are also found which form infinite dimensional invariant linear manifolds.