Global well-posedness for the dynamical Q-tensor model of liquid crystals

We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Global attractor for a low order ODE model problem for transition to turbulence

Many researchers have studied simple low order ODE model problems for fluid flows in order to gain new insight into the dynamics of complex fluid flows. We investigate the existence of a global attractor for a low order ODE system that has served as a model problem for transition to turbulence in viscous incompressible fluid flows. The ODE system has a linear term and an energy‐conserving, non‐quadratic nonlinearity. Standard energy estimates show that solutions remain bounded and converge to a global attractor when the linear term is negative definite, that is, the linear term is energy decreasing; however, numerical results indicate the same result is true in some cases when the linear term does not satisfy this condition. We give a new condition guaranteeing solutions remain bounded and converge to a global attractor even when the linear term is not energy decreasing. We illustrate the new condition with examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Global existence and time decay estimate of solutions to the Keller–Segel system

We consider the chemotaxis‐Navier–Stokes system 1.1-1.4 (Keller–Segel system) in the whole space, which describes the motion of oxygen‐driven bacteria, eukaryotes, in a fluid. We proved the global existence and time decay estimate of solutions to the Cauchy problem 1.1-1.2 in with the small initial data. Moreover, when the fluid motion is described by the Stokes equations, we established the global weak solutions to 1.3-1.4 in with the potential function ? is small and the initial density n₀(x) has finite mass.     

Global weak solutions to the Nordström-Vlasov system

The Nordström-Vlasov system is a Lorentz invariant model for a self-gravitating collisionless gas. We establish suitable a priori bounds on the solutions of this system, which together with energy estimates and the smoothing effect of “momentum averaging” yield the existence of global weak solutions to the corresponding initial value problem. In the process we improve the continuation criterion for classical solutions which was derived recently. The weak solutions are shown to preserve mass.     

On Solutions to a Two‐Component Generalized Camassa‐Holm Equation

We consider a two‐component Camassa–Holm system which arises in shallow water theory. We analyze a wave breaking mechanism and the global existence of solutions. First, we discuss the local well posedness and a blow up mechanism, then establish some new blow up criteria for this system formulated either on the line or with space‐periodic initial conditions. Finally, the existence of global solutions is analyzed.     

Global existence of weak solutions to a weakly dissipative modified two‐component Dullin–Gottwald–Holm system

We consider a weakly dissipative modified two‐component Dullin–Gottwald–Holm system. The existence of global weak solutions to the system is established. We first give the well‐posedness result of viscous approximate problem and obtain the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the system. Copyright © 2014 John Wiley & Sons, Ltd.     

Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity

ABSTRACT

We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier–Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.     

On the initial‐boundary value problem of the incompressible viscoelastic fluid system

In this paper, we shall establish the local well‐posedness of the initial‐boundary value problem of the viscoelastic fluid system of the Oldroyd model. We shall also prove that the local solutions can be extended globally and that the global solutions decay exponentially fast as time goes to infinity provided that the initial data are sufficiently close to the equilibrium state. © 2007 Wiley Periodicals, Inc.     

Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.     

The long‐time behaviour of the thermoconvective flow in a porous medium

For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. Copyright © 2004 John Wiley & Sons, Ltd.     

Analysis of nonlocal model of compressible fluid in 1‐D

We study a nonlocal modification of the compressible Navier–Stokes equations in mono‐dimensional case with a boundary condition characteristic for the free boundaries problem. From the formal point of view, our system is an intermediate between the Euler and Navier–Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient, we obtain the local and global existence of solutions. Particularly, we show the uniform in time bound on the density of fluid. Copyright © 2010 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.     

Existence of global weak solutions to 2D reduced gravity two-and-a-half layer model

We study the global existence of weak solutions to a reduced gravity two-and-a-half layer model appearing in oceanic fluid dynamics in two-dimensional torus. Based on Faedo–Galerkin method and weak convergence method, we construct the global weak solutions which are renormalized in velocity variable, where the technique of renormalized solutions was introduced by Lacroix-Violet and Vasseur (2018). Besides, we prove that the renormalized solutions are weak solutions, which satisfy the basic energy inequality and Bresch–Desjardins entropy inequality, but not the Mellet–Vasseur type inequality. In the proof, we use the reduced gravity two-and-a-half layer model with drag forces and capillary term as approximate system. It should be pointed out that only when the capillary term vanishes, we prove the existence of renormalized solution to the approximation system, which is different from Lacroix-Violet and Vasseur (2018) with the quantum potential.     

On the linearized system of equations for the condensate–normal fluid interaction at very low temperature

The linearization around one of its equilibrium of a system that describes the correlations between the superfluid component and the normal fluid part of a condensed Bose gas in the approximation of very low temperature and small condensate density is studied. A simple and transparent argument gives a necessary and sufficient condition for the existence of global solutions satisfying the conservation of the total number of particles and energy. The global solutions describe the evolution in time of the density of the thermal cloud and, unlike in previous work, that of the condensate's density. Their convergence to a suitable stationary state is also shown and rates of convergence for the normal fluid and superfluid components are obtained.     

Existence of Weak Solutions Up to Collision for Viscous Fluid‐Solid Systems with Slip

We study in this paper the movement of a rigid solid inside an incompressible Navier‐Stokes flow within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no‐slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents obtaining global H¹ bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions inappropriate. © 2014 Wiley Periodicals, Inc.     

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.     

Stability of nonconstant steady‐state solutions for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor

We consider the periodic problem for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor. By choosing a suitable symmetrizers and using an induction argument on the order of the time‐space derivatives of solutions in energy estimates, we obtain the global stability of solutions with exponential decay in time near the nonconstant steady‐states for 2‐fluid nonisentropic Euler‐Poisson equations. This improves the results obtained for models with temperature diffusion terms by using the pressure functions p^ν in place of the unknown variables densities n^ν.     

On a micro‐macro model for polymeric fluids near equilibrium

In this paper, we study a micro‐macro model for polymeric fluid. The system involves coupling between the macroscopic momentum equation and a microscopic evolution equation describing the combined effects of the microscopic potential and thermofluctuation. We employ an energetic variation procedure to explore the relation between the macroscopic transport of the particles and the induced elastic stress due to the microscopic structure. For the initial data not far from the equilibrium, we prove the global existence and uniqueness of classical solutions to the system. © 2006 Wiley Periodicals, Inc.