Global Existence Of Regular Solutions Research Articles

Incompressible Navier–Stokes limit from nonlinear Vlasov–Fokker–Planck equation

The aim of this paper is to justify the rigorous derivation of the incompressible Navier–Stokes equations from the nonlinear Vlasov–Fokker–Planck (VFP) equation with a constant temperature. Under the incompressible Navier–Stokes scaling, we first establish the global existence of regular solutions to the rescaled nonlinear VFP equation near the Maxwellian, obtaining some uniform bound estimates. We then show the strong convergence of solution to the nonlinear VFP equation towards the incompressible Navier–Stokes system.

Global Regular Axially Symmetric Solutions to the Navier–Stokes Equations: Part 2

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder Ω⊂R3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry, and S2 is perpendicular to it. We have two parts of S2. On S1 and S2, we impose vanishing of the normal component of velocity and the angular component of vorticity. Moreover, we assume that the angular component of velocity vanishes on S1 and the normal derivative of the angular component of velocity vanishes on S2. We prove the existence of global regular solutions. To prove this, the coordinate of velocity along the axis of symmetry must vanish on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The paper is a generalization of Part 1, where the periodic boundary conditions are prescribed on S2. The transformation is not trivial because it needs to examine many additional boundary terms and derive new estimates.

Global Regular Axially Symmetric Solutions to the Navier–Stokes Equations: Part 1

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder Ω⊂R3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry and S2 is perpendicular to it. We have two parts of S2. For simplicity, we assume the periodic boundary conditions on S2. On S1, we impose the vanishing of the normal component of velocity, the angular component of velocity, and the angular component of vorticity. We prove the existence of global regular solutions. To prove this, it is necessary that the coordinate of velocity along the axis of symmetry vanishes on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The weighted spaces used are such that the stream function divided by the radius must vanish on the axis of symmetry. Currently, we do not know how to relax this restriction. In part 2 of this topic, the periodic boundary conditions on S2 are replaced by the conditions that both the normal component of velocity and the angular component of vorticity must vanish. Moreover, it is assumed that the normal derivative of the angular component of velocity also vanishes on S2. A transformation from part 1 to part 2 is not trivial because it needs new boundary value problems, so new estimates must be derived.

Reacting Multi-component Fluids: Regular Solutions in Lorentz Spaces

The paper deals with analysis of a model of a multi-component fluid admitting chemical reactions. The flow is considered in the incompressible regime. The main result shows global existence of regular solutions under assumption of suitable smallness conditions. In order to control the solutions a special structure condition on the derivatives of chemical production functions determining the reactions is required. The existence is shown in a new critical functional framework of Lorentz spaces of type $L_{p,r}(0,T;L_q)$, which allows to control the integral $\int_0^\infty \|\nabla u(t)\|_{\infty} dt$.

Existence of weak and regular solutions for Keller–Segel system with degradation coupled to fluid equations

We establish the global well-posedness for the following chemotaxis-fluid system{∂tn+u⋅∇n=Δn−∇⋅(n∇c)−μnq,∂tc+u⋅∇c=Δc−c+n,∂tu+κ(u⋅∇)u+∇P=Δu−n∇ϕ,∇⋅u=0, in Rd, d=2,3, where μ>0, q>2−1d and κ∈{0,1}. For either q≥2, (κ,d)=(1,2) or q>2, (κ,d)=(0,3), we prove the global existence of regular solutions. In case that q>2−1d and κ=0, very weak solutions are constructed as well.

Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces

This paper is devoted to the study of the modified quasi-geostrophic equation \begin{document}$ \partial_t\theta+u\cdot\nabla\theta+\nu\Lambda^\alpha\theta = 0 \ \ \mbox{ with } \ \ u = \Lambda^\beta\mathcal{R}^\perp\theta $\end{document} in $ \mathbb{R}^2 $. By the Littlewood-Paley theory, we obtain the local well-posedness and the smoothing effect of the equation in critical Besov spaces. These results are applied to show the global existence of regular solutions for the critical case $ \beta = \alpha-1 $ and the existence of regular solutions for large time $ t>T $ with respect to the supercritical case $ \beta >\alpha -1 $ in Besov spaces. Earlier results for the equation in Hilbert spaces $ H^s $ spaces are improved.

Global classical solutions to 1D full compressible Navier–Stokes equations with the Robin boundary condition on temperature

In this paper, we consider the initial boundary value problem for the one-dimensional Navier–Stokes equations for viscous compressible and heat-conducting fluids in a bounded domain with the Robin boundary condition on temperature. There are few results until now about global existence of regular solutions to the full Navier–Stokes equations with the Robin boundary condition on temperature. By the analysis of the effect of boundary dissipation, we derive the global existence of classical solution to the corresponding initial boundary value problem with large initial data and vacuum. This result could be viewed as the first one on the global well-posedness of classical solutions to the full Navier–Stokes equations in a bounded domain with the Robin boundary condition on temperature.

Three-dimensional thermo-visco-elasticity with the Einstein-Debye $(\theta^3+\theta)$-law for the specific heat. Global regular solvability

A three-dimensional thermo-visco-elastic system for the Kelvin-Voigt type material at small strain is considered. The system involves the constant heat conductivity and the specific heat satisfying the Einstein-Debye $(\theta^3+\theta)$-law. Such a nonlinear law, relevant at relatively low temperatures, represents the main novelty of the paper. The existence of global regular solutions is proved without the small data assumption. The crucial part of the proof is the strictly positive lower bound on the absolute temperature $\theta$. The problem remains open in the case of the Debye $\theta^3$-law. The existence of local in time solutions is proved by the Banach successive approximations method. The global {\em a priori} estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time.

Global well posedness for a 2D drift–diffusion–Maxwell system

We prove global existence of regular solutions to the drift–diffusion–Maxwell system in two space dimension. We also provide an exponential growth estimate for the norm of the solution when the time goes to infinity.

Blowup and global solutions in a chemotaxis–growth system

We study a Keller–Segel type of system, which includes growth and death of the chemotactic species and an elliptic equation for the chemo-attractant. The problem is considered in bounded domains with smooth boundary as well as in the whole space. In case the random motion of the chemotactic species is neglected, a hyperbolic–elliptic problem results, for which we characterize blow-up of solutions in finite time and existence of regular solutions globally in time, in dependence on the systems parameters. In this case, convexity of the domain is needed. For the parabolic–elliptic problem in dimensions three and higher, we establish global existence of regular solutions in a limiting case, which is an extension of the results given by Tello and Winkler (2007).

GLOBAL SOLUTIONS TO THE NONLINEAR HYPERBOLIC PROBLEM

We prove the existence and uniqueness of global regular solutions to the mixed problem for the nonlinear hyperbolic equation with nonlinear damping. Utt a (u) u + I u t IP Ut = f (x .t ) in (O, 1) x (O, T ) = Q, u (o.. ) = u (l,t ) = 0, u(x,O) = Uo(x), ut (x,O) = ui (x), Where a (u ) ~ ao > 0, p > 1.No restrictions on a size of uo, ui, f are imposed. It is well-known that quasilinear hyperbolic equations, generally speaking, do not have regular solutions for all t >0_Their solutions can blow up at a finite period of time. See examples of such singularities in [1, 3]. On the other hand,it was observed that adding a linear damping to the nonlinear hyperbolic equations one can expect the existence of global regular solutions provided initial conditions and right-hand side have sufficiently small appropriate norms, [2]. Moreover, in [3] was shown that the presence of the nonlinear damping allows to prove the existence of regular solutions for the equation without restrictions on a size ofthe initial data and f. Later, using the idea of [3], we proved in [4] the existence of regular solutions for the damped Carrier equation. Utt -M ~u(t)12)L1U + alu, IP u, = f without smallness conditions for the initial data and f . • Supported by CNP-Brazil as Visiting Profesor as the State University ofMaringa. Here, we continue to exploit this idea and consider the following nonlinear mixed problem.

On Global Existence of Solutions of the Neumann Problem for Spherically Symmetric Nonlinear Viscoelasticity in a Ball

We examine spherically symmetric solutions to the viscoelasticity system in a ball with the Neumann boundary conditions. Imposing some growth restrictions on the nonlinear part of the stress tensor, we prove the existence of global regular solutions for large data in the weighted Sobolev spaces, where the weight is a power function of the distance to the centre of the ball. First, we prove a global a priori estimate. Then existence is proved by the method of successive approximations and appropriate time extension.

Global Classical Large Solutions to Navier--Stokes Equations for Viscous Compressible and Heat-Conducting Fluids with Vacuum

We consider the one-dimensional Navier--Stokes equations for viscous compressible and heat-conducting fluids (i.e., the full Navier--Stokes equations). Because of the stronger nonlinearity and degeneracies of the equations due to the temperature equation and vanishing of density (i.e., appearance of vacuum), respectively, there are few results until now about global existence of regular solutions to the full Navier--Stokes equations. In the paper, we get a unique global classical solution to the equations with large initial data and vacuum. Our analysis is based on some delicate energy estimates together with some new ideas which were used in our previous paper [Ding, Wen, and Zhu, J. Differential Equations, 251 (2011), pp. 1696--1725] to get the high-order estimates of the solutions. This result could be viewed as the first one on the global well-posedness of regular (classical) solutions to the Navier--Stokes equations for viscous compressible and heat-conducting fluids when initial data may be large and vacuum could appear.

Notes on the Global Well-Posedness for the Maxwell-Navier-Stokes System

Masmoudi (2010) obtained global well-posedness for 2D Maxwell-Navier-Stokes system. In this paper, we reprove global existence of regular solutions to the 2D system by using energy estimates and Brezis-Gallouet inequality. Also we obtain a blow-up criterion for solutions to 3D Maxwell-Navier-Stokes system.

Global attractor for the Navier–Stokes equations with the heat convection in a cylindrical pipe

Global existence of regular solutions to the Navier–Stokes equations coupled with the heat convection in a cylindrical pipe has already been shown. In this paper, we prove the existence of the global attractor to the equations and convergence of their solutions to a stationary one. Copyright © 2012 John Wiley & Sons, Ltd.

On the heat flow equation of surfaces of constant mean curvature in higher dimensions

In this paper, we consider the heat flow for the H-system with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2 2 - n as time goes to infinity.

Global Existence and Regularity for the Full Coupled Navier–Stokes andQ-Tensor System

In this paper we study the full system of incompressible liquid crystals, as modeled in the Q-tensor framework. Under certain conditions we prove the global existence of weak solutions in dimension two or three and the existence of global regular solutions in dimension two. We also prove the weak-strong uniqueness of the solutions, for sufficiently regular initial data.

On the heat flow of equation of surfaces of constant mean curvatures

We consider the initial and boundary value problem of heat flow of equation of surfaces of constant mean curvatures. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solutions to the heat flow that converges to zero in H1 exponentially as time goes to infinity.

Regularity and Blow up for Active Scalars

We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissi- pation. We also recall what is known about the possibility of finite time blow up in the supercritical regime.

Global well posedness for the Maxwell–Navier–Stokes system in 2D

We prove global existence of regular solutions to the full MHD system (or more precisely the Maxwell–Navier–Stokes system) in 2D. We also provide an exponential growth estimate for the H s norm of the solution when the time goes to infinity.