On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems

Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data

In this paper we prove the global well-posedness for a three-dimensional Boussinesq system with axisymmetric initial data. This system couples the Navier–Stokes equation with a transport-diffusion equation governing the temperature. Our result holds uniformly with respect to the heat conductivity coefficient κ?0$\kappa ?0$ which may vanish.     

Solitary-wave solutions to a dual equation of the Kaup–Boussinesq system

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup–Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.     

On the global well-posedness of 3-D axi-symmetric Navier–Stokes system with small swirl component

In this paper, we prove the local well-posedness of 3-D axi-symmetric Navier–Stokes system with initial data in the critical Lebesgue spaces. We also obtain the global well-posedness result with small initial data. Furthermore, with the initial swirl component of the velocity being sufficiently small in the almost critical spaces, we can still prove the global well-posedness of the system.     

On well-posedness for the Benjamin–Ono equation

We prove existence and uniqueness of solutions for the Benjamin–Ono equation with data in \(H^{s}({\mathbb{R}})\) , s > 1/4. Moreover, the flow is hölder continuous in weaker topologies.     

On the asymptotic behaviour of a pantograph–type difference equation

In this paper we study the asymptotic behaviour of solutions of the pantograph-type differnce equation, and obtain aymptotic estimates, which can imply asymptotic stability or stability of solutions     

Global well-posedness of the three dimensional incompressible anisotropic Navier–Stokes system

In this paper, we study the global well-posed problem for the three dimensional incompressible anisotropic Navier–Stokes system (ANS) with initial data in the scaling invariant Besov–Sobolev type spaces. We prove that (ANS) has a unique global solution provided that the initial vertical velocity is large while initial horizontal data are sufficiently small compared with the horizontal viscosity. In particular, our result implies the global well-posedness of (ANS) with highly oscillating initial data.     

On the existence of standing waves for a davey–stewartson system

We consider the standing waves for the Davey–Stewartson system in R² and R³. By reducing this system to a single nonlinear equation of Schrödinger type, we study the existence, the regularity and asymptotics of ground states.     

On the solutions of the cross-coupled Camassa–Holm system

In this paper, we study the Cauchy problem for a recently derived system of two cross-coupled Camassa–Holm equations. We firstly establish the local well-posedness result of this system in Besov spaces by using Littlewood–Paley decomposition and the transport equation theory, and then present a precise blow-up scenario for strong solutions.     

On the Arkhipov–Karatsuba multivariate system of congruences

The Arkhipov–Karatsuba multivariate system of congruences modulo any prime greater than the degrees of forms in this system is solvable for any right-hand sides and any number of variables larger than 8(n + 1)mlog₂(rn) + 12(n + 1)m + 4(n + 1), where n is the degree of the forms in the system and \(m = \left( {\begin{array}{*{20}{c}} {n + r - 1} \\ {r - 1} \end{array}} \right)\) is the number of congruences.     

On a Hamiltonian version of a three-dimensional Lotka–Volterra system

In this paper we present some relevant dynamical properties of a three-dimensional Lotka–Volterra system from the Poisson dynamics point of view.     

On a model of thermoviscoelasticity of Jeffreys–Oldroyd type

In the plane case, the initial–boundary value problem for a thermoelastic medium model with a rheological relation determined by the Jeffreys–Oldroyd model is shown to be nonlocally weakly solvable. The study is based on separating the system, reducing it to an operator equation, and performing an iterative process.     

On a homoclinic manifold of a coupled long-wave–short-wave system

A Bäcklund transformation is obtained for linearly unstable spatially independent plane-wave solutions of a system of coupled long-wave–short-wave resonance equations. Explicit expressions are constructed for the periodic orbits lying on a homoclinic manifold of a torus of planewaves by evaluating the Bäcklund transformation at double points of an irreducible factor of the Floquet spectral curve of the associated scattering problem.     

On the well-posedness of the vacuum Einstein��s equations

The Cauchy problem of the vacuum Einstein’s equations aims to find a semi-metric g _αβ of a spacetime with vanishing Ricci curvature R _α,β and prescribed initial data. Under the harmonic gauge condition, the equations R _α,β  = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric h _ab and a second fundamental form K _ab . A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h _ab , K _ab ) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of the present article is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.