In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence,...

In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We...

In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence,...

We consider the homogeneous time-dependent Oseen system in the whole space ${ℝ}^3$. The initial data is assumed to behave as $O\left(\right|x{|}^{-1-ϵ})$, and its gradient as $O\left(\right|x{|}^{-3/2-ϵ})$, when $\left|x\right|$ tends to infinity, where $ϵ$ is a fixed positive number. Then we show that the velocity $u$ decays according to the equation $\left|u(x,t)\right|=O\left(\right|x{|}^{-1})$, and its spatial gradient ${\nabla }_{x}u$ decreases with the rate ${\left|x\right|}^{-3/2}$, for $\left|x\right|$ tending to infinity, uniformly with respect to the time variable $t$. Since these decay rates are optimal even in the stationary case, they should also be the best possible...

The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p,1}^{n/p-1}\left({ℝ}^n\right)\times {\dot{B}}_{p,1}^{n/p}\left({ℝ}^n\right)$ with $n<p<2n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

We study the uniqueness and regularity of Leray-Hopf's weak solutions for the MHD equations with dissipation and resistance in different frameworks. Using different kinds of space-time estimates in conjunction with the Littlewood-Paley-Bony decomposition, we present some general criteria of uniqueness and regularity of weak solutions to the MHD system, and prove the uniqueness and regularity criterion in the framework of mixed space-time Besov spaces by applying Tao's trichotomy method.

The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. In the 3D case Guo [7] first constructed a global smooth irrotational solution by using the dispersive Klein-Gordon effect. It has been conjectured that same results should hold in the two-dimensional case. In our recent work [13], we proved the existence of a family of smooth solutions by constructing the wave operators for the 2D system....

In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index s &gt; 2 to obtain local solutions for any initial data, and global solutions for small initial data.

The Rayleigh-Bénard convection for a couple-stress fluid with a thermorheological effect in the presence of an applied magnetic field is studied using both linear and non-linear stability analysis. This problem discusses the three important mechanisms that control the onset of convection; namely, suspended particles, an applied magnetic field, and variable viscosity. It is found that the thermorheological parameter, the couple-stress parameter, and the Chandrasekhar number influence the onset of...

The Navier-Stokes system is studied on a family of domains with rough boundaries formed by oscillating riblets. Assuming the complete slip boundary conditions we identify the limit system, in particular, we show that the limit velocity field satisfies boundary conditions of a mixed type depending on the characteristic direction of the riblets.

We consider the magneto-micropolar fluid flow in a bounded domain Ω ⊂ ℝ². The flow is modelled by a system of PDEs, a generalisation of the two-dimensional Navier-Stokes equations. Using the Galerkin method we prove the existence and uniqueness of weak solutions and then using the ℓ-trajectories method we prove the existence of the exponential attractor in the dynamical system associated with the model.