In this paper, we introduce and study the weighted variable Hardy space on
domains. We prove the atomic decompositions of this space, and as application, we
figure out its dual space.

The current paper is principally motivated by establishing the global wellposedness to the three-dimensional Boussinesq system with zero diffusivity in thesetting of axisymmetric flows without swirling with v0 ∈ H12(R3) ∩ B˙ 30,1(R3) anddensity ρ0 ∈ L2(R3)∩B˙ 30,1(R3). This respectively enhances the two results recentlyaccomplished in Danchin and Paicu (2008) and Hmidi and Rousset (2010). Ourformalism is inspired, in particular for the first part from Abidi (2008) concerningthe axisymmetric Navier–Stokes equations once v0 ∈ H12(R3) and external forcef ∈ L2 loc(R+; Hβ(R3)), with β > 1 4 . This latter regularity on f which is thedensity in our context is helpless to achieve the global estimates for Boussinesqsystem. This technical defect forces us to deal once again with a similar proof tothat of Abidi (2008) but with f ∈ Lβ loc(R+; L2(R3)) for some β > 4. Second, weexplore the gained regularity on the density by considering it as an external forcein order to apply the study already obtained to the Boussinesq system.

In this paper, we consider a nonlinear wave equation with damping and source terms of variable-exponent types. First, we use the stable-set method to prove a global result. Then, by applying an integral inequality due to Komornik, we obtain the stability result.

In this paper, we present a new technique for constructing a nonstationary wavelet. The key idea relies on the following: for each wavelet level, we solve the Bezout’s equation and we propose a positive solution over the interval [–1, 1]. Using the Bernstein’s polynomials we approximate this proposed positive solution with the intention to perform a spectral factorization.

In this paper, we consider a coupled system of two nonlinear
hyperbolic equations with variable-exponents in the damping and source
terms. Under suitable assumptions on the intial data and the variable
exponents, we prove a global existence theorem, using the Stable-set
method. Then, we establish a decay estimate of the solution energy, by
Komornik’s integral approach.

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in [19, 20]. Roughly speaking, we show essentially that if the initial data (v0; ρ0) is axisymmetric
and (!0; ρ0) belongs to the critical space L1(Ω) ×L1(R3), with !0 is the initial
vorticity associated to v0 and Ω = f(r; z) 2 R2 : r > 0g, then the viscous
Boussinesq system has a unique global solution.

This paper presents a new procedure for designing a fractional order unknown input observer (FOUIO) for nonlinear systems represented by a fractional-order Takagi–Sugeno (FOTS) model with unmeasurable premise variables (UPV). Most of the current research on fractional order systems considers models using measurable premise variables (MPV) and therefore cannot be utilized when premise variables are not measurable. The concept of the proposed is to model the FOTS with UPV into an uncertain FOTS model by presenting the estimated state in the model. First, the fractional-order extension of Lyapunov theory is used to investigate the convergence conditions of the FOUIO, and the linear matrix inequalities (LMIs) provide the stability condition. Secondly, performances of the proposed FOUIO are improved by the reduction of bounded external disturbances. Finally, an example is provided to clarify the proposed method. The obtained results show that a good convergence of the outputs and the state estimation errors were observed using the new proposed FOUIO.

This paper presents a new procedure for designing a fractional order unknown input observer (FOUIO) for nonlinear systems represented by a fractional-order Takagi–Sugeno (FOTS) model with unmeasurable premise variables (UPV). Most of the current research on fractional order systems considers models using measurable premise variables (MPV) and therefore cannot be utilized when premise variables are not measurable. The concept of the proposed is to model the FOTS with UPV into an uncertain FOTS model by presenting the estimated state in the model. First, the fractional-order extension of Lyapunov theory is used to investigate the convergence conditions of the FOUIO, and the linear matrix inequalities (LMIs) provide the stability condition. Secondly, performances of the proposed FOUIO are improved by the reduction of bounded external disturbances. Finally, an example is provided to clarify the proposed method. The obtained results show that a good convergence of the outputs and the state estimation errors were observed using the new proposed FOUIO.

The current paper deals with the local well-posedness problem for the twodimensional partial viscous Boussinesq system when the initial vorticity belongs to the patch class. We prove in particular some results concerning the regularity persistence of the patch boundary and establish the convergence towards the inviscid limit when the molecular diffusivity goes to zero. 

We propose a primal-dual interior-point algorithm for linear optimisation based on a class of kernel functions which is eligible. New search directions and proximity measures are defined based on these functions. We derive the complexity bounds for large and small-update methods respectively. These are currently the best known complexity results for such methods.