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Absorption effects for some elliptic equations with singularities

A. Porretta (2005)

Bollettino dell'Unione Matematica Italiana

We give an expository review of recent results obtained for elliptic equations having natural growth terms of absorption type and singular data. As a new result, we provide an application to the case of lower order terms of subcritical growth, proving a general solvability result with measure data for a class of equations modeled on (1.6).

Abstract parabolic problem with non-Lipschitz nonlinearity

Jan Cholewa, Tomasz Dlotko (2000)

Banach Center Publications

An abstract parabolic equation with sectorial operator and continuous nonlinearity is studied in this paper. In particular, the asymptotic behavior of solutions is described within the framework of the theory of global attractors. Examples included in the final part of the paper illustrate the presented ideas.

Abstract parabolic problems in ordered Banach spaces

Alexandre N. Carvalho, Jan W. Cholewa, Tomasz Dlotko (2001)

Colloquium Mathematicae

We consider abstract parabolic problems in ordered Banach spaces and give conditions under which they have global attractors. Our approach is via comparison of solutions. Within this approach abstract comparison principles are obtained and bounds on the attractors are given by order intervals in Banach spaces. These results are applied to ordinary differential equations and to parabolic equations for which the main part is given by a sum of fractional powers of sectorial operators having increasing...

Abstract parabolic problems with non-Lipschitz critical nonlinearities

Konrad J. Wąs (2010)

Colloquium Mathematicae

The Cauchy problem for a semilinear abstract parabolic equation is considered in a fractional power scale associated with a sectorial operator appearing in the linear main part of the equation. Existence of local solutions is proved for non-Lipschitz nonlinearities satisfying a certain critical growth condition.

Actuator fault tolerant control design based on a reconfigurable reference input

Didier Theilliol, Cédric Join, Youmin Zhang (2008)

International Journal of Applied Mathematics and Computer Science

The prospective work reported in this paper explores a new approach to enhance the performance of an active fault tolerant control system. The proposed technique is based on a modified recovery/trajectory control system in which a reconfigurable reference input is considered when performance degradation occurs in the system due to faults in actuator dynamics. An added value of this work is to reduce the energy spent to achieve the desired closed-loop performance. This work is justified by the need...

Addition de variables et application à la régularité

Bernard Helffer (1978)

Annales de l'institut Fourier

On montre dans cet article comment des théorèmes récents d’hypoellipticité ou de propagation des singularités peuvent être améliorés par une méthode d’addition de variables qui permet dans certains cas de “désingulariser” l’ensemble caractéristique.

Additional note on partial regularity of weak solutions of the Navier-Stokes equations in the class L ( 0 , T , L 3 ( Ω ) 3 )

Zdeněk Skalák (2003)

Applications of Mathematics

We present a simplified proof of a theorem proved recently concerning the number of singular points of weak solutions to the Navier-Stokes equations. If a weak solution 𝐮 belongs to L ( 0 , T , L 3 ( Ω ) 3 ) , then the set of all possible singular points of 𝐮 in Ω is at most finite at every time t 0 ( 0 , T ) .

Adiabatic approximation for a two-level atom in a light beam

Amandine Aftalion, Francis Nier (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Following the recent experimental realization of synthetic gauge potentials, Jean Dalibard addressed the question whether the adiabatic ansatz could be mathematically justified for a model of an atom in 2 internal states, shone by a quasi resonant laser beam. In this paper, we derive rigorously the asymptotic model guessed by the physicists, and show that this asymptotic analysis contains the information about the presence of vortices. Surprisingly, the main difficulties do not come from the nonlinear...

Admissibly integral manifolds for semilinear evolution equations

Nguyen Thieu Huy, Vu Thi Ngoc Ha (2014)

Annales Polonici Mathematici

We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation u ( t ) = U ( t , s ) u ( s ) + s t U ( t , ξ ) f ( ξ , u ( ξ ) ) d ξ when the evolution family ( U ( t , s ) ) t s has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(t) belongs to some classes of admissible function spaces. These manifolds are formed by trajectories of the solutions belonging...

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