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Familles de convexes invariantes et équations de diffusion-réaction

Christine Reder (1982)

Annales de l'institut Fourier

Pour localiser la solution d’un système de diffusion-réaction, il suffit de construire une famille de convexes ( K t ) t ≥ 0 , invariante par rapport au champ de vecteurs associé à ce système; la solution est alors incluse dans K t à l’instant t dès qu’elle est contenue dans K 0 à l’instant zéro. Les fonctions d’appui associées à de telles familles de convexes sont solutions d’un système différentiel, mais celui-ci peut également engendrer des familles non invariantes.

Feedback stabilization of Navier–Stokes equations

Viorel Barbu (2003)

ESAIM: Control, Optimisation and Calculus of Variations

One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a L Q control problem associated with the linearized equation.

Feedback stabilization of Navier–Stokes equations

Viorel Barbu (2010)

ESAIM: Control, Optimisation and Calculus of Variations

One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a LQ control problem associated with the linearized equation.

Fine topology and quasilinear elliptic equations

Juha Heinonen, Terro Kilpeläinen, Olli Martio (1989)

Annales de l'institut Fourier

It is shown that the ( 1 , p ) -fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the p -Laplace equation div ( | ∇ u | p - 2 ∇ u ) = 0 continuous. Fine limits of quasiregular and BLD mappings are also studied.

Finite-dimensional Pullback Attractors for Non-autonomous Newton-Boussinesq Equations in Some Two-dimensional Unbounded Domains

Cung The Anh, Dang Thanh Son (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We study the existence and long-time behavior of weak solutions to Newton-Boussinesq equations in two-dimensional domains satisfying the Poincaré inequality. We prove the existence of a unique minimal finite-dimensional pullback D σ -attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms.

Finite-dimensional pullback attractors for parabolic equations with Hardy type potentials

Cung The Anh, Ta Thi Hong Yen (2011)

Annales Polonici Mathematici

Using the asymptotic a priori estimate method, we prove the existence of a pullback -attractor for a reaction-diffusion equation with an inverse-square potential in a bounded domain of ℝ N (N ≥ 3), with the nonlinearity of polynomial type and a suitable exponential growth of the external force. Then under some additional conditions, we show that the pullback -attractor has a finite fractal dimension and is upper semicontinuous with respect to the parameter in the potential.

Finite-dimensionality of 2-D micropolar fluid flow with periodic boundary conditions

Piotr Szopa (2007)

Applicationes Mathematicae

This paper is devoted to proving the finite-dimensionality of a two-dimensional micropolar fluid flow with periodic boundary conditions. We define the notions of determining modes and nodes and estimate their number. We check how the distribution of the forces and moments through modes influences the estimate of the number of determining modes. We also estimate the dimension of the global attractor. Finally, we compare our results with analogous results for the Navier-Stokes equation.

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica Musso, Frank Pacard, Juncheng Wei (2012)

Journal of the European Mathematical Society

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δ u - u + f ( u ) = 0 in ℝ N , u ∈ H 1 ( ℝ N ) , where N ≥ 2 . Under natural conditions on the nonlinearity f , we prove the existence of 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 in any dimension N ≥ 2 . Our result complements earlier works of Bartsch and Willem ( N = 4 𝚘𝚛 N ≥ 6 ) and Lorca-Ubilla ( N = 5 ) where solutions invariant under the action of O ( 2 ) × O ( N - 2 ) are constructed. In contrast, the solutions we construct are invariant under the action of D k × O ( N - 2 ) where D k ⊂ O ( 2 ) denotes the dihedral group...

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