关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

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Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Dynamical systems: stability, symbolic dynamics and chaos

In the present chapter we consider the well-posedness of an abstract boundary-value problem for differential equations of elliptic type

$$- \upsilon ''\left( t \right) + A\upsilon \left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant T} \right),\upsilon \left( 0 \right) = {{\upsilon }_{0}},\upsilon \left( T \right) = {{\upsilon }_{T}}$$

in an arbitrary Banach spaceEwith the positive operator A. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on three points for the numerical solutions of this problem are presented. The well-posedness of these difference schemes in various Banach spaces are studied. The stability and coercive stability estimates in Holder norms for solutions of the high order of accuracy difference schemes of the mixed type boundary-value problems for elliptic equations are obtained.

Partial Differential Equations of Elliptic Type

Equations of evolution

In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and Hormander. Some other consequences are also given for the controllability and the existence of a compact attractor.

/pdf/stabilization-for-the-semilinear-wave-equation-with-39elx95jua.pdf

Stabilization for the semilinear wave equation with geometric control condition

In this paper, we show that, for scalar reaction–diffusion equations u t = u x x + f ( x , u , u x ) on the circle S 1 , the Morse–Smale property is generic with respect to the non-linearity f. In Czaja and Rocha (2008) [13] , Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In Joly and Raugel (2010) [31] , we have shown that, generically with respect to the non-linearity f, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to f, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincare–Bendixson property, allow us to deduce that, generically with respect to f, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits. The main tools in the proofs include the lap number property, exponential dichotomies and the Sard–Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits.

/pdf/generic-morse-smale-property-for-the-parabolic-equation-on-2tq2zdijue.pdf

Generic Morse–Smale property for the parabolic equation on the circle

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilization and control.

https://hal.archives-ouvertes.fr/hal-01058120/file/Exp-decay-unbounded-Arxiv.pdf

Exponential decay for the damped wave equation in unbounded domains

In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to $f$, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincare-Bendixson property, allow us to deduce that, generically with respect to $f$, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits . The main tools in the proofs include the lap number property, exponential dichotomies and the Sard-Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits.

Generic Morse-Smale property for the parabolic equation on the circle

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control.

Romain Joly

Papers

Stabilization for the semilinear wave equation with geometric control condition

Generic Morse–Smale property for the parabolic equation on the circle

Exponential decay for the damped wave equation in unbounded domains

Generic Morse-Smale property for the parabolic equation on the circle

Exponential decay for the damped wave equation in unbounded domains