Pour localiser la solution d’un système de diffusion-réaction, il suffit de construire une famille de convexes $(K_t{)}_{t\ge 0}$, 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.

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.

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.

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.

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f\left(u\right)=0$ in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\mathrm{𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4\mathrm{𝚘𝚛}N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O\left(2\right)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k\times O(N-2)$ where $D_k\subset O\left(2\right)$ denotes the dihedral group...

We consider a system of two linear conservative wave equations, with a nonlinear coupling, in space dimension three. Spherical pulse like initial data cause focusing at the origin in the limit of short wavelength. Because the equations are conservative, the caustic crossing is not trivial, and we analyze it for particular initial data. It turns out that the phase shift between the incoming wave (before the focus) and the outgoing wave (past the focus) behaves like $ln\epsilon$, where $\epsilon$ stands for the wavelength....

We study spherical pulse like families of solutions to semilinear wave equattions in space time of dimension 1+3 as the pulses focus at a point and emerge outgoing. We emphasize the scales for which the incoming and outgoing waves behave linearly but the nonlinearity has a strong effect at the focus. The focus crossing is described by a scattering operator for the semilinear equation, which broadens the pulses. The relative errors in our approximate solutions are small in the L∞ norm.

We investigate for which metric $g$ (close to the standard metric $g_0$) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (i.e., non integrable) decay conditions on $g-g_0$; in particular, $g-g_0$ decays like $t^{-\frac{1}{2}-\epsilon }$ along wave cones.

We establish the uniqueness of fundamental solutions to the p-Laplacian equationut = div (|Du|p-2 Du),   p > 2,defined for x ∈ RN, 0 < t < T. We derive from this result the asymptotic behavoir of nonnegative solutions with finite mass, i.e., such that u(*,t) ∈ L1(RN). Our methods also apply to the porous medium equationut = ∆(um),   m > 1,giving new and simpler proofs of known results. We finally introduce yet another method of proving asymptotic results based on the...