Abstract. We investigate the long-term dynamics of the three-dimensional NavierStokes-Voight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view we consider the Navier-Stokes-Voight model as a non-viscous (inviscid) regularization of the three-dimensional Navier-Stokes equations. Furthermore, we also show that the weak solutions of the Navier-StokesVoight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient ν → 0. MSC Classification: 37L30, 35Q35, 35Q30, 35B40

No abstract

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.1991 Mathematics Subject Classification. 35B40, 35B41, 35Q35.

We present a new method of investigating the so-called quasilinear strongly-damped wave equationsdomains. This method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity φ is less than 6 and f may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established. In a particular case φ ≡ 0 which corresponds to the so-called semi-linear strongly-damped wave equation, our result allows to remove the long-standing growth restriction | f (u)| C (1 + |u| 5 ).

Abstract. The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.