Search

This paper deals with the asymptotic compactness of the solutions of the two-dimensional Navier-Stokes equations defined in unbounded channel-like domains. In order to overcome the non-compactness of Sobolev embeddings in unbounded domains, we establish the uniform tail-ends estimates of solutions by showing all the solutions are uniformly small outside a sufficiently large bounded domain for all large time and bounded initial data, which implies the asymptotic compactness and hence the existence of global attractors in the natural energy space. The uniform tail-ends estimates are derived by the scalar stream function of the Navier-Stokes equations which is distinct from the reaction-diffusion equation due to the divergence free condition. [ABSTRACT FROM AUTHOR]

This paper is concerned with uniform large deviation principles of fractional stochastic reaction-diffusion equations driven by additive noise defined on unbounded domains where the solution operator is non-compact and hence the result of [32] does not apply. The nonlinear drift is assumed to be locally Lipschitz continnous instead of being globally Lipschitz continuous. We first prove a large deviation principle for a fractional linear stochastic equation by the weak convergence method, and then show a uniform large deviation principle for the fractional nonlinear equation by a uniform contraction principle, despite the Sobolev embeddings are non-compact in unbounded domains. The result of the paper regrading the uniform large deviations can be applied to investigate the exit time and exit place of the solutions of the stochastic reaction-diffusion equations from a given domain in the phase space. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long term dynamics of the supercritical stochastic wave equation driven by multiplicative noise on unbounded domains. By introducing an appropriate variable, we convert the stochastic equation into a pathwise deterministic wave equation and prove the existence and uniqueness of solutions by the uniform Strichartz estimates. Then we prove the existence and uniqueness of pullback random attractors of the non-autonomous random dynamical system associated with the stochastic equation. The pullback asymptotic compactness of solutions is established by the idea of uniform tail-ends estimates as well as the decomposition method of the solution operator in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]