2014/02/26 by Marina Ghisi, Massimo Gobbino, Ghisi, Marina +3
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems
paper · doi:10.48550/arxiv.1402.6595
We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup, and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution. What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term. We also provide counterexamples in order to show the optimality of our results.