We consider the Navier-Stokes equations with constant/variable density in a 2D slab domain, horizontally periodic.  We will study those two nonlinear perturbed problems around special equilibrium, under which Navier-slip boundary conditions are imposed, with constant slip coefficients. In this talk, the influence of slip coefficients on linear instability will be presented to obtain infinitely many normal mode solutions, following the operator method of Lafitte and Nguyen. Hence, we prove the nonlinear instability with a wide class of initial data, thanks to the abstract frameworks of Guo-Strauss and of Desjardins-Grenier. 

We study the propagation of coherent states in self-interacting bosonic quantum field theories in the semi-classical (mean-field) regime. Our work focuses in particular on the spatially cutoff P(\phi)_2 model, under standard assumptions ensuring essential self-adjointness of the Hamiltonian. For this model, the classical field equation is given by a non-linear Klein-Gordon equation, whose well-posedness constitutes the first step of our analysis. We then apply Hepp’s method, combined with a detailed study of both the classical and quantum dynamics, in order to construct an asymptotic expansion of arbitrary order for the quantum evolution of coherent states.