The Stochastic Compressible Navier-Stokes-Korteweg system: The one-dimensional case

The thesis concerns the mathematical theory of the Navier-Stokes-Korteweg equations for capillary fluids. More precisely, we focus on the study of local and global well-posedness problems for the one-dimensional stochastic Quantum-Navier-Stokes equations, which fall into the class of the Navier-Stokes-Korteweg equations, driven by random initial data and stochastic forcing term. The main results achieved in the thesis concern the local and global existence of strong and weak solutions of the one-dimensional stochastic Quantum-Navier-Stokes equations with degenerate viscosity μ(ρ) = ρ^α. In particular we first prove a local well-posedness result for all the values of the viscosity parameter α ≥ 0 in the class of strong pathwise solutions which are strong solutions in both PDEs and probability sense. The proof is based on a multi-layer approximation technique and a stochastic compactness argument. After having established the local well-posedness, we then explore the possibility to extend the solution from a local to a global one. The continuity argument relies on the control of the vacuum regions. This is achieved in the range of the viscosity exponent α ∈ [0, 1/2 ], and thus in this case we prove the global well-posedness of the solutions. Furthermore, in the range 1/2 < α ≤ 1, we prove the existence of global weak dissipative martingale solutions for the one-dimensional stochastic Quantum-Navier-Stokes equations. These solutions are weak in both PDEs and probability sense and may have vacuum regions. The proof relies on the construction of an approximating system which provides extra dissipation properties and admits more regular solutions. The convergence is based on an appropriate truncation of the velocity field in the momentum equation and a stochastic compactness argument. As a byproduct of our analysis, the results hold also in the deterministic setting that is without stochastic forcing term. The thesis also contains an introduction to the derivation of the Navier-Stokes-Korteweg equations in the barotropic regime, a discussion on the choice of the viscosity coefficients, and a detailed description of the stochastic tools used in our analysis.