Global weak solutions for quantum isothermal fluids

Abstract

We construct global weak solutions to isothermal quantum Navier-Stokes equations, with or without Korteweg term, in the whole space of dimension at most three. Instead of working on the initial set of unknown functions, we consider an equivalent reformulation, based on a time-dependent rescaling, that we introduced in a previous paper to study the large time behavior, and which provides suitable a priori estimates, as opposed to the initial formulation where the potential energy is not signed. We proceed by working on tori whose size eventually becomes infinite. On each fixed torus, we consider the equations in the presence of drag force terms. Such equations are solved by regularization, and the limit where the drag force terms vanish is treated by resuming the notion of renormalized solution developed by I. Lacroix-Violet and A. Vasseur. We also establish global existence of weak solutions for the isothermal Korteweg equation (no viscosity), when initial data are well-prepared, in the sense that they stem from a Madelung transform.

Assumptions of Proposition 1.7 improved. More references and comments

References in corpus (4)

• On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics
• Symmetries and global solvability of the isothermal gas dynamics equations
• About the barotropic compressible quantum Navier-Stokes equations
• Large-time behavior of compressible polytropic fluids and nonlinear Schr{ö}dinger equation

Cited by in corpus (3)

• Global existence of finite energy weak solutions to the Quantum Navier-Stokes equations with non-trivial far-field behavior
• Large-time behavior of compressible polytropic fluids and nonlinear Schr{ö}dinger equation
• Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation