Highly rotating viscous compressible fluids in presence of capillarity effects

In this paper we study a singular limit problem for a Navier-Stokes-Korteweg system with Coriolis force, in the domain $\R^2\times\,]0,1[\,$ and for general ill-prepared initial data. Taking the Mach and the Rossby numbers to be proportional to a small parameter \veps going to $0$, we perform the incompressible and high rotation limits simultaneously. Moreover, we consider both the constant capillarity and vanishing capillarity regimes. In this last case, the limit problem is identified as a $2$-D incompressible Navier-Stokes equation in the variables orthogonal to the rotation axis. If the capillarity is constant, instead, the limit equation slightly changes, keeping however a similar structure. Various rates at which the capillarity coefficient can vanish are also considered: in most cases this will produce an anisotropic scaling in the system, for which a different analysis is needed. The proof of the results is based on suitable applications of the RAGE theorem.Comment: Version 2 includes a corrigendum, which fixes errors contained in the proofs to Theorems 6.5 and 6.

Existence of global strong solutions for the shallow-water equations with large initial data

This work is devoted to the study of a viscous shallow-water system with friction and capillarity term. We prove in this paper the existence of global strong solutions for this system with some choice of large initial data when $N\geq 2$ in critical spaces for the scaling of the equations. More precisely, we introduce as in \cite{Hprepa} a new unknown,\textit{a effective velocity} v=u+\mu\n\ln h ($u$ is the classical velocity and $h$ the depth variation of the fluid) with $\mu$ the viscosity coefficient which simplifies the system and allow us to cancel out the coupling between the velocity $u$ and the depth variation $h$. We obtain then the existence of global strong solution if $m_{0}=h_{0}v_{0}$ is small in $B^{\N-1}_{2,1}$ and $(h_{0}-1)$ large in $B^{\N}_{2,1}$. In particular it implies that the classical momentum $m_{0}^{'}=h_{0} u_{0}$ can be large in $B^{\N-1}_{2,1}$, but small when we project $m_{0}^{'}$ on the divergence field. These solutions are in some sense \textit{purely compressible}. We would like to point out that the friction term term has a fundamental role in our work inasmuch as coupling with the pressure term it creates a damping effect on the effective velocity

Global Existence and Vanishing Dispersion Limit of Strong/Classical Solutions to the One-dimensional Compressible Quantum Navier-Stokes Equations with Large Initial Data

We are concerned with the global existence and vanishing dispersion limit of strong/classical solutions to the Cauchy problem of the one-dimensional barotropic compressible quantum Navier-Stokes equations, which consists of the compressible Navier-Stokes equations with a linearly density-dependent viscosity and a nonlinear third-order differential operator known as the quantum Bohm potential. The pressure $p(\rho)=\rho^\gamma$ is considered with $\gamma\geq1$ being a constant. We focus on the case when the viscosity constant $\nu$ and the Planck constant $\varepsilon$ are not equal. Under some suitable assumptions on $\nu,\varepsilon, \gamma$, and the initial data, we proved the global existence and large-time behavior of strong and classical solutions away from vacuum to the compressible quantum Navier-Stokes equations with arbitrarily large initial data. This result extends the previous ones on the construction of global strong large-amplitude solutions of the compressible quantum Navier-Stokes equations to the case $\nu\neq\varepsilon$. Moreover, the vanishing dispersion limit for the classical solutions of the quantum Navier-Stokes equations is also established with certain convergence rates. The proof is based on a new effective velocity which converts the quantum Navier-Stokes equations into a parabolic system, and some elaborate estimates to derive the uniform-in-time positive lower and upper bounds on the specific volume.Comment: 40page