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Register a new userOn Navier-Stokes-Korteweg and Euler-Korteweg Systems: Application to Quantum Fluids Models
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In this paper, the main objective is to generalize to the Navier-Stokes-Korteweg (with density dependent viscosities satisfying the BD relation) and Euler-Korteweg systems a recent relative entropy [proposed by D. Bresch, P. Noble and J.--P. Vila, (2016)] introduced for the compressible Navier-Stokes equations with a linear density dependent shear viscosity and a zero bulk viscosity. As a concrete application, this helps to justify mathematically the convergence between global weak solutions of the quantum Navier-Stokes system [recently obtained simultaneously by I. Lacroix-Violet and A. Vasseur (2017)] and dissipative solutions of the quantum Euler system when the viscosity coefficient tends to zero: This selects a dissipative solution as the limit of a viscous system. We also get weak-strong uniqueness for the Quantum-Euler and for the Quantum-Navier-Stokes equations. Our results are based on the fact that Euler-Korteweg systems and corresponding Navier--Stokes-Korteweg systems can be reformulated through an augmented system such as the compressible Navier-Stokes system with density dependent viscosities satisfying the BD algebraic relation. This was also observed recently [by D. Bresch, F. Couderc, P. Noble and J.--P. Vila, (2016)] for the Euler-Korteweg system for numerical purposes. As a by-product of our analysis, we show that this augmented formulation helps to define relative entropy estimates for the Euler Korteweg systems in a simplest way compared to recent works [See D. Donatelli, E. Feireisl, P. Marcati (2015) and J. Giesselmann, C. Lattanzio, A.-E. Tzavaras (2017)] with less hypothesis required on the capillary coefficient.
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In this paper, we consider global weak solutions to com-pressible Navier-Stokes-Korteweg equations with density dependent viscosities , in a periodic domain $Omega = mathbb T^3$, with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition introduced in D. Bresch, A. Vasseur and C. Yu [12]. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, we conclude the exponential decay to equilibrium. Note that our result covers the quantum Navier-Stokes system with a drag term.
We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods is consistent with the energy dissipation of the continuous PDE systems.
We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banachs fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the $L^2(mathbb{R}^d)$-framework.
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.
In the first main result of this paper we prove that one can approximate discontinious solutions of the 1d Navier Stokes system with solutions of the 1d Navier-Stokes-Korteweg system as the capilarity parameter tends to 0. Moreover, we allow the viscosity coefficients $mu$ = $mu$ ($rho$) to degenerate near vaccum. In order to obtain this result, we propose two main technical novelties. First of all, we provide an upper bound for the density verifing NSK that does not degenerate when the capillarity coefficient tends to 0. Second of all, we are able to show that the positive part of the effective velocity is bounded uniformly w.r.t. the capillary coefficient. This turns out to be crucial in providing a lower bound for the density. The second main result states the existene of unique finite-energy global strong solutions for the 1d Navier-Stokes system assuming only that $rho$0, 1/$rho$0 $in$ L $infty$. This last result finds itself a natural application in the context of the mathematical modeling of multiphase flows.
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