Global existence of weak solutions to the Navier–Stokes–Korteweg equations

Abstract

In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove global existence of finite energy weak solutions for large initial data. Contrary to previous results regarding this system, vacuum regions are considered in the definition of weak solutions and no additional damping terms are considered. The convergence of the approximating solutions is obtained by introducing suitable truncations of the velocity field and the mass density at different scales in the momentum equations and use only the a priori bounds obtained by the energy and the Bresch-Desjardins entropy. Moreover, the approximating solutions enjoy only a limited amount of regularity, and the derivation of the truncations of the velocity and the density is performed by a suitable regularization procedure.

Decay of higher order derivatives for $L^p$ solutions to the compressible fluid model of Korteweg type

“…up to a change of the pressure term P into ℓ 2 P , Danchin and Desjardins [16] investigated the global well-posedness of (1.1) in critical Besov spaces for initial data close enough to stable equilibrium (ρ * , 0) with the stability assumption P ′ (ρ * ) > 0. Antonelli and Spirito [1] established the global existence of finite energy weak solutions for large initial data, where vacuum regions are allowed in the definition of weak solutions. Charve, Danchin and the second author [9] investigated the global existence and Gevrey analyticity of (1.1) in more general critical L p framework, which exhibits Gevrey analyticity for a model of compressible fluids.…”

“…where Λ 1 stands for the Fourier multiplier with symbol |ξ| 1 = d i=1 |ξ i |. 1 Furthermore, choosing a suitable regularity (for instance, σ = d/2) enables us to get the optimal decay estimates like (1.3) for solution and its high order derivatives in the L 2 norm. It should point out here that the approach can be applied to a wide range of dissipative parabolic systems of Gevrey analyticity.…”

High Mach number limit for Korteweg fluids with density dependent viscosity

“…In [2] the authors prove the existence of arbitrarily large, global-in-time finite energy weak solutions to (1.1) -(1.2) by considering test functions of the form ̺ϕ, with ϕ smooth and compactly supported. As stressed in [1], this is somehow equivalent to considering test functions that are supported where the mass density is positive. In [1], the authors improve the result in [2] by removing the requirement on the test functions and by considering a more natural definition of weak solutions (see Definition 2.1 in [1]).…”

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

“…The viscous version of (1.11), that is the Navier-Stokes-Korteweg system, was also studied in the mathematical literature [5,49]. In particular, several papers are concerned with the case of the nonlocal equation, where −∆ρ is approximated by the nonlocal operator B η .…”