74 research outputs found

    Stability properties of the Euler-Korteweg system with nonmonotone pressures

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    We establish a relative energy framework for the Euler-Korteweg system with non-convex energy. This allows us to prove weak-strong uniqueness and to show convergence to a Cahn-Hilliard system in the large friction limit. We also use relative energy to show that solutions of Euler-Korteweg with convex energy converge to solutions of the Euler system in the vanishing capillarity limit, as long as the latter admits sufficiently regular strong solutions

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

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    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≥2N\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 (uu is the classical velocity and hh the depth variation of the fluid) with μ\mu the viscosity coefficient which simplifies the system and allow us to cancel out the coupling between the velocity uu and the depth variation hh. We obtain then the existence of global strong solution if m0=h0v0m_{0}=h_{0}v_{0} is small in B2,1N−1B^{\N-1}_{2,1} and (h0−1)(h_{0}-1) large in B2,1NB^{\N}_{2,1}. In particular it implies that the classical momentum m0′=h0u0m_{0}^{'}=h_{0} u_{0} can be large in B2,1N−1B^{\N-1}_{2,1}, but small when we project m0′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

    Highly rotating viscous compressible fluids in presence of capillarity effects

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    In this paper we study a singular limit problem for a Navier-Stokes-Korteweg system with Coriolis force, in the domain R2× ]0,1[ \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 00, 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 22-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.

    Relative entropy in multi-phase models of 1d elastodynamics: Convergence of a non-local to a local model

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    In this paper we study a local and a non-local regularization of the system of nonlinear elastodynamics with a non-convex energy. We show that solutions of the non-local model converge to those of the local model in a certain regime. The arguments are based on the relative entropy framework and provide an example how local and non-local regularizations may compensate for non-convexity of the energy and enable the use of the relative entropy stability theory -- even if the energy is not quasi- or poly-convex
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