Boris Haspot

Existence of Global Weak Solution for Compressible Fluid Models of Korteweg Type

This work is devoted to proving existence of global weak solutions for a general isothermal model of capillary fluids derived by Dunn and Serrin (Arch Rational Mech Anal 88(2):95–133, 1985) which can be used as a phase transition model. We improve the results of Danchin and Desjardins (Annales de l’IHP, Analyse non linéaire 18:97–133, 2001) by showing the existence of global weak solution in dimension two for initial data in the energy space, close to a stable equilibrium and with specific choices on the capillary coefficients. In particular we are interested in capillary coefficients approximating a constant capillarity coefficient κ. To finish we show the existence of global weak solution in dimension one for a specific type of capillary coefficients with large initial data in the energy space.

Cauchy problem for viscous shallow water equations with a term of capillarity

In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

Existence of a Global Strong Solution and Vanishing Capillarity-Viscosity Limit in One Dimension for the Korteweg System

In the first part of this paper, we prove the existence of a global strong solution for the Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a

Global well-posedness of the Euler-Korteweg system for small irrotational data

\gamma

Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids

law for the pressure (

Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces

P(\rho)=a\rho^{\gamma}

Existence of strong solutions for nonisothermal Korteweg system

,

Global existence of strong solution for viscous shallow water system with large initial data on the irrotational part

\gamma>1

On a Lagrangian method for the convergence from a non-local to a local Korteweg capillary fluid model

) to a weak-entropy solution of the compressible Euler equations. In particular it justifies that the Korteweg system is suitable for selecting the physical solutions in the case where the Euler system is strictly hyperbolic. The problem remains open for a van der Waals pressure; indeed in this case the system is not strictly hyperbolic and in particular the classical theory of Lax [Comm. Pure Appl. Math., 10 (1957), pp. 537--566] and Glimm [Comm. Pure Appl. Math., 18 (1965), pp. 697--715] cannot be used.

Existence of global weak solutions for compressible fluid models with a capillary tensor for discontinuous interfaces

The Euler–Korteweg equations are a modification of the Euler equations that take into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni–Danchin–Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension